Relativistic HD/MHD Flow for GRB Jets

Jin Matsumoto
RIKEN
Astrophysical Big Bang Laboratory
Collaborators: Nagataki, Ito, Mizuta, Barkov,Dainotti, Teraki (RIKEN),
Masada (Kobe University)

What a relativistic jet?
collimated bipolar outflow from gravitationally bounded object
active galactic nuclei (AGN) jet: γ ~ 10
microquasar jet: v ~ 0.9c
Gamma-ray burst: γ > 100
3C 31

Lorentz factor

schematic picture of the GRB jet

Meszaros 01

AGN jetA relativistic jet is considered to be launched form
the central engine and propagates the progenitor star.

z1(a) Side Viewz3z2many numerical works inorder to investigate thepropagation dynamics ofthe relativistic jet(e.g., Marti+ 97, Aloy+ 00,
Zhang+ 03,04, Mizuta+ 06,
Perucho+ 08, Morsony+07,
Lazzati+ 09, LopezCamara+ 13)
reconfinement shock in thecollimated jet

Z

JM& Masada 13

un-shocked
ambient medium
shockedambient mediumbow
cocoon

shock

P3
P2

Jet

P1

reconfinement
region(Norman et al. 1982;
Sanders 1983)

contact
discontinuity (CD)(b) Top View(b1) Expansion Phase [z=z1]
expanding CDradial oscillating motion andrepeated excitation of thereconfinement region (e.g.,Gomez+ 97, JM+ 12)(b2) Contraction Phase (I) [z=z2]
contracting CD(b3) Contraction Phase (II) [z=z3]
contracting CD

adiabatic cooling

P3
P2

P1

expanding shock[P1 < P2 < P3]

contracting shock[P1 < P2 < P3]

expanding shock[P2 < P3 < P1]

Motivation of Our Study
radial inertia force

reconfinement
regioncross section of the jet<jet interface

Rayleigh-Taylor instability grows?

To investigate the propagation dynamics and stability ofthe relativistic jet- using relativistic hydrodynamic simulations
focus on the transverse structure of the jet
2D simulations: evolution of the cross section of the relativistic jet
3D simulation: propagation of the relativistic jet

2D simulations:evolution of the cross section of the relativistic jet
JM & Masada, ApJL, 772, L1 (2013)

Numerical Setting: 2D Toy Model
relativistic jet

1% perturbationin the pressure

1225
outflowboundarycylindrical coordinate: (r - θ plane)
relativistically hot jet (z-direction)ideal gas
numerical scheme: HLLC (Mignone & Bodo 05)
Δr = 20/640 (0 < r < 20), Δlogr = const. (20 < r < 225)
Δθ = 2π/200

Basic Equations
mass
conservation
momentumconservation:r:θ:z
energyconservation
specific enthalpy

ratio of specific heats

Lorentz factor

Time Evolution of Jet Cross Section
The effective inertia is important.
relativistically hot plasma:
effective inertia:

The effective inertia of thejet is lager than the external
medium although the densityof the jet is smaller than theexternal medium in oursetting.

jet cross section

The amplitude of the corrugated jet interface grows as time passes.
A finger-like structure is a typical outcome of the Rayleigh-Taylor instability.

Growth of RTI in SN Explosion
The condition for the RTI
heavy fluid, (Chevalier 76)

light fluid

It is well known that a contact discontinuity, formed where a heavy
fluid is supported above a light fluid against gravity, is unstable toperturbations to the boundary between the two fluids.
Although the gravity is negligible in SN explosions, the accelerationof the gas works as a gravity.

The mechanism of the growth of RTIThe effective inertia is important.
relativistically hot plasma:
effective inertia:

In addition to the growth of the Rayleigh-Taylor instability, the growth of theRichtmyer-Meshkov instability is also contributed to the finger like structures.

Richtmyer-Meshkov Instability
15105

contact discontinuity
The Richtmyer-Meshkov instability is induced by impulsive acceleration
due to shock passage.
The perturbation amplitude grows linearly in time (Richtmyer 1960),0Time Evolution of Jet Cross Section

Richtmyer-Meshkov instability is secondary excited between the RTI fingers.
Almost all finger-like structures in panel (f) have their origin in the RMI.

Synergetic Growth of Rayleigh-Taylor andRichtmyer-Meshkov Instabilities
10−2
Model A1
Model A4|vθ |ave /c

10−3Richtmyer-Meshkov
Rayleigh-Taylor

10−4
10−5
10−6

050

100

150
t

200

250

300

development of the Rayleigh-Taylor
instability at the jet interface
increases exponentially.

excitation of the Richtmyer-Meshkov
instability at the jet interface
grows linearly with time.

The transverse structure of the jet is dramatically deformed by a synergetic growth ofthe Rayleigh-Taylor and Richtmyer-Meshkov instabilities once the jet-external
medium interface is corrugated in the case with the pressure-mismatched jet.

3D simulation:
propagation of the relativistic jet
focusing on the impact of the oscillation-induced
Rayleigh-Taylor and Richtmyer-Meshkov instabilities on
the 3D jet propagation.

Numerical Setting: 3D Toy Model
Hydrodynamic variables are the same as theprevious calculation in the 2D oscillation model.

r100

jet

outflowboundaryoutflowboundary1000

cylindrical coordinate
relativistic jet (z-direction)ideal gas
numerical scheme: HLLC (Mignone & Bodo 05)
uniform grid: Δr = 0.0666, Δθ = 2π/160, Δz=1z

Basic Equations
mass
conservation
momentumconservationenergyconservation

specific enthalpy

ratio of specific heats

Lorentz factor

In 3D case, we calculated the numerical flux to all directions.

Result: Density
Only the jet component is shown.
The amplitude of the corrugated jet interface
grows due to the oscillation-induced RayleighTaylor and Richtmyer-Meshkov instabilities.

Since the relativistic jet is continuously injected into thecalculation domain, standing reconfinement shocks are formed.

Result: Density
Only the jet component is shown.
We confirmed that the Rayleigh-Taylor and Richtmyer-Meshkov
instabilities grows at the interface of the jet. The material mixing due
to the Rayleigh-Taylor and Richtmyer-Meshkov instabilities between
the jet and surrounding medium leads to the deformation of the jet.

the transversal cuts of the rest-mass density

Spatial Growth Rate of Azimuthal Velocity

1st 2nd 3rdreconfinement region
Growth of the volume-averaged azimuthal velocity is saturated around the 3rdreconfinemet region.

non-Axisymmetric vs Axisymmetric
non-axisymmetric case

2D cut of jet velocity along the propagation axis
The numerical settings of the axisymmetric case are same as the 3D case
except the azimuthal direction.
axisymmetric casedeceleration of the jet due to the mixing between the jet and surrounding
medium in the non-axisymmetric case.
Three-dimensionality is essential for the non-linear dynamics andstability of the relativistic jet.

Deceleration of the jet due to mixing

axisymmetric
non-axisymmetric

1D profile of the jet velocity along the jet axis
first reconfinement
region1D profile of the Lorentz factor along the jet axis
axisymmetric
non-axisymmetric

deceleration of the jet due to the mixing between the jet and surrounding
medium in the non-axisymmetric case.
Three-dimensionality is essential for the non-linear dynamics andstability of the relativistic jet.

Deceleration of the jet due to mixing
non-axisymmetric caseaxisymmetric caserelativistic Bernoulli equation:
gives the maximum Lorentz factor of the jet after adiabatic expansion.
However,drops to ~ 10 due to the mixing in this case.

Deceleration of the jet due to mixing
non-axisymmetric case

1D profile ofalong the jet axis

axisymmetric
non-axisymmetric

relativistic Bernoulli equation:
gives the maximum Lorentz factor of the jet after adiabatic expansion.
However,drops to ~ 10 due to the mixing in this case.

Magnetically Driven Jet?
schematic picture of the GRB jet

Meszaros 01

Central engine
MHD process
Extraction of the rotational energyof the BH or accretion disk throughB-field
thermal process-annihilation-annihilation
fireballObservation
polarization in prompt phase(Yonetoku et al. 2011, 2012)
polarization in afterglow phase(Mundell et al. 2014,Wiersema et al. 2014)

Propagation of MHD jet for GRB
axisymmetricprogenitor: 16TI model (Woosley & Heger 06)
jet:Summary
Basic physics of the propagation of the relativistic jet isinvestigated though 3D HD/2D MHD numerical simulations.
A pressure mismatch between the jet and surrounding medium leads to
the radial oscillating motion of the jet.
The jet-external medium interface isunstable due to the oscillation-induced
Rayleigh-Taylor instability
Richtmyer-Meshkov instabilitydeceleration of the jet due to the mixing between
the jet and surrounding medium

Next Study:
more realistic situation for relativistic MHD jets in the context of GRBs