Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock...

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Condition for the growth of the Rayleigh-Taylor instability at the relativistic jet interface Jin Matsumoto RIKEN

Transcript of Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock...

Page 1: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

Condition for the growth of the Rayleigh-Taylor instability at the relativistic jet interface

Jin Matsumoto RIKEN

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collimated bipolar outflow from gravitationally bounded object

microquasar jet: v ~ 0.9c active galactic nuclei (AGN) jet: γ ~ 10

Gamma-ray burst: γ > 100

Lorentz factor

What is a relativistic jet?

Meszaros 01

schematic picture of the GRB jet

t

v ~ 0.92c

microquasar: GRS 1915+105

Kovalev et al. 2007

Mirabel et al. 1998

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Morphological Dichotomy of the Jet3C 31

Cygnus A,

Morphology is one of the most fundamental property of the relativistic jet.

FR I FR II

- A complex combination of several intrinsic and external factors A morphological dichotomy between FR I and FR II

Instabilities play an important role in the morphology and stability of the jet through the interaction between the jet and external medium.

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Why is stability of the jet interface so important? The stability of the jet interface is also related to the inhomogeneity of the

jet and the evolution of the turbulence inside/outside the jet.

They may affect on the radiation from the jet associated with the particle and/or photon acceleration.

Multiple outflow layers inside the relativistic jet are essential to reproduce the typical observed spectra of GRBs (Ito et al. 2014)

The development of the turbulence inside the jet is important point in order to discuss the mechanism of the particle acceleration in the context of the GRB (Asano & Terasawa 2015) and blazer (Asano & Hayashida 2015; Inoue & Tanaka 2016).

A promising mechanism to make the interface of the jet unstable is thought to be the Kelvin-Helmholtz instability. Many authors have investigated the growth of the Kelvin-Helmholtz instability. However, the growth of the Rayleigh-Taylor instability at the interface of the relativistic jet is not still understood well.

Page 5: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

contact discontinuity

Modeling for the growth of RTIforward shock

reverse shock

cocoon

jet

contact discontinuity

reconfinement shock

jet head

jet cross-section

cocoon

jet

jet boundary

in the rest frame of the decelerating jet interface

restoring force of the oscillationeffective gravity

initially hydrostatic equilibrium

if y = 0, then P = P0

�@P

@y= �2⇢hg = �2

✓⇢+

�� 1

P

c2

◆g

P = P0e�y/H +

⇢c2(�� 1)

�(e�y/H � 1)

1

H⌘ �

�� 1

�2g

c2

vy =

�v

4

(1 + cos(2⇡x))(1 + cos(2⇡x/10))

initial perturbation

�v = 10�4

�j = 5

⇢j = 0.1

⇢c = 1

P0 = 1

g

c2/Lx

= 10�4

h = 1 +�

�� 1

P

⇢c2

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Estimation of the pressure scale heightSince the effective gravity has its origin in the radial oscillating motion of the jet, assuming the amplitude of the jet oscillation is roughly equal to the jet radius, the effective gravity is estimated as follows;

g ⇠ rjet

⌧2osci

Here τ is the typical oscillation time of the jet and given by the propagation time of the sound wave over the typical oscillating radius of the jet (JM et al. 2012);

⌧osci

= �jet

rjet

/Cs

Pressure scale height: H ⌘ �� 1

c2

�2g⇠ rjet

harmonic oscillator

Since we neglect the impact of the curvature of the jet radius in this study, the position at y = H is far from the jet-cocoon interface (y = 0).

a =@2r

@t2= � r

T 2

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Derivation of linear growth of RTI

basic equations:

equation of motion: x

equation of motion: y

incompressible condition

x

gjet

cocoon

�2⇢h

@v

@t+ (v ·r)v

�+rP +

v

c2@P

@t� �2⇢hg = 0

yequation of motion:

@v

x

@x

+@v

y

@y

= 0

vx

, vy

⌧ csassumption:

temporal variation of the pressure is negligible

continuity

conservation of entropy

effective inertia force

incompressible fluid

@

@t

(�⇢) + v

x

@

@x

(�⇢) + v

y

@

@y

(�⇢) = 0

2⇢h

✓@v

x

@t

+ v

x

@v

x

@x

+ v

y

@v

x

@y

◆= �@P

@x

2⇢h

✓@v

y

@t

+ v

x

@v

y

@x

+ v

y

@v

y

@y

◆= �@P

@y

� �

2⇢hg

@s

@t

+ v

x

@s

@x

+ v

y

@s

@y

= 0

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Derivation of linear growth of RTI

linearized equations:

equation of motion: x

equation of motion: y

incompressible condition

x

gjet�2⇢h

@v

@t+ (v ·r)v

�+rP +

v

c2@P

@t� �2⇢hg = 0

yequation of motion:

2⇢h

@v

x

@t

= �@�P

@x

@v

x

@x

+@v

y

@y

= 0

vx

, vy

⌧ csassumption:

temporal variation of the pressure is negligible

�2⇢h@vy@t

= �@�P

@y� �2

✓�⇢+

�� 1

�P

c2

◆g

@(��⇢)

@t= �vy

@(�⇢)

@y@

@t

✓�P

P� �

�⇢

◆+ vy

✓1

P

@P

@y� �

1

@⇢

@y

◆= 0

continuity

conservation of entropy

effective inertia force

incompressible fluidcocoon

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Derivation of linear growth of RTI

x

gjet�2⇢h

@v

@t+ (v ·r)v

�+rP +

v

c2@P

@t� �2⇢hg = 0

yequation of motion:

�⇢, �P, vx

, vy

/ ei(kx�!t)

vx

, vy

⌧ cs

i!�2⇢hvx

= ik�P

ikvx

= �@vy

@y

assumption:temporal variation of the pressure is negligible

i!��⇢ = vy@(�⇢)

@y

i!�2⇢hvy =@�P

@y+ �2

✓�⇢+

�� 1

�P

c2

◆g

�P

P= �

�⇢

⇢� vy

i!

✓�2⇢hg

P+ �

1

@⇢

@y

equation of motion: x

equation of motion: y

incompressible condition

continuity

conservation of entropy

effective inertia force

incompressible fluidcocoon

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Derivation of linear growth of RTI

x

gjet�2⇢h

@v

@t+ (v ·r)v

�+rP +

v

c2@P

@t� �2⇢hg = 0

yequation of motion:

vx

, vy

⌧ csassumption:

temporal variation of the pressure is negligible

Dy(!2�2⇢hDyvy)� k2

✓!2 � g

H

◆�2⇢hvy = k2�

✓Dy(�⇢) +Dy�

�2

�� 1

P

c2

◆gvy

⇢h =

✓⇢+

�� 1

P0

c2

◆e�y/H

D2yvy �

1

HDyvy � k2

✓1� g

H!2

◆vy = 0

Dy =@

@y

effective inertia force

incompressible fluidcocoon

Since the density and Lorentz factor are constant in the jet and cocoon regions,

using ,

the differential equation for both jet and cocoon regions of the fluid reduces to

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Derivation of linear growth of RTI

x

gjet�2⇢h

@v

@t+ (v ·r)v

�+rP +

v

c2@P

@t� �2⇢hg = 0

yequation of motion:

vx

, vy

⌧ csassumption:

temporal variation of the pressure is negligibleDy =

@

@y

effective inertia force

incompressible fluidcocoon

vy = Ae↵1y +Be↵2ygeneral solution:

D2yvy �

1

HDyvy � k2

✓1� g

H!2

◆vy = 0

↵1 =1

2

✓1

H+

s1

H2+ 4k2

✓1� g

H!2

◆◆, ↵2 =

1

2

✓1

H�

s1

H2+ 4k2

✓1� g

H!2

◆◆

k � 1/H ⇠ 1/rjet

g/H ⇠ 1/⌧2osci

, g/H!2 ⌧ 1

vy ! 0 when y ! ±1boundary condition: vy =

Ae�ky, y > 0

Aeky, y < 0

Page 12: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

Derivation of linear growth of RTI

x

gjet

effective inertia force

�2⇢h

@v

@t+ (v ·r)v

�+rP +

v

c2@P

@t� �2⇢hg = 0

yequation of motion:

vx

, vy

⌧ csassumption:

temporal variation of the pressure is negligible

! = i

s

gk�2j ⇢jh

0j � �2

c⇢ch0c

�2j ⇢jhj + �2

c⇢chc! = i

rgk

⇢1 � ⇢2⇢1 + ⇢2

Dy(!2�2⇢hDyvy)� k2

✓!2 � g

H

◆�2⇢hvy = k2�

✓Dy(�⇢) +Dy�

�2

�� 1

P

c2

◆gvy

dispersion relation non-relativistic case

h0 = 1 +�2

�� 1

P0

⇢c2

incompressible fluidcocoon

The dispersion relation is obtained by integrating the differential equation across the interface and dropping integrals of non-divergent terms.

Dy =@

@y

Page 13: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

contact discontinuity

Modeling for the growth of RTIforward shock

reverse shock

cocoon

jet

contact discontinuity

reconfinement shock

jet head

jet cross-section

cocoon

jet

jet boundary

1e-06

1e-05

0.0001

0.001

0.01

0 50 100 150 200 250 300 350 400 450 500

Ux

t

simulationtheory

0 100 200 300 400 500t10-6

10-5

10-4

10-3

10-2

10-1

Ux,/c

growth of the RTIdispersion relation:

! = i

s

gk�2j ⇢jh

0j � �2

c⇢ch0c

�2j ⇢jhj + �2

c⇢chc

estimation of inertia�2j ⇢jh

0j ⇠ �2

jPj

�2c⇢ch

0c ⇠ Pc

Pj ⇠ Pc

�2j ⇢jh

0j � �2

c⇢ch0c > 0

RTI grows at relativistic jet interface.

h0 = 1 +�2

�� 1

P0

⇢c2

g

c2/Lx

= 10�4

�j = 5

⇢j = 0.1

⇢c = 1

P0 = 1

in the rest frame of the decelerating jet interface

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r

z

jet150

300

cylindrical coordinate

ideal gas numerical scheme: HLLC (Mignone & Bodo 05)

uniform grid:

relativistic jet (z-direction)

Numerical Setting: 3D Toy Model

outflow boundary

outflow boundary

Pa/Pj = 0.1⇢ac

2 = 1

�r = �z = 0.1, �✓ = 2⇡/160

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Jet models

two key parameters

- the effective inertia ratio of the jet to the ambient medium:⌘j,a =�2j ⇢jhj

⇢aNeglecting the multi-dimensional effect, the propagation velocity of the jet head through a cold ambient medium can be evaluated by the balancing the momentum flux of the jet and the ambient medium in the frame of the jet head (Marti+ 97, Mizuta+ 04):

vh =

p⌘j,a

1 +p⌘j,a

vj

- dimension less specific enthalpy of the jet: hj

Page 16: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

specific enthalpy ratio of specific heats Lorentz factor

energy conservation

mass conservation

momentum conservation

Basic Equations

@

@t(�⇢) +r · (�⇢v) = 0

@

@t(�2⇢hv) +r · (�2⇢hvv + P I) = 0

@

@t(�2⇢hc2 � P ) +r · (�2⇢hc2v) = 0

h = 1 +�

�� 1

P

⇢c2� =

4

3� =

1p1� (v/c)2

Page 17: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

Result: Density

The amplitude of the corrugated jet interface grows due to the oscillation-induced Rayleigh-Taylor and Richtmyer-Meshkov instabilities.

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

Only the jet component is shown.

Page 18: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

Distribution of the Modified Effective Inertia

As predicted analytically, the effective inertia of the jet becomes larger than the cocoon envelope for all the models.

�2⇢h0

Page 19: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

3D Rendering of the Tracer

@

@t(�⇢f) +r · (�⇢fv) = 0

passive tracer: f

f = 1

f = 0

: jet material

: ambient medium

The oscillation-induced Rayleigh-Taylor instability is responsible for the distortion of the cross section at z = 30.

Finger-like structures appeared in the cross-section at z = 65 and 90 are outcome of both the Rayleigh-Taylor and Richtmyer-Meshkov instabilities.

Page 20: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

Inherent Property of Relativistic Jet

Although the relativistic jet shows a rich variety of the propagation dynamics depending on its launching condition, the oscillation-induced Rayleigh-Taylor instability and secondary Richtmyer-Meshkov instability grow commonly at the jet interface and then induce a lot of finger-like structures.

- radial expansion (hot models)

initial evolution

- radial contraction (cold models)

After initial stage, the cold jet also follows the same evolution path as the hot jet and thus excites the Rayleigh-Taylor and Richtmyer-Meshkov instabilites.

Page 21: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

Kelvin-Helmholtz instability?

798 P. Rossi et al.: Dynamical structures in relativistic jets

Fig. 1. Volume rendering of the tracer distribution for case A at t = 240 (top panel), B at t = 760 (central panel) and E at t = 150 (bottom panel).Dark colors are due to the opacity of the material in the rendering procedure

Jet material is dark brown, external medium material white, andthe level of mixing corresponds to the scale of orange. Fromthese figures one can gain a first qualitative view about the roleof the parameters on the evolution and structure of the jet: incases A and E the jets propagate straight with moderate andlow dispersion of jet particles, whereas in case B the jet has, atleast partially, lost its collimation and the particle dispersion isconsiderably greater.

This is more clearly represented by the 2D cuts in the x, yplane (at z = 0) of the density and Lorentz factor distributionsshown in Fig. 2. From the density (left) panels one can note thatin both cases A and E the jet seems to be weakly affected by theperturbation growth and entrainment. This is not the situationfor case B, where the beam structure is heavily modified by thegrowth of disturbances beyond ∼20 jet radii. The distributionsof the Lorentz factor (right panels) again indicate that the per-turbation slightly affects the system in case A after ∼100 radii,leaves the jet almost unchanged in case E, and strongly influ-ences the propagation after ∼20 radii for case B, where the max-imum value of γ has reduced approximately to half of its initialvalue. However, even in case B a well collimated high velocitycomponent along the axis still displays.

A quantitative estimate of the jet deceleration can be ob-tained by plotting the Lorentz factor as a function of thelongitudinal coordinate y. Figure 3 shows the maximum value

of γ at constant y-planes together with its volume average, de-fined as

γav =

∫γg(γ) dxdz

∫g(γ) dxdz

, (5)

where g(γ) is a filter function to select the relativistic flows:

g(γ) =

⎧⎪⎪⎨⎪⎪⎩

1 for γ ≥ 2 ,0 for γ < 2 . (6)

For case B one can see that the deceleration occurs both in γmax(the flow velocity, although still relativistic, shows a strong de-crease from its initial value) and in γav, which indicates a globaleffect, whereas in cases A and E the central part of the jet con-tinues to be almost unperturbed and only thin external layers aredecelerated.

As previously mentioned, the interaction between jet andsurrounding medium involves different phenomena, reflectingthe characteristics of the deceleration process: the growth rateof the perturbations, the type of perturbations that dominates thejet structure, the possibility of mixing through the backflowingmaterial and the density of the ambient medium relative to the jetmaterial. Long wavelength modes will tend to produce global jetdeformations like jet wiggling more than mixing, while modeson a shorter scale may be more efficient in promoting the mix-ing process. On the other hand, it is clear that the mixing witha denser environment is more effective for the jet deceleration.

Rossi et al. 2008 3D pure hydro simulation for jet propagation

tracer

They claimed that the entrainment process takes place from the interaction between the jet beam and the cocoon, promoted by the development of Kelvin-Helmholtz instabilities at the beam interface.

Rayleigh-Taylor instability is also expected to grow at the interface of the jet.

Which instability is dominant??

Page 22: Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock reverse shock cocoon jet contact discontinuity reconfinement shock jet head jet cross-section

contact discontinuityforward shock

reverse shock

cocoon

jet

contact discontinuity

reconfinement shock

jet head

cocoon

jet

jet boundary

Summary

1e-06

1e-05

0.0001

0.001

0.01

0 50 100 150 200 250 300 350 400 450 500

Ux

t

simulationtheory

0 100 200 300 400 500t10-6

10-5

10-4

10-3

10-2

10-1

Ux,/c

growth of the RTIdispersion relation

! = i

s

gk�2j ⇢jh

0j � �2

c⇢ch0c

�2j ⇢jhj + �2

c⇢chc

estimation of inertia�2j ⇢jh

0j ⇠ �2

jPj

�2c⇢ch

0c ⇠ Pc

Pj ⇠ Pc

�2j ⇢jh

0j � �2

c⇢ch0c > 0

RTI grows at relativistic jet interface.