Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock...
Transcript of Condition for the growth of the Rayleigh-Taylor …...Modeling for the growth of RTI forward shock...
Condition for the growth of the Rayleigh-Taylor instability at the relativistic jet interface
Jin Matsumoto RIKEN
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
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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.
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.
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??
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.