Eﬀect of Interacting Rarefaction Waves on Relativistic Jets
Jin Matsumoto1 , Youhei Masada2 , and Kazunari Shibata3
1
National Astronomical Observatory of Japan, Tokyo, Japan
Graduate School of System Informatics, Kobe University, Japan
3
Kwasan and Hida Observatories, Kyoto University, Kyoto, Japan
2

1Introduction

Relativistic jets are collimated bipolar outﬂows thathave a velocity almost equal to light speed. They are ubiquitous among astrophysical systems consisting of a compact object surrounded by an accretion disk, e.g., activegalactic nuclei [2], microquasars [4] and the central engineof gamma-ray bursts [5, 3]. Although there are a lot ofworks that try to determine the mechanism by which relativistic jets are accelerated and collimated, it is still notunderstood.
Aloy & Rezzolla (2006) recently reported that relativistic jets can be powerfully boosted along the interface between the jet and ambient medium, if the jet has suﬃciently large velocity and speciﬁc enthalpy, and is overpressured. A rarefaction wave excited at the interface converts
the relativistic thermal energy of the plasma into kinetic
energy and yields an ampliﬁcation of the Lorentz factor
in the jet-ambient medium interface. This type of boost,which we label rarefaction acceleration, is not possible in
Newtonian dynamics, but is an inherent process in relativistic hydrodynamics.
In this paper, we study in detail how the rarefaction
acceleration aﬀects the propagation dynamics of relativistically hot jets through one-dimensional (1D) and twodimensional (2D) relativistic hydrodynamic simulations.
The nonlinear interaction of rarefaction waves excited at
the interface between the jet and ambient medium is especially focused upon because it might have a potential
impact on the boosting process and even alter the dynamics and structure of the jet.
2medium ρamb,0 c2 . The computational domain spans 0 <r < 2. A uniform grid with a grid size ∆r = 10−3 isadopted for our calculations. We use a reﬂecting boundary condition on the axis r = 0. The outer boundary of thegrid uses the outﬂow (zero gradient) boundary condition.
3
3.1

Results
1D calculation

Fig. 1. Panel (a): Temporal evolution of the Lorentz factorin 1D model. Panel (b): Spatial distribution of the Lorentz
factor in 2D model.

Numerical Models and Setup

In order to investigate the interaction of rarefaction
waves excited at the jet-ambient medium interface, we initially set a relativistically hot jet surround by ambient gas
in the calculation domain. The initial density and pressure in the jet are chosen as ρjet,0 = 0.1 and Pjet,0 = 1,respectively. Those of the ambient medium are ρamb,0 = 1and Pamb,0 = 0.1. In addition, the jet velocity to the zdirection is relativistic vjet,0 = 0.99c, with a Lorentz factorof γjet,0 ∼ 7. The ambient medium does not move and theradial velocity vr is set to be zero initially in the calculation domain. In the 1D model, we calculate the evolutionof the jet only in the radial direction. On the other hand,in the 2D model, axisymmetric simulations of the jet propagation are carried out in cylindrical coordinates.
The normalization units in length, velocity, time, andenergy density are chosen as the initial jet width Wjet,0 ,light speed c, light crossing time over the initial jet width
Wjet,0 /c, and rest mass energy density in the ambient

Figure 1(a) shows the temporal evolution of our jetambient medium system. The color contour represents
the spatial distribution of the Lorentz factor. In the early
evolutionary stage (0 < t < 5), the single rarefaction acceleration of the gas is observed in the jet-ambient mediuminterface. Subsequently at t ∼ 5 the rarefaction waves
converge on the central region of the jet, bringing a substantial change in the dynamics. The gas pressure in theinteracting region of rarefaction waves is then further reduced and becomes lower than that of the ambient gas.
Since the thermal energy is converted to the bulk kinetic
energy of the gas, the gas in the interacting region is further boosted. The peak Lorentz factor of the gas inside
the rarefaction region reaches ∼ 60 at t ∼ 40, which is afactor ∼ 5 higher than that due to the single rarefaction
acceleration.
The interaction of rarefaction waves generates a strong
inward pressure gradient behind the jet-ambient medium
interface which acts to decelerate the radial expansion of

the jet, turning expansion of the jet into contraction ataround time t = 25. The contraction of the contact discontinuity results in converging ﬂows inside the jet.
When t = 43, the converging ﬂows collide with eachother at the center of the jet and excite shock waves that
propagate outward. The gas bounded by the shock is compressed and heated. Since a time-increasing pressure decelerates the tangential velocity of the jet, the Lorentz factorof the jet reduces.
When the shock wave encounters the contact discontinuity at around t = 58, the system has almost returned to its
initial state. The jet still has suﬃciently larger tangential
velocity and speciﬁc enthalpy than the ambient medium.
Since the system is restored to a condition that is almost
the same as the initial conditions, the three types of hydrodynamic wave, an outward propagating shock wave, acontact discontinuity (the edge of the jet), and a converging rarefaction wave, appear at the jet-ambient medium
interface. Therefore the contracting radial motion of thejet becomes an expanding motion.
After returning to conditions similar to the initial conditions, the radial motion of the jet changes between expansion and contraction until the pressure of the jet becomesequal to that of the ambient gas. Using the energy conservation, we can give the scaling law for the oscillationtimeτ =√(3γjet,0Wjet,0c)(Pjet,0Pamb,0)1/2.

(1)

In Figure 2, we plot the oscillation time averaged overten cycles for numerical runs with diﬀerent initial pressure
ratios. The solid line represents the analytic scaling wederived. This indicates that our numerical results are wellcaptured by our simple scaling law shown in Equation (1).
1000√

3γjet,0Pjet,0Pamb,01/2

The spatial distribution of the Lorentz factor near thejet injection point is shown in Figure 1(b) when the jet
is injected into the uniform ambient medium. The sizeof the cusp-shaped rarefaction region in both 1D and 2D
cases seems to be almost the same, although the verticalaxis is the time in Figure 1(a). This suggests that thetypical size of the region where the Lorentz factor of thejet is powerfully boosted is essentially determined by themodulation caused by the interaction of rarefaction waves
which propagate toward the center of the jet.
4Conclusion

The nonlinear evolution of the interacting rarefaction
waves excited at the cylindrical jet-ambient medium interface is studied through one-dimensional relativistic hydrodynamic simulations. It is found that an enhanced decrease in the relativistic pressure due to the interactionof rarefaction waves transiently yields a more powerfulboost of the Lorentz factor of the bulk jet than that expected from a single rarefaction wave. The cyclic in-situ
energy conversion between thermal energy and bulk kinetic energy is a natural relativistic outcome of the jet
scenario studied and responsible for the radial oscillating motion of the jet. The oscillation timescale is characterized by the initial pressure ratio of the jet to theambient medium, and follows a simple scaling relation
τoscillation ∝ (Pjet,0 /Pamb,0 )1/2 .
It is conﬁrmed from extended two-dimensional simulations that repeated excitation and convergence of rarefaction waves result in the alignment of the interacting regionsof rarefaction waves, conﬁned by oblique shocks, along thepropagation direction of the jet when a relativistically hot
jet propagates through an ambient medium. The typicalsize of the reconﬁnement region inside the jet is essentiallydetermined by the modulation caused by the interactionof rarefaction waves.

τReferences
100

101

10100
Pjet,0 /Pamb,01000

Fig. 2. Relation between the oscillation time scale of thesystem and initial pressure ratio of the jet to the ambientgas. Diamonds denote the oscillation time averaged overten cycles for each parameter.

3.2

2D calculation

We investigate how the oscillating motion found in the1D model aﬀects the propagation dynamics and the structure of the jet in a more realistic two dimensional axisymmetric system. Our axisymmetric simulation of the jet
propagation has been carried out in cylindrical coordinates(r, z), where the z-axis coincides with the symmetric axis.
Relativistically hot ﬂow is continuously injected into an
uniform ambient medium from the lower boundary of thecomputational domain. Hydrodynamic parameters in 2Dmodel are the same as the 1D model.

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