Rayleigh-Taylor and Richtmyer-Meshkov
Instabilities in Relativistic Hydrodynamic Jets

Jin Matsumoto
RIKEN
Astrophysical Big Bang Laboratory
Collaborator: Youhei 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 View

z3z2

Z

JM& Masada 13

un-shocked
ambient medium
shockedambient mediumbowcocoonshockP3P2Jet

P1

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

contact
discontinuity (CD)(b) Top View(b1) Expansion Phase [z=z1]
expanding CD

many numerical works inorder to investigate thepropagation dynamics ofthe relativistic jet(e.g., Marti+ 97, Aloy+ 00,
Zhang+ 03,04, Mizuta+ 06,
Morsony+ 07, Lazzati+ 09,
Nagakura+ 11, LopezCamara+ 13)
reconfinement shock
radial 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

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 of
the 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 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
Matsumoto & Masada, ApJL, 772, L1 (2013)

Numerical Setting: 2D Toy Model
relativistic jet
0.5

10
outflowboundarycylindrical coordinate: (r - θ plane)
relativistically hot jet (z-direction)ideal gas
numerical scheme: HLLC (Mignone & Bodo 05)
uniform grid: Δr = Δz = 10/320, Δθ = 2π/200

Basic Equations
mass
conservation
momentumconservation:r:θ
:zenergy
conservation
specific enthalpy

ratio of specific heats

Lorentz factor

Time Evolution of Jet Cross Section

relativistically hot plasma:
effective inertia:

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

Richtmyer-Meshkov Instability
15105contact 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

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 of
the Rayleigh-Taylor and Richtmyer-Meshkov instabilities once the jet-external
medium interface is corrugated in the case with the pressure-mismatched jet.

Stability Condition of the Jet
2D simulations of transverse structure of the jet is excluding
the destabilization effects by the Kelvin-Helmholtz mode
fixed

the stability criterion of the jet

jet cross section

10−2

102
A4

Model A1
Model A4
Richtmyer-Meshkov
Rayleigh-Taylor

10−4
10−5
10−6

A2

A1

101
hjet,0 − 1|vθ |ave /c10−3

A3

100

B4B3

B2

B1C4C3

C2

C1

D4D3D2D1stable

unstable

10−1

050

100

150t

200

250

300

10−2
10−2

10−1100101
η0102103

3D simulations:evolution of the cross section of the relativistic jetNumerical Setting: 3D Toy Model 1
outflowboundary

10

periodic boundary

cylindrical coordinate
relativistic jet (z-direction)ideal gas
numerical scheme: HLLC (Mignone & Bodo 05)
uniform grid: Δr = Δz = 10/320, Δθ = 2π/200
periodic boundary

10Basic Equations
mass
conservation
momentumconservationenergyconservation

specific enthalpy

ratio of specific heats

Lorentz factor

Result: Density
finger-like structure emerges at
the jet-external medium interface

radial oscillating motionof the jetlog ρ0.5

-1.0

-2.5
the interface deformation
gradually grows.

unit in time:-4.0

Synergetic Growth of Rayleigh-Taylor andRichtmyer-Meshkov Instabilities

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.

We can not find the destabilization effect by the Kelvin-Helmholtz
mode in the case of the radial oscillation motion of the jet althoughsuch effect is not excluded in the settings.

Without Oscillation
outflowboundary

The total energy of the jet issame as the oscillation case.

periodic boundary

jetperiodic boundaryWithout Oscillation
Kelvin-Helmholtz instability
grows at the jet interface.

The interface deformation
gradually grows.

Trigger the Deformation of the Jet
oscillation-induced RTI and RMI
non-oscillation

The growth rate of the volume-averaged azimuthal velocity due tothe Kelvin-Helmholtz instability is greater than the oscillationinduced Rayleigh-Taylor and Richtmyer-Meshkov instabilities.
The synergetic growth of the Rayleigh-Taylor and RichtmyerMeshkov instabilities trigger the deformation of the radiallyoscillating jet.

3D simulation:
propagation of the relativistic jetNumerical Setting: 3D Toy Model 2r100

outflowboundaryjetoutflowboundary1000

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

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

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

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

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

3D vs Axisymmetric
3D caseaxisymmetric casedeceleration of the jet due to the mixing between the jet
and surrounding medium in the 3D case.

Deceleration of the jet due to mixing

axisymmetric
3D

axisymmetric
3D

deceleration of the jet due to the mixing between the jet
and surrounding medium in the 3D case.

Deceleration of the jet due to mixing
3D 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
3D caseaxisymmetric3D

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.

Summary
Propagation dynamics and stability of the relativisticallyhot jet is studied through 2D and 3D relativistichydrodynamic simulations.
A pressure mismatch between the jet and surrounding medium leads to
the radial oscillating motion of the jet.
The jet-ambient medium interface isunstable due to the oscillation-induced
Rayleigh-Taylor instability
Richtmyer-Meshkov instability
deceleration of the jet due to the mixing between
the jet and surrounding medium

Next Study:
more realistic situation for relativistic jets such as AGN jets and GRBs
effect of the magnetic field on the dynamics and stability of the jetComparison of Grid Points

In higher resolution case, you can find smaller structuresdue to the growth of the Rayleigh-Taylor and RichtmyerMeshkov instabilities.

Comparison of Numerical Scheme

HLLC

HLL

It is not easy to find Rayleigh-Taylor and Richtmyer-Meshkov
fingers in the model with HLL scheme although the completelysame initial settings and grid spacing (320 x 200 zones r- and\theta directions) are adopted in both models.

Propagation of Rarefaction Wave through the Origin

Propagation of Shock Wave through the Origin

Relaxation of Initial non-equilibrium State
470γmaxγ̄

60

350γjetWjet /Wjet,0(a)

2(b)

40
30
20

1

10000100

200300400

5000100200tt
50

Transition stage (oscillation)

initialvalue
initialvalue (c)

h̄jet

40

Steady state
Hydrostatic balance

3020

Energy conversion from thermal energy
into bulk motion energy

100

initial value
300
400
5000100200300t400

500

relativistic Bernoulli:Scaling for the Oscillation Timescale
oscillation timescale: propagation time of the sound wave over the jet width

total energy conservation neglecting rest mass energyτoscillation timescale

1000

proportional to the square root of
the initial pressure ratio between
the jet and ambient medium.

√3γjet,0!

Pjet,0Pamb,0"1/2

10010110100
Pjet,0 /Pext

1000