Download Two-stream instability in collisionless shocks and foreshock

INSTITUTE OF PHYSICS PUBLISHING
PLASMA PHYSICS AND CONTROLLED FUSION
Plasma Phys. Control. Fusion 48 (2006) B303–B311
doi:10.1088/0741-3335/48/12B/S29
Two-stream instability in collisionless shocks
and foreshock
M E Dieckmann1,4 , B Eliasson1 , P K Shukla1 , N J Sircombe2 and
R O Dendy2,3
1 Institute of Theoretical Physics IV and Centre for Plasma Science and Astrophysics,
Ruhr-University Bochum, D-44780 Bochum, Germany
2 Centre for Fusion, Space and Astrophysics, Department of Physics, Warwick University,
Coventry CV4 7AL, UK
3 UKAEA Culham Division, Culham Science Centre, Abingdon, Oxfordshire, OX14 3DB, UK
E-mail: [email protected]
Received 23 June 2006
Published 13 November 2006
Online at stacks.iop.org/PPCF/48/B303
Abstract
Shocks play a key role in plasma thermalization and particle acceleration in
the near Earth space plasma, in astrophysical plasma and in laser plasma
interactions. An accurate understanding of the physics of plasma shocks is thus
of immense importance. We give an overview over some recent developments
in particle-in-cell simulations of plasma shocks and foreshock dynamics. We
focus on ion reflection by shocks and on the two-stream instabilities these beams
can drive, and these are placed in the context of experimental observations,
e.g. by the Cluster mission. We discuss how we may expand the insight gained
from the observation of proton beam driven instabilities at near Earth plasma
shocks to better understand their astrophysical counterparts, such as ion beam
instabilities triggered by internal and external shocks in the relativistic jets of
gamma ray bursts, shocks in the accretion discs of micro-quasars and supernova
remnant shocks. It is discussed how and why the peak energy that can be reached
by particles that are accelerated by two-stream instabilities increases from keV
energies to GeV energies and beyond, as we increase the streaming speed to
relativistic values, and why the particle energy spectrum sometimes resembles
power law distributions.
(Some figures in this article are in colour only in the electronic version)
4
On leave of absence from ITN, Linköping University, Campus Norrköping, SE-60174 Norrköping, Sweden.
0741-3335/06/SB0303+09$30.00
© 2006 IOP Publishing Ltd
Printed in the UK
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1. Introduction
Shocks exist in environments that range from Q-machine plasma [1] through Solar system
plasma, e.g. the Earth bow shock [2–4], shocks forming at coronal mass ejections [5,6] and the
Solar wind termination shock [6] to astrophysical plasma shocks, such as supernova remnant
(SNR) shocks [7,8], pulsar wind termination shocks [9] and the internal and external shocks of
jets ejected by micro-quasars [10], active galactic nuclei [11, 12] and gamma ray bursts [13].
Lasers give rise to shocks on a laboratory scale [14–16] and the processes underlying unbounded
shocks can be examined in situ at the Earth bow shock [2–4,17]. Important shock parameters
are the magnetisation (electron gyrofrequency ωc = eB/me ), the plasma density (plasma
1/2
frequency ωp = (ne e2 /me 0 ) ) and how the plasma flow speed vs compares with the
proton sound speed CS ≈ (kB T /mp )1/2 , the Alfven speed VA = (ωc /ωp )(me /mp )1/2 c, the
1/2
magnetosonic speed CMS = (VA2 + CS2 ) and the speed of light c. The satellite observations
of Solar system shocks, e.g. by the Cluster mission [18–20] and numerous others, have shed
considerable light on the shock dynamics. The Solar wind has a typical magnetic field of ≈5 nT
and a plasma number density of ≈5 cm−3 . Its speed can vary in the range vSW ≈ 105 –106 m s−1 ,
resulting in Alfvenic Mach numbers of the Earth bow shock in the interval vSW ≈ 2–20 VA
and thus in a supercritical shock.
The Earth bow shock does not produce cosmic ray radiation, in contrast to astrophysical
shocks that support Fermi acceleration [21–25]. We concentrate on electron Fermi acceleration,
which requires a seed of hot electrons. It must be produced by an injection mechanism that may
be facilitated by the shock-reflected beams of ions [26–31]. The ion beams drive waves that
accelerate particles nonlinearly. The ion beams in the foreshock of the Earth bow shock have
energies of 1–10 keV and move at the speed vb ≈ 106 m s−1 in the upstream reference frame.
The thermal speed of the Solar wind electrons is vte ≈ 106 m s−1 . If we assume that SNR
shocks expand into the stellar wind of the progenitor star, we may assume that the upstream vte
is similar to that in the Solar wind. SNR shocks expand, however, at 107 –3 × 107 m s−1
and a specularly reflected ion beam would have the speed vb /vte ≈ 20—60 vte in the
upstream frame. Violent two-stream instabilities can develop in the foreshock of SNR shocks
[6, 26], whereas the slower ion beams close to the Earth’s bow shock produce predominantly
magnetohydrodynamic waves [2] and the two-stream instability and the modified twostream instability [32, 33] do not yield electrons that are energetic enough to undergo Fermi
acceleration.
Particle-in-cell (PIC) simulations have been applied with great success to the study of
shocks in a fully kinetic treatment. In section 2, we discuss some of the results that have been
obtained by recent simulations of several research groups [34–36], and which are discussed
in detail in the review articles [37, 38]. We demonstrate how a shock and the shock-reflected
ion beam form, if two plasma slabs collide at a superAlfvenic speed perpendicularly to the
ambient magnetic field. We discuss the interaction of the shock-reflected proton beam with
the upstream plasma [32, 33,39–43] and, in particular, we demonstrate that electrons can be
accelerated to high energies by two-stream instabilities and wakefield acceleration and that
they can develop power law distributions due to their interaction with an ambient magnetic
field [44] in section 3 and electrostatic (ES) turbulence [45–47] in section 4. Such processes
can exist in the presence of a strong guiding magnetic field, a high plasma temperature and a
moderately relativistic flow speed, since then the competing relativistic Weibel instability is
suppressed [48–50]. Otherwise, the Weibel and the mixed mode instabilities outrun the twostream instability [51,52], unless the flow speed is non-relativistic [53], leading to the formation
of current channels delaying a full thermalization. We omit here a discussion of the Weibel and
Two-stream instability in collisionless shocks and foreshock
B305
2
3.5
3
A
–4
v /V
–2
x
0
1
2.5
2
1.5
1
–6
–8
a)
2
0
0.5
Xω
CI
vy / VA
3
2
0
–2
–4
–6
–8
1
/v
0.5
1.5
b
3.5
3
2.5
1
2
1.5
1
b)
2
0
0.5
1
X ωCI / vb
0.5
1.5
Figure 1. The front end of the forward shock: panel (a) shows the distribution in the x—vx plane
and panel (b) shows that in the x—vy plane. The colour scale indicates the 10-logarithm of the
number of computational protons.
filamentation instabilities and refer the interested reader instead to [48,49,54–57]. In section 5,
we discuss the findings in the context of astroparticle acceleration.
2. PIC simulation of a 1D perpendicular shock
We simulate with a PIC simulation the collision of two identical non-relativistically moving
plasma slabs consisting of electrons and protons, thereby reproducing some aspects of the
previous shock simulations discussed in [34–38]. Each of the four species has the number
density ne . We model the full proton-to-electron mass ratio mp /me = 1836, as in [34]. The
plasma slabs are magnetized such that ωp = 2ωc and β = 0.05. The magnetic field B is
aligned with the z-axis. The slabs move along the x-direction and collide at a speed of 4.5 VA .
A forward and reverse shock develop, as shown by the supplementary movie 1. The plasma in
the overlap region of the plasma slabs (downstream) moves at the speed of 2.25 VA relative to
each slab, which is half the speed considered in [34]. A snapshot of the proton distribution at
time t with ωc me t/mp = 4.8 is shown in figure 1. Three characteristic and important plasma
structures are visible. Structure 1 is the beam of shock-reflected protons. Structure 2 originates
from protons that have been and are still trapped in the shock potential. The trapped protons
are transported across B and undergo shock surfing acceleration along the vy direction [27,58].
Structure 3 is an ion hole [59, 60], which gives rise to bipolar ES fields that are often found
at shocks, e.g. the Earth bow shock [61]. The energy dissipation by the proton beam forces
the shock to restructure periodically, in line with previous simulations [34–38] and shown by
figure 2. The plasma parameters are not representative for most geospace and astrophysical
shocks, due to the β 1. Decreasing |B|, however, increases the computational cost and the
more realistic values β ≈ 1 are out of range for our computer resources. Instead, we focus on
the thermalization of the shock-reflected ion beam. In what follows, we limit our discussion
to the two-stream instability as one of several potential electron injection mechanisms,
e.g. [32, 33, 62, 63].
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Figure 2. The time evolution of the Bz component. The amplitude is normalized to the initial
amplitude and CI is the proton gyrofrequency. The reformation of Bz is evidence for the shock
restructuring.
3. The mildly relativistic two-stream instability in magnetized plasma
The proton beam driven two-stream instability probably cannot yield strong electron
acceleration if the beam speed is non-relativistic [39–42], although it may be sufficient to inject
these into Fermi acceleration across a perpendicular SNR shock [26–29]. This is a consequence
of the sideband and coalescence instabilities, that limit the life-time of the saturated ES wave.
For relativistic proton beam speeds, however, the ES waves stabilize [64], resulting in efficient
electron surfing acceleration [44]. The stability of the ES waves is demonstrated by figure 3,
which shows the modulus of the ES wave fields that are driven by two counterpropagating
proton beams. The latter move at the speed vb = 0.9c through a background electron–proton
plasma and they have each 0.1 times the background proton density. A perpendicular magnetic
field yields ωp /ωc = 100 (red curve) and ωp /ωc = 10 (blue curve). The proton beam driven
ES waves are stable over several hundreds of plasma periods Tp = 2π/ωp . This stability,
together with the relativistic phase speed of the ES waves, implies that the trapped electrons
must undergo a substantial acceleration. This is confirmed by the multimedia supplement
movie 2 of [44], which shows the time-evolution of the electrons for ωp /ωc = 10. The
electrons are accelerated by electron surfing acceleration; they detrap and gyrate freely around
B. The electrons thermalize, once the electrons are in resonance again with the ES wave due
to stochastic interactions arising from a coupling of their gyromotion and their oscillation
in the wave electric field [65] and a power law distribution develops at |p| > 100me c, as
demonstrated by the curve for ωp /ωc = 10 in figure 4.
4. The ultrarelativistic two-stream instability in magnetized plasma.
By increasing the ion beam velocity to a value, that corresponds to a gamma factor of a few,
the coalescence and sideband instabilities become ineffective and the ES wave is completely
stabilized. A new limit for the electron acceleration by cross-field transport emerges,
γ ≈ mp /me [66, 67]. We exemplify some consequences of the presence of ultraenergetic
Two-stream instability in collisionless shocks and foreshock
B307
E (k=–ku, t)
200
(a)
100
0
0
200
400
600
800
t/T
p
(b)
u
E (k=k , t)
200
100
0
0
200
400
600
800
t / Tp
Figure 3. (Taken from [44]) Panel (a) shows the electric field modulus in V m−1 at the wave
number k = −ku due to the proton beam 1, where ku is the two-stream unstable wave number
modulus. Panel (b) shows the electric field at k = ku , which is driven by the proton beam 2 that
moves in the opposite direction.
4
N(|p|)
10
2
10
0
10
0
10
1
10
|p| / me c
2
10
Figure 4. (Taken from [44]) The final electron momentum distributions in the simulation with
ωp /ωc = 10 (blue curve) and in the simulation with ωp /ωc = 100 (red curve).
electrons in the simulation. The two-stream instability modelled in [66] is driven by two
counter-propagating proton beams with a γ (vb ) = 10 that move through a background plasma
that consists of electrons and protons. The magnetic field is oriented at 45◦ relative to the beam
velocity vectors and has the strength ωp /ωc = 71. The electron acceleration is not spatially
homogeneous and phase space bunches of ultrarelativistic electrons can modulate the proton
beam through wakefield acceleration, as demonstrated by figure 5. Wakefield acceleration is
a well-known process [68] and the proton velocity modulation it results in has been shown to
destabilize ion beams also in Q-machine experiments [1]. The electric field power associated
with this structure exceeds that of the two-stream instability by a factor of 103 , as depicted by
figure 6. Such fields may yield strong bremsstrahlung [69]. It saturates by the formation of an
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Figure 5. (Taken from [45]) Panel (a) shows the 10-logarithm of the phase space distribution of
high energy electrons. A boundary is separating an interval with a high electron density from a
void. This boundary is locally accelerating the protons of one beam by wakefield acceleration, as
displayed by panel (b), where the colour signifies the 10-logarithm of the proton beam density.
Figure 6. (Taken from [45]) The 10-logarithm of the ES wave power that is normalized to
(ωp cme /e)2 . Here, k is the minimum resolved wavenumber, k is an integer value and ku = ωp /vb .
At kk ≈ ku the two-stream mode develops and the wakefield accelerator results in the waves
with kk ku .
ultrarelativistic phase space hole in the proton distribution [45,47]. This wakefield acceleration
is not limited to oblique magnetic fields. A similar acceleration has been observed for a strictly
perpendicular magnetic field [67] and in unmagnetized plasma [46, 47]. The electric fields
accelerate the plasma to GeV energies in the reference frame of the background (upstream)
plasma. If this process would develop at an internal shock of a gamma ray burst jet that is
Two-stream instability in collisionless shocks and foreshock
B309
Figure 7. (Taken from [45]) The energy distributions of electrons (red) and protons (blue) after
the saturation of the strong ES wave. Panel (a) shows the distributions in the rest frame of the
background plasma. Panel (b) shows the distributions in a reference frame moving at γ = 103 .
moving with a γ ≈ 103 , the jet could produce, by a relativistic addition of the velocities,
cosmic ray particles with 1014 eV in the Earth frame of reference, as demonstrated by figure 7.
The electrons follow a power law distribution, in response to their interaction with the strong
ES waves at low wavenumbers.
5. Discussion
We have summarized in this work some aspects of the role the two-stream instability has in
the dynamics of geospace and astrophysical shocks. A simulation of two colliding plasma
slabs, which propagate through a perpendicular magnetic field, has reproduced the finding
in the [34–38] that a shock is forming. The energy dissipation of the shock is well known
to be accomplished by a shock-reflected proton beam [30, 31], shock surfing acceleration of
protons [27,58] and the formation of holes in the proton phase space distribution [59,60]. This
is true also for Solar system shocks, as experimental observations show [2, 30, 31, 61]. The
speed with which these ion beams move through the upstream plasma is comparable to the
electron thermal speed and the initial two-stream instability is relatively weak due to Landau
damping. The electron acceleration by the two-stream instability can be boosted by secondary
instabilities [32, 33], however, but probably not to the extent of SNR shocks. The ion beams
further relax through the growth of magnetohydrodynamic waves [2], which probably do not
constitute efficient particle injection mechanisms at the Earth’s bow shock.
The much faster expansion speed of SNR shocks and an upstream electron thermal speed
that is, presumably, comparable to that of the Solar wind, imply that their shock-reflected
ion beams may excite stronger two-stream and modified two-stream instabilities. The nonrelativistic ion beam speeds imply, however, that mechanisms like electron surfing acceleration
may not be efficient [39–42]. However, other mechanisms like parametric instabilities [29],
gyroresonant surfing [62] and Alfven wave energy dissipation [63] may inject electrons into
Fermi acceleration.
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Once the ion beams reach relativistic speeds, the cross-field acceleration can become
highly effective. Proton beam speeds of 0.6 c–0.9 c that we may find close to shocks in the
jets [10,70] and accretion discs [71] of micro-quasars can accelerate electrons to 6 < γ < 200
and their energy may increase further by stochastic interactions with the ES field and B [65].
The proton beam is, however, not thermalized.
At even higher beam speeds that we may find close to the internal shocks of gamma ray
bursts, the electrons can be accelerated up to γ ≈ mp /me . Their large relativistic mass implies
that electron bunches can give rise to a wakefield acceleration of the proton beam; the latter
secondary instability can rapidly release the beam energy and accelerate particles to cosmic
ray energies during short spatio-temporal intervals. We can consider this system as an example
of a modified two-stream instability, similar to that in non-relativistic systems [32, 33]. Future
work must, however, address the interplay of the two-stream instability with the mixed mode
and filamentation instabilities [50–52] in three-dimensional simulations to correctly represent
the colliding plasma.
Acknowledgment
This work has been financed by an individual DFG grant SH21/1-1 and the
Sonderforschungsbereich SFB 591. The computer time has been provided by the Swedish
NSC and HPC2N supercomputer centres.
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