Acta Polytechnica


doi:10.14311/AP.2013.53.0683
Acta Polytechnica 53(Supplement):683–686, 2013 © Czech Technical University in Prague, 2013

available online at http://ojs.cvut.cz/ojs/index.php/ap

MODELING JET INTERACTIONS WITH THE AMBIENT
MEDIUM

J.H. Bealla,b,c,∗, J. Guilloryc, D.V. Rosed, Michael T. Wolffb

a St. John’s College, Annapolis, MD
b Space Sciences Division, Naval Research Laboratory, Washington, DC
c College of Sciences, George Mason University, Fairfax, VA
d Voss Scientific, Albuquerque, NM
∗ corresponding author: beall@sjca.edu

Abstract. Recent high-resolution (see, e.g., [13]) observations of astrophysical jets reveal complex
structures apparently caused by ejecta from the central engine as the ejecta interact with the surrounding
interstellar material. These observations include time-lapsed “movies” of both AGN and microquasars
jets which also show that the jet phenomena are highly time-dependent. Such observations can be
used to inform models of the jet–ambient-medium interactions. Based on an analysis of these data, we
posit that a significant part of the observed phenomena come from the interaction of the ejecta with
prior ejecta as well as interstellar material. In this view, astrophysical jets interact with the ambient
medium through which they propagate, entraining and accelerating it. We show some elements of the
modeling of these jets in this paper, including energy loss and heating via plasma processes, and large
scale hydrodynamic and relativistic hydrodynamic simulations.

Keywords: jets, active galaxies, blazars, intracluster medium, non-linear dynamics, plasma astro-
physics.

1. Introduction
Large scale hydrodynamic simulations of the interac-
tion of astrophysical jets with the ambient medium
through which they propagate can be used to illumi-
nate a number of interesting consequences of the jets’
presence. These include acceleration and entrainment
of the ambient medium, the effects of shock structures
on star formation rates, and other effects originating
from ram pressure and turbulence generated by the jet
(see, e.g., [1, 10, 16, 23]. We present results for large
scale hydrodynamic simulations and initial relativistic
hydrodynamic simulations in this paper.

2. Modeling the jet interaction
with the ambient medium

Hydrodynamic sumulations neglect an important
species of physics: the microscopic interactions that
occur because of the effects of particles on one another
and of particles with the collective effects that accom-
pany a fully or partially ionized ambient medium
(i.e. a plasma).

While the physical processes (including plasma pro-
cesses) in the ambient medium can be modeled in
small regions by PIC (Particle-in-Cell) codes for some
parameter ranges, simulations of the larger astrophysi-
cal jet structure with such a PIC code are not possible
with current or foreseeable computer systems. For
this reason, we have modeled these plasma processes
in the astrophysical regime by means of a system of
coupled differential equations which give the wave

populations generated by the interaction of the astro-
physical jet with the ambient medium through which
it propagates. A detailed discussion of these efforts
can be found, variously, in [6, 8, 18, 19, 22].
The system of equations used to solve for the nor-

malized wave energies is very stiff. Solving the system
of equations yields a time-dependent set of normal-
ized wave energies (i.e., the ratio of the wave energy
divided by the thermal energy of the plasma) that
are generated as a result of jets interaction with the
ambient medium. As we will show, these solutions can
yield an energy deposition rate (dE/dt), an energy
deposition length (dE/dx), and ultimately, a momen-
tum transfer rate (dp/dt) that can be used to estimate
the effects of plasma processes on the hydrodynamic
evolution of the jet.
For this analysis, we posit a relativistic jet of ei-

ther e±, p–e−, or more generally, a charge-neutral,
hadron–e− jet, with a significantly lower density than
the ambient medium. The primary energy loss mech-
anism for the electron–positron jet is via plasma pro-
cesses, as Beall [8] notes. Kundt [11, 12] also discusses
the propagation of electron–positron jets.
The principal plasma waves generated by the

jet–ambient-medium interaction can be characterized
as follows: the two stream instability waves, W1, in-
teract directly with the ambient medium, and are
“predated” (principally) by the oscillating two stream
instability waves, W2, and the ion-density fluctuations
or ion-acoustic waves, WS. The waves produced in
the plasma by the jet produce regions of high elec-

683

http://dx.doi.org/10.14311/AP.2013.53.0683
http://ojs.cvut.cz/ojs/index.php/ap


J.H. Beall, J. Guillory, D.V. Rose, Michael T. Wolff Acta Polytechnica

tric field strength and relatively low density, the so-
-called “cavitons” (after solitons or solitary waves)
which propagate like wave packets. These cavitons
mix, collapse, and reform, depositing energy into the
ambient medium, transferring momentum to it, and
entraining (i.e., dragging along and mixing) the ambi-
ent medium within the jet. The typical caviton size
when formed is of order 10’s of Debye lengths, where
a Debye length, λD = 7.43 × 102

√
T/np cm, T is the

electron temperature in units of eV, and np is the
number density of the ambient medium in units of
cm−3.

2.1. Energy loss, energy deposition
rate, and momentum transfer

In order to determine the energy deposition rate, the
momentum transfer rate, and heating, we model the
plasma interaction as a system of very stiff, coupled
differential equations (see, e.g., [7]), which simulate
the principal elements of the plasma processes that
draw energy out of the jet. As a test of the fealty of
this method, we “benchmark” (see [15]) the wave pop-
ulation code by using the PIC code in regions of the
parameter space where running the PIC code simula-
tion is practicable. We then use the wave population
code for regions of more direct astrophysical interest.
A more detailed discussion of the comparisons between
the PIC-code simulations and the wave-population
model can be found in [20, 21]. The coupling of
these instability mechanisms is expressed in the model
through a set of rate equations. These equations are
discussed in some detail by Rose et al. [18, 19], and
Beall [8].
Beall et al. [7] illustrate two possible solutions for

the system of coupled differential equations that model
the jet–ambient-medium interaction: a damped os-
cillatory and an oscillatory solution. The Landau
damping rate for the two-temperature thermal dis-
tribution of the ambient medium is used for these
solutions. As noted in the figure caption, transitions
toward chaotic solutions have been observed for very
large growth rates for the two-stream instability.

In order to benchmark the wave-population code, we
use that code to calculate the propagation length of an
electron–positron jet as described above. Specifically,
we model the interaction of the relativistic jet with
the ambient medium through which it propagates by
means of a set of coupled, differential equations which
describe the growth, saturation, and decay of the three
wave modes likely to be produced by the jet–medium
interaction. First, two-stream instability produces a
plasma wave, W1, called the resonant wave, which
grows initially at a rate Γ1 ≤ (

√
3/2γ)(nb/2np)1/3ωp,

where γ is the Lorentz factor of the beam, nb and np
are the beam and cloud number densities, respectively,
in units of cm−3 and ωp is the plasma frequency, as
described more fully in [18].
The average energy deposition rate, 〈d(α�1)/dt〉,

of the jet energy into the ambient medium via

plasma processes can be calculated as 〈d(α�1)/dt〉 =
npkT 〈W〉(Γ1/ωp)ωp erg cm−3 s−1, where k is Boltz-
mann’s constant, T is the plasma temperature, 〈W〉 is
the average (or equilibrium) normalized wave energy
density obtained from the wave population code, Γ1
is the initial growth rate of the two-stream instability,
and α is a factor that corrects for the simultaneous
transfer of resonant wave energy into nonresonant
and ion-acoustic waves. The energy loss scale length,
dEplasma/dx = −(1/nbvb)(dα�1/dt), can be obtained
by determining the change in γ of a factor of 2 with
the integration

∫
dγ = −

∫
[d(α�1)/dt]dl/(vbnbm′c2)

as shown in [8, 17], where m′ is the mass of the beam
particle. Thus, Lp = ((1/2)γcnbmc2)/(dα�1/dt) cm is
the characteristic propagation length for collisionless
losses for an electron or electron–positron jet, where
dα�1/dt is the normalized energy deposition rate (in
units of thermal energy) from the plasma waves into
the ambient plasma. In many astrophysical cases, this
is the dominant energy loss mechanism. We can there-
fore model the energy deposition rate (dE/dt) and the
energy loss per unit length (dE/dx), and ultimately
the momentum loss per unit length (dp/dx) due to
plasma processes.
Beall, Guillory, and Rose (2009) have com-

pared the results of a PIC code simulation of an
electron–positron jet propagating through an ambient
medium of an electron–proton plasma with the solu-
tions obtained by the wave population model code,
and have found good agreement between the two re-
sults (see Fig. 1 from that paper). At the same time,
that paper demonstrates that the ambient medium
is heated and entrained into the jet. That analy-
sis also shows that a relativistic, low-density jet can
interpenetrate an ambient gas or plasma.
Initially, and for a significant fraction of its propa-

gation length, the principal energy loss mechanisms
for such a jet interacting with the ambient medium
are via plasma processes [8, 18].

As part of our research into the micro-physics of the
interaction of jets with an ambient medium, we con-
tinue to investigate the transfer of momentum from
the jet. Understanding how such a transfer is accom-
plished is essential to understanding the manner in
which the ambient medium (for example, from inter-
stellar clouds) is accelerated and eventually entrained
into the large-scale astrophysical jet.
In order to proceed to a more detailed analysis

of the issue of momentum transfer, we have used
modern PIC code simulations to study the dynamics
of caviton formation, and have confirmed the work of
Robinson and Newman (1990) in terms of the cavitons’
formation, evolution, and collapse.
We are in the process of developing a multi-scale

code which uses the energy deposition rates and
momentum transfer rates from the PIC and wave-
population models as source terms for the highly par-
allelized hydrodynamic code currently running on the
NRL SGI Altix machine.

684



vol. 53 supplement/2013 Modeling Jet Interactions with the Ambient Medium

Figure 1. Simulation using a highly-parallelized ver-
sion of the VH-1 hydrodynamics code. The figure
shows the x–z cross-section (Figure 1a) and the y–
z cross-section (Figure 1b) for a fully 3-dimensional
hydrodynamic simulation of a jet with v = 1.5 ×
109 cm s−1. The simulation length is approximately
64 kpc on the long axis.

Figure 2. 2D simulation of an axially symmetric,
relativistic hydrodynamic jet with v/c = 0.995, show-
ing jet density (top) and axial velocity along the jet
axis and radially (bottom) using the PLUTO code
(Mignone et al. 2009)

Plasma effects can have observational consequences.
Beall [8] has noted that plasma processes can slow the
jets rapidly, and Beall and Bednarek [4] have shown
that these effects can truncate the low-energy portion
of the γ-ray spectrum (see their Fig. 3). In the in-
terests of brevity, we do not go into the (reasonable)
assumptions for these calculation. Please see [4] for a
detailed discussion. A similar effect will occur for neu-
trinos and could also reduce the expected neutrino flux
from AGN. The presence of plasma processes in jets
can also greatly enhance line radiation species by gen-
erating high-energy tails on the Maxwell–Boltzmann
distribution of the ambient medium, thus abrogating
the assumption of thermal equilibrium.

An analytical calculation of the boost in energy of
the electrons in the ambient medium to produce such a
high energy tail, with Ehet ∼ 30÷100 kT , is confirmed
by PIC-code simulations. Aside from altering the
Landau damping rate, such a high-energy tail can
greatly enhance line radiation over that expected for

a thermal equilibrium calculation (see [5, 7] for a
detailed discussion).

If the beam is significantly heated by the jet–cloud
interaction, the beam will expand transversely as it
propagates, and will therefore have a finite opening
angle. These “warm beams” result in different growth
rates for the plasma instabilities, and therefore pro-
duce somewhat different propagation lengths (see,
e.g., [7, 9, 20]). A “cold beam” is assumed to have
little spread in momentum. The likely scenario is that
the beam starts out as a cold beam and evolves into
a warm beam as it propagates through the ambient
medium. This scenario is clearly illustrated by the
PIC simulations we have used to benchmark the wave
population codes appropriate for the astrophysical
parameter range (see, e.g., [5, 21]).

3. Hydrodynamic Simulations
As an example of the large scale hydrodynamic simu-
lations we are conducting, we show a jet simulation in
the x–z cross-section (Figure 1a) and the y–z cross-
section for a 3-dimensional hydrodynamic simulation
of a jet with v = 1.5 × 109 cm s−1. The simulation
length is approximately 64 kpc on the long axis. The
simulation shows the detailed structure of the shocks
generated by the jet as well as the Kelvin–Helmholtz
instabilities of the jet itself, which generate additional
shock structures. These shocks produce Jeans-length
structures which will ultimately collapse to form stars
that in turn may feed the central engine in the AGN.
In Fig. 2, we show the density and axial velocity plot-
ted radially and along the jet axis for a 2D simulation
of a relativistic jet using the PLUTO code [14]. We
plan to develop a multi-scale code with the plasma
processes as momentum and energy source terms for
these codes.

Acknowledgements
HB and MTW gratefully acknowledge the support of the
Office of Naval Research for this project. Thanks also to
colleagues at various institutions for their continued inter-
est and collaboration, including Kinwah Wu, Curtis Sax-
ton, S. Schindler and W. Kapferer, and S. Colafrancesco.

References
[1] Basson, J.F., and Alexander, P., 2002, MNRAS, 339,
353

[2] Beall, J.H., et al. 1978: Ap.J., 219, 836

[3] Beall, J.H. and Rose, W.K. 1981: Ap.J., 238, 539

[4] Beall, J.H. and Bednarek, W. 1999: Ap.J., 510, 188

[5] Beall J.H., Guillory, J., & Rose, D.V., 1999, Journal
of the Italian Astronomical Society, 70, 1235

[6] Beall J.H., Guillory, J., & Rose, D.V., 2003, C.J.A.A.,
Vol. 3, Suppl., 137

[7] Beall J.H., Guillory, J., Rose, D.V., Schindler, S.,
Colafrancesco, S., 2006, C.J.A.A., Supplement 1, 6, 283

685



J.H. Beall, J. Guillory, D.V. Rose, Michael T. Wolff Acta Polytechnica

[8] Beall J.H., 1990, “Energy Loss Mechanisms for Fast
Particles,” in Physical Processes in Hot Cosmic
Plasmas, (Kluwer Academic Publishers:
Dordrecht/Boston/London: W. Brinkmann, A.C.
Fabian, and F. Giobannelli, eds.)

[9] Kaplan, S.A., & Tsytovich, V.N., 1973 Plasma
Astrophysics, (Pergamon Press: Oxford, New York)

[10] Krause, M, & Camenzind, M., 2003, New Astron.
Rev., proceedings of the conference in Bologna: “The
Physics of Relativistic Jets in the CHANDRA and
XMM Era”, Sep.,2002 Journal-ref: New Astron.Rev. 47
(2003) 573–576

[11] Kundt, W., 1987, In Astrophysical Jets & Their
Engines: Erice Lectures, ed. W. Kundt (Reidel:
Dordrecht. Netherlands)

[12] Kundt, W. 1999: in Multifrequency Behaviour of
High Energy Cosmic Sources, F. Giovannelli & L.
Sabau-Graziati (eds.), Mem. S.A.It. 70, 1097

[13] Lister et al., 2009, AJ, 137, 3718
[14] Mignone et al., 2007, ApJS, 170, 228
[15] Oreskes, N., Shrader-Frechette, K., and Belitz, K.,
1996, Science, 263, 641

[16] Perucho et al., 2012, MmSAI 83, 297
[17] Rose, D.V., 1997, PhD Dissertation, George Mason
University, Fairfax, VA, USA

[18] Rose, W.K., Guillory, J., Beall, J.H., & Kainer, S.,
1984, Ap.J., 280, 550

[19] Rose, W.K., Beall, J.H., Guillory, J., & Kainer, S., et
al. 1987, Ap.J., 314, 95

[20] Rose, D.V., Guillory, J. & Beall, J.H., 2002, Physics
of Plasmas, 9, 1000

[21] Rose, D.V., Guillory, J. & Beall, J.H., 2005, Physics
of Plasmas, 12, 014501

[22] Scott, J.H., Holman, G.D., Ionson, and
Papadopoulos, K., 1980, Ap.J., 239, 769

[23] Zanni, C., Murante, G., Bodo, G., Massaglia, S., Rossi,
P., Ferrari, A., 2005, Astronomy and Astrophysics, 429

Discussion
Maria Diaz Trigo — Two questions: 1. Given that
winds are slower than jets, do you expect that the interac-
tion with the ambient medium, and therefore the energy
deposition, is more efficient for winds? 2. At which temper-
atures is the ambient medium heated after the interaction
with the jet?
Jim Beall — As to the first question, the winds being
slower than the jet, I think they would tend to be less
energetic, and therefore would tend to have a less dramatic
effect on the ambient medium than do the jets. Regarding
the temperatures the ambient medium can reach, PIC code
simulations as well as analytical estimates indicate that
the plasma processes can heat a 104 K ambient medium
to 105 K using the plasma processes along.
Maurice van Putten — As you know, knotted struc-
tures are observationally quite generic. In your simulations
in hydro, they do not form. In my simulations, with rel-
ativistic MHD (RMHD) in Ap.J., 467, L57 (1996), they
form by dynamically significant magnetic field strengths.
Adding “test” or “tracer” magnetic fields to hydro simula-
tions does not seem to be a valid approximation.
Jim Beall — I agree that we need to move to MHD and
RMHD simulations, and it is our plan of research to do
this. But if you look closely at the structure of the jets in
our simulations, you can see knotted structures along the
jet’s longitudinal axis that are similar to those observed
in the sky.

686


	Acta Polytechnica 53(Supplement):683–686, 2013
	1 Introduction
	2 Modeling the jet interaction with the ambient medium
	2.1 Energy loss, energy deposition rate, and momentum transfer

	3 Hydrodynamic Simulations
	Acknowledgements
	References