A&A 452, 371-381 (2006)
DOI: 10.1051/0004-6361:20054074
N. J. Sircombe1, -
M. E. Dieckmann1,
- P. K. Shukla1 - T. D. Arber2
1 - Ruhr-University Bochum, Institute of Theoretical Physics IV,
NB 7/56, 44780 Bochum, Germany
2 - University of Warwick,
Department of Physics, Coventry, W Midlands CV4 7AL,
UK
Received 19 August 2005 / Accepted 9 February 2006
Abstract
Context. We examine plasma thermalisation processes in the foreshock region of astrophysical shocks within a fully kinetic and self-consistent treatment. We concentrate on proton beam driven electrostatic processes, which are thought to play a key role in the beam relaxation and the particle acceleration. Our results have implications for the effectiveness of electron surfing acceleration and the creation of the required energetic seed population for first order Fermi acceleration at the shock front.
Aims. We investigate the acceleration of electrons via their interaction with electrostatic waves, driven by the relativistic Buneman instability, in a system dominated by counter-propagating proton beams.
Methods. We adopt a kinetic Vlasov-Poisson description of the plasma on a fixed Eulerian grid and observe the growth and saturation of electrostatic waves for a range of proton beam velocities, from 0.15c to 0.9c.
Results. We can report a reduced stability of the electrostatic wave (ESW) with increasing non-relativistic beam velocities and an improved wave stability for increasing relativistic beam velocities, both in accordance with previous findings. At the highest beam speeds, we find the system to be stable again for a period of 160 plasma periods. Furthermore, the high phase space resolution of the Eulerian Vlasov approach reveals processes that could not be seen previously with PIC simulations. We observe a, to our knowledge, previously unreported secondary electron acceleration mechanism at low beam speeds. We believe that it is the result of parametric couplings to produce high phase velocity ESW's which then trap electrons, accelerating them to higher energies. This allows electrons in our simulation study to achieve the injection energy required for Fermi acceleration, for beam speeds as low as 0.15c in unmagnetised plasma.
Key words: acceleration of particles - waves - plasmas - supernovae: general
First order Fermi acceleration has been proposed as a mechanism for the acceleration of electrons to ultra-relativistic velocities in SNR shocks (Bell 1978a,b; Blandford & Ostriker 1978). Electrons gain energy by repeated crossings of the shock front, and by their scattering off MHD waves on either side of the shock.
The orientation of the ambient magnetic field
relative to the shock normal has implications for the efficiency of electron acceleration (Galeev 1984). We consider the case where the ambient field is orthogonal to the shock normal, this geometry provides an efficient electron acceleration mechanism for high Mach number shocks (Treumann & Terasawa 2001). First order Fermi acceleration at such shocks requires a seed population of electrons that have Larmor radii comparable to the shock thickness. Since the shock thickness is, for perpendicular shocks,
of the order of the ion Larmor radius, the electrons of this seed
population must have mildly relativistic initial speeds.
A kinetic energy comparable to 100 keV is believed to be sufficient
(Treumann & Terasawa 2001). Such electrons are
unlikely to be present, neither in the interstellar medium (ISM) nor
in the stellar wind of the progenitor star of the supernova, but may be created by a pre-acceleration mechanism at the SNR shock front. This pre-acceleration, commonly referred to as the injection problem, is not well understood.
As the shock front of a supernova remnant (SNR) expands into the ISM, it reflects a substantial fraction of the ISM ions as observed in simulations (Shimada & Hoshino 2000; Schmitz et al. 2002) and in-situ at the Earth's bow shock (Eastwood et al. 2005). If the shock normal is quasi-perpendicular to the magnetic field in a high Mach number shock, as many as 20% of the ions can be reflected (Sckopke et al. 1983; Galeev 1984; Lembege & Savoini 1992; Lembege et al. 2004). The reflected ions form a beam that can reach a peak speed comparable to twice the shock speed in the ISM frame of reference as, for example, discussed by McClements et al. (1997). The shock-reflected plasma particles are a source of free energy, similar to the shock-generated cosmic rays (Zank et al. 1990) which are also thought to heat the inflowing plasma. However, the shock reflected ion beam is considerably more dense than the cosmic rays and each particle carries less energy. The developing plasma thermalisation mechanisms are thus likely to be different. Since binary collisions between charged particles in the dilute ISM plasma are negligible, these ion beams relax by their interaction with electrostatic waves and electromagnetic waves. In what follows, we focus on the interaction of electrons with high-frequency electrostatic waves (ESWs) that are driven by two-stream instabilities.
Recent particle-in-cell (PIC) simulation studies (Shimada & Hoshino 2003, 2004; Dieckmann et al. 2000, 2004a; McClements et al. 2001) have examined these mechanisms with a particular focus on how and up to what energies the ESWs driven by non-relativistic or mildly relativistic ion beams can accelerate the electrons in the foreshock region.
The maximum energy the electrons can reach by such wave-particle interactions, depends on the life-time of the
saturated ESW and on the strength and the orientation of .
Initially
a stable non-linear wave known as BGK mode (Bernstein et al. 1957;
Manfredi 1997; Brunetti et al. 2000) develops if the
plasma is unmagnetised or if the magnetic field is weak (McClements et al.
2001; Dieckmann et al. 2002; Eliasson et al.
2005). Such modes are associated with phase space holes - islands of trapped electrons. These BGK modes are destabilised by the
sideband instability, a resonance between the electrons that
oscillate in the ESW potential and secondary ESWs (Kruer et al. 1969;
Tsunoda & Malmberg 1989; Krasovsky 1994). This
resonance transfers energy from the trapped electrons to the secondary
ESWs. The initial BGK mode collapses, once these secondary ESWs grow to
an amplitude that is comparable to that of the initial wave.
Many previous simulations of two-stream instabilities in the context of electron injection and of shocks have employed PIC simulation codes, which suffer from high noise levels (Dieckmann et al. 2004c) and from a dynamical range for the plasma phase space distribution that is limited by the number of computational particles. It is thus possible that certain instabilities can not develop, due to a lack of computational particles in the relevant phase space interval, or that the phase space structures (e.g. BGK modes) are destabilised by the noise (Schamel & Korn 1996) and may benefit from a plasma model based on the direct solution of the Vlasov equation. Such effects have been demonstrated in the non-relativistic limit, for example by Eliasson et al. (2005) and Dieckmann et al. (2004b), in which the results computed by PIC codes have been compared to those of Vlasov codes. The particle species in a Vlasov code are represented by a continuous distribution function, conventionally evolved on a fixed Eulerian grid, rather than by simulation macro-particles. The comparison of results from these two methodologies has shown significant differences in the life-time of the BGK modes both for unmagnetized and magnetized plasma.
SNR shocks typically expand into the ISM at speeds ranging between a
few and twenty percent of c (Kulkarni et al. 1998). The
shock-reflected ions can thus reach a speed of
if the
reflection is specular (McClements et al. 1997) or even higher
speeds if we take into account shock surfing acceleration (Ucer & Shapiro
2001; Shapiro & Ucer 2003). The proton-beam driven
ESWs have a phase speed similar to
(Buneman 1958;
Thode & Sudan 1973). Since the ISM electrons can be accelerated
to a maximum speed well in excess of the phase speed of the ESW
(Rosenzweig 1988), we must consider relativistic modifications
of the life-time of the BGK modes in SNR foreshock plasma. It has been found
by Dieckmann et al. (2004a) that the BGK mode is stabilised if the
phase speed of the ESW is relativistic in the ISM frame of reference. The
likely reason is that the change in the relativistic electron mass introduces
a strong dependence of the electron bouncing frequency on the electron speed
in the ESW frame of reference. This decreases the coherency with which the
electrons interact with the secondary ESWs and thus the efficiency of the
sideband instability. The stabilization is clearly visible in
PIC simulations for ESWs moving with a phase speed of
0.9c (Dieckmann et al. 2004a). For speeds
the
relativistic modifications of the BGK mode stability have been small in
the PIC simulations (Dieckmann et al. 2004a). The better
representation of the phase space density afforded by a Vlasov simulation
may, however, yield a different result and it requires a further
examination.
We thus focus in this work on the modelling of electrostatic instabilities by means of relativistic Vlasov simulations and we assess the impact of these instabilities as a potential pre-acceleration mechanism. To this end, we neglect magnetic field effects and consider only the one-dimensional electrostatic system, which is equivalent to the approach taken by Dieckmann et al. (2004a) but at a much larger dynamical range for the plasma phase space distribution. This description serves as simple model for the stability of ESWs by which we identify similarities and differences between the results provided by PIC and Vlasov simulations. Future work will expand the simulations to include magnetic fields, which introduces electron surfing acceleration (ESA) (Katsouleas & Dawson 1983; McClements et al. 2001) and stochastic particle orbits (Mohanty & Naik 1998), and multiple dimensions.
More specifically, we examine by relativistic electrostatic
Vlasov simulations (Arber & Vann 2002; Sircombe et al.
2005) how the BGK mode life-time and its collapse in an
unmagnetised plasma depend on
and thus on the phase speed of the
ESW. The purpose is twofold. First, we extend the comparison of results
provided by PIC and by Vlasov codes beyond the nonrelativistic
regime in Dieckmann et al. (2004b). We perform Vlasov
simulations for initial conditions that are identical to those in
Dieckmann et al. (2004a) where a PIC simulation code
(Eastwood 1991) has been used. We find a good agreement
of the results of the relativistic Vlasov code and the PIC code for
relativistic
and with the corresponding result provided by
the nonrelativistic Vlasov code (Eliasson 2002;
Dieckmann et al. 2004b). We thus bring forward further evidence
for both, an increasing destabilisation of BGK modes for increasing
nonrelativistic phase speeds of the wave and a stabilisation for
increasing relativistic phase speeds.
Secondly we want to exploit the much higher dynamical range of
Vlasov simulations to examine the wave spectrum and the particle
energy spectrum that we obtain by the considered nonlinear
interactions. We find that the collapse of the BGK modes
couples energy to three families of waves. Firstly a continuum of
electrostatic waves that move with approximately the beam speed.
These waves are connected to the turbulent electron phase space
flow. Secondly we find for high beam speeds the growth of
quasi-monochromatic modes with frequency comparable
to the Doppler-shifted bouncing frequency of the trapped electrons
in the wave potential. These waves would be sideband modes (Kruer et
al. 1969). Thirdly we find waves that do not have a clear
connection to any characteristic particle speed. We believe that
these modes are produced by parametric instabilities. These modes
can reach phase speeds well above the maximum speed the initial
trapped electron population reaches, and they grow to amplitudes at
which they can trap electrons, i.e. a BGK mode cascade to high speeds
develops. By this trapping cascade, the electrons can reach momenta
well in excess of those reported previously (Dieckmann et al.
2000, 2004a). This result
would imply that the ion beams, that are reflected by shocks that
expand at speeds comparable to SNR shocks, can accelerate electrons
to energies in excess of 100 keV, at which they can undergo Fermi
acceleration to higher energies.
![]() |
Figure 1:
As the SNR shock expands it reflects a fraction of the ISM
protons. These protons move back into the upstream region and form beam 1.
The upstream ![]() |
Open with DEXTER |
We set
,
and thus, exclude electromagnetic
instabilities, e.g. Whistler waves (Kuramitsu & Krasnoselskikh
2005), MHD waves, and electrostatic waves in magnetised
plasma, e.g. electron cyclotron waves. However, by this choice we
decouple the development of competing instabilities and we can
consider them separately. In this work we focus on ESWs and the
nonlinear BGK modes, which are important phase space structures in
the foreshocks of Solar system plasma shocks (Treumann & Terasawa 2001).
The system is, with the choice ,
suitable for modelling with an electrostatic and relativistic
Vlasov-Poisson solver.
The size of the simulation box is small compared to the distance
across which the beam parameters change.
Thus we can take periodic boundary conditions for the
simulation and spatially homogeneous Maxwellian distributions for
both beams. Both beams have the same mean speed modulus,
,
but move into opposite directions which gives a zero net current
in the simulation box. We take a higher temperature for beam 2
than for beam 1 to reflect the scattering of the beam protons as
they move through the foreshock. We show the velocity distributions
in Fig. 2.
![]() |
Figure 2:
The initial velocity distribution showing the ISM electrons
and protons, proton beam 1 and proton beam 2. The expanding SNR shock
reflects a fraction of the ISM protons which form beam 1. The upstream
![]() ![]() |
Open with DEXTER |
The choice of
is a critical limitation of the model. The initial conditions described above could not fully describe a perpendicular shock and with
one could not account for the presence of a returning proton beam. We assume that, while the field is neglected over the simulation box, the field outside the box is sufficient to produce the proton beam structure described.
Therefore, the results presented here are not directly applicable to the foreshock dynamics of high Mach number shocks. However, they provide an overview of processes that require a large dynamical
range for the plasma phase space distribution and which may, therefore,
not have been observed in previous PIC simulations. Our system is applicable to
parts of the foreshock of perpendicular shocks, where
magnetic field fluctuations (Jun & Jones 1999) cause the
magnetic field to vanish or to be beam aligned. It may also apply to
the field aligned ion beams that are observed in the foreshock region
of the Earth's bow shock (Eastwood et al. 2005) and which
may be present also at SNR shocks. Here, the second proton beam in our
initial conditions takes the role of the return current in the plasma
which is typically provided by all plasma species (Lovelace & Sudan
1971). Interactions between two BGK modes
in unmagnetized plasma affect primarily the velocity interval that is
confined by the phase speeds of the two waves (Escande 1982).
Our results, which focus on the developing high energy tails of the plasma
distribution, may thus not depend on the exact setting of the initial
return current and may be more universally applicable.
This L is short compared to the typical size of the foreshock which
justifies our spatially homogeneous initial conditions and periodic
boundary conditions. We use the amplitudes of the initial ESW and
that of the sideband modes as indicators for the wave collapse.
We define l as the index of the simulation cell,
,
and Nx,
as the number of simulation cells in x. The index m refers
to the data time step
,
where
is the time interval
between outputs rather than the simulation time step. We Fourier Transform
the spatio-temporal ESW field E(x,t) as
To analyse the nonlinear and time dependent processes developing after
the saturation of the ESW, we introduce a Window Fourier Transform. We define
the window size, in time steps, as Nt and we introduce
as the frequency defined in a Fourier time
window with a size
.
This frequency is limited by the sampling
theorem to
.
The Window
Fourier transform can be written as
In the absence of a magnetic field the one dimensional relativistic
Vlasov-Poisson system of electrons and protons is given by the Vlasov
equation for the electron distribution function
![]() |
Figure 3:
The logarithmic amplitude of the most unstable mode
![]() ![]() ![]() |
Open with DEXTER |
We set the temperatures of the four species to
K,
T2 = T3 = 10 T1 and
T4 = 100 T2. We thus obtain
.
Each species is described by a Maxwellian momentum distribution
of the form
In order to excite the linear instability, we add to the proton distribution
(species 2, 3 and 4) a density perturbation at the most unstable wavenumber
of the form
,
where a is small, typically of the order
of 1% of the background density. The minimum beam speed of
we
examine here corresponds to the slow beam in Dieckmann et al. (2004b).
The maximum beam speed
equals the maximum beam speed in
Dieckmann et al. (2004a).
Figure 3 shows the initial growth stage of the most unstable mode,
,
for a range of initial beam velocities. From these we estimate the growth rate of the
instability (
), in normalised units, to be
0.0256, 0.0264, 0.0270, 0.0246, 0.0201
and 0.0137 for beam speeds of
and 0.9c, respectively.
These compare favourably with the linear theory. Writing
as the ratio between
observed and theoretical growth rates we find;
,
,
,
,
and
.
As explained in Dieckmann et al. (2004a),
where a similar systematic reduction has been observed in PIC simulations (in this case by
15-20%), this might be connected with the fast growth rate of the instability itself since
it results in a considerable spread in frequency for the unstable wave. This makes the
treatment of the instability in terms of single frequencies (
)
inaccurate. In truth,
the growth rate should be lower since the energy of the unstable wave is spread over damped
frequencies.
The ESWs saturate by the trapping of electrons and the formation of BGK
modes. This is shown in Fig. 4, for the case of
,
and Fig. 5 for
.
Here we see the appearance of
phase-space holes characteristic of particle trapping. While the saturation
mechanism is the same in both cases, for the high velocity beam we observe
the development of two counter-propagating BGK modes. This is because at
lower beam speeds the ESW instability driven by the cooler beam (at
)
dominates whereas for higher, relativistic, beam speeds forwards and
backwards propagating ESWs grow in unison. At lower beam speeds the increased
temperature of beam 2 (species 4) is more significant, since the thermal
velocity of the beam is a larger proportion of the beam speed than is the
case at
.
Thus, the growth rate for the hot beam is sufficiently
reduced for the system to be dominated by the growth of the cooler proton
beam (species 3). We do observe the growth of a counter propagating ESW which
begins to trap electrons after t=150. However, the final momentum
distribution is clearly dominated by electrons accelerated by the ESW with
positive phase velocity.
At
,
the thermal spread of both proton beams is negligible in
comparison to the beam velocities and we observe the growth and saturation
of ESWs associated with both beams.
![]() |
Figure 4:
Contour plot of
![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
![]() |
Figure 5:
Contour plot of
![]() ![]() ![]() ![]() |
Open with DEXTER |
Trapping of electrons in electrostatic waves produces BGK modes which eventually collapse via the sideband instability. The sideband instability is due to the nonlinear
oscillations of the electrons in the potential of the ESW. For electrons
close to the bottom of the wave potential, their oscillation is that of
a harmonic oscillator. The monochromatic bouncing frequency is Doppler
shifted, due to the phase speed of the ESW. These sidebands can couple
the electron energy to secondary high-frequency ESWs, which must have
a wave number
(Krasovsky 1994).
The sideband instability is a limiting factor for the lifetime of the
ESW which has implications in the presence of an external
magnetic field in particular. To demonstrate this we show in Fig. 6 the
amplitude of the ESW driven by beam 1 at
for beam speeds ranging
from 0.15c to 0.9c.
We find that the ESWs moving at nonrelativistic phase speeds saturate
smoothly, which is in line with the wave saturation of the ESW in the
Vlasov simulation in Dieckmann et al. (2004b). Their lifetime
is significant and, in the presence of a weak external magnetic field
orthogonal to the wave vector
,
the trapped electrons
would undergo substantial ESA (Eliasson et al.
2005). The ESA is proportional to
and
the comparatively low
would limit the maximum energy the
electrons can reach.
![]() |
Figure 6:
The time evolution of the ESW amplitudes with
![]() ![]() ![]() ![]() |
Open with DEXTER |
The ESW driven by the beam with
grows to a larger amplitude
and then collapses abruptly, which confirms the finding in Dieckmann et al.
(2004b) that the BGK modes become more unstable the larger
the ratio between beam speed to electron thermal speed becomes. The Lorentz
force
excerted by a
would
be significantly higher than for the case
,
however the short
lifetime of the wave would here prevent electrons from reaching highly relativistic
speeds. By increasing the beam speed to
we obtain a stabilisation
of the ESW, in line with the results in Dieckmann et al. (2004a).
Here the Lorentz force is strong and the lifetime of the saturated ESW is long constituting a formidable electron accelerator, provided
the field evolution is not strongly influenced by
.
We now summarise the principal results from each beam speed.
Note that in particular the ESWs driven by the mildly relativistic proton
beams show oscillations after their initial saturation. To identify the
origin of these fluctuations we apply a Window Fourier Transform to the
amplitudes of the ESWs with the wavenumbers
and
.
In Fig. 7 we find a strongly asymmetric ESW growth
for
.
Here the thermal spread of the individual plasma
species is not small compared to
.
Therefore the ESW driven by the cooler
beam 1 grows and saturates first. It rapidly collapses and this
collapse inhibits a further growth of the ESW driven by beam 2.
The initial monochromatic ESW collapses into a broad wave continuum
centred around
.
We find the equivalent broad
wave continuum for
centred at
.
These waves propagate at phase speeds comparable to
and represent
the structures remaining of the initial BGK mode. The ESW spectrum at
further shows two wave bands that are separated by a
frequency of
from the main peak. These two wave
bands could be pumped by a beat between the turbulent structure
centred at
,
,
the turbulent
structure centred at
and
and the
Langmuir wave with
.
Evidence
for this is the correlation between both turbulent structures
and the wave bands at
.
At this time most wave power at
is absorbed at
.
At the
same time wave power at
is absorbed at
while the power in the wave bands with
and
grows at
.
The faster of these
two has a phase speed of
.
This beat wave is thus considerably faster than the initial ESW. By
its large amplitude it could trap electrons. This is confirmed by
Fig. 8 where we find a BGK mode in the electron distribution
centred around a momentum
which corresponds
to a speed 0.46c. The fastest electrons of this BGK mode reach
or a speed of
.
![]() |
Figure 7:
The ESW spectrum for a beam speed of
![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
![]() |
Figure 8:
Contour plot of
![]() ![]() ![]() ![]() |
Open with DEXTER |
![]() |
Figure 9:
The ESW spectrum for a beam speed of
![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
The large amplitudes of both bands suggests that they might also be
trapping electrons. This is confirmed by Fig. 10 where we
show the electron momentum distribution at t=2000. We find BGK modes
centred at
or a speed of 0.6c. The
BGK modes extend up to a peak momentum of
or
a speed of 0.87c respectively equivalent to
.
![]() |
Figure 10:
Contour plot of
![]() ![]() ![]() |
Open with DEXTER |
![]() |
Figure 11:
The ESW spectrum for a beam speed of
![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
![]() |
Figure 12:
The ESW spectrum for a beam speed of
![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
The phase speeds of these ESWs is higher than the peak speed the
electrons reach in the inital BGK mode as shown in Fig. 13.
We can thus not explain its growth by a streaming instability between
the trapped electrons and, for example, the untrapped electrons. We can
obtain phase speeds of the ESW bands that are higher than the speed of
the trapped electron beam, however, by applying the relativistic Doppler
shift to the electron bouncing frequency in the ESW wave potential.
The rest frame of the ESWs moves with the speed
.
The
frequency of the ESWs in the observer frame is, according to Fig. 12,
.
With the relativistic
Doppler equation we would obtain a bouncing frequency of the electrons
in the ESW frame of reference of
.
We use the nonrelativistic estimate of the electron bouncing frequency
in a parabolic electrostatic potential
and the corresponding width of the trapped electron island
to eliminate the electric field E. We
obtain the relation
.
Since for
our cold plasma species the velocity width of the island of trapped
electrons must be comparable to
to trap the bulk electrons
we get an estimate for
which is close to
.
We may thus indeed interpret the two
sidebands observed in Fig. 12 as the Doppler shifted
bouncing frequency of the electrons.
![]() |
Figure 13:
Contour plot of
![]() ![]() ![]() |
Open with DEXTER |
These sideband modes driven by the beams of trapped electrons have a phase
speed that is just below c and a large amplitude. Since their phase
speed is comparable to the fastest speed the electrons reach in the initial
BGK mode, they should be capable of trapping some of these electrons.
This is confirmed by Fig. 14 where we find BGK
modes centred at the momentum
which accelerate
electrons up to the peak momentum
or a speed of 0.99c.
![]() |
Figure 14:
The electron momentum distribution for
![]() ![]() |
Open with DEXTER |
![]() |
Figure 15:
The ESW spectrum for a beam speed of
![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
We observe the same growth of sideband modes for the fastest beam speed
of
in Fig. 16. The sideband modes at
have
a frequency modulus
and phase
speeds just above c. As for
the sideband
mode at
has a frequency of
and thus
a superluminal phase speed.
![]() |
Figure 16:
The ESW spectrum for a beam speed of
![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
For both beam speeds
and
the sideband modes
appear to have a superluminal phase speed and they can therefore not
trap electrons. No secondary BGK modes should develop for these
beam speeds.
This is confirmed by Fig. 17 where we show the phase space
distribution of the electrons for
at the simulation's end.
The phase space distribution at high momenta shows no evidence of a
BGK mode despite the strong sideband modes in Fig. 16.
![]() |
Figure 17:
Contour plot of
![]() ![]() |
Open with DEXTER |
In contrast to the PIC simulations in Dieckmann et al. (2004a)
the Vlasov simulation code shows the growth of sideband modes and what
appears to be waves resulting from a parametric instability. These
secondary waves grow to a large amplitude at which they can nonlinearly
interact with the electrons. For the beam speeds up to
the waves generated by the parametric interaction have been strongest.
Here the turbulent wave fields interact with the waves with
to produce a wave with a higher frequency. In contrast to plasma beat
wave accelerators, which have recently been reviewed by Bingham et al. (2004), for which two high-frequency electromagnetic
waves beat
to yield a low frequency ESW that can accelerate the electrons, our parametric
coupling couples low frequency ESWs to an ESW with a higher frequency.
Its larger phase speed can accelerate the trapped electrons to higher peak
speeds. We may thus call it the "inverse plasma beat wave accelerator''.
For a beam speed of
the turbulent wave fields do not
noticably interact with
.
Instead a sideband unstable mode
with
develops, i.e. at the largest allowed wave number for
nonrelativistic BGK modes (Krasovsky 1994). This mode has
for
a phase speed below c and the electron phase space
distribution shows a fast BGK mode driven by it. For even higher
the probably superluminal phase speed of the sideband modes
suppresses their interaction with the electrons.
The complex spectrum of secondary waves and their nonlinear interactions
with the electrons, which has not been observed clearly by Dieckmann
et al. (2004a), suggests a stronger dependence of the electron
heating on
than for the simulations by Dieckmann et al.
(2004a). There the momentum distributions could be matched
if one were to scale the momentum axis to
.
We thus
do the same here and we integrate the electron phase space distribution
over x. The result is shown in Fig. 18.
We find that the final momentum distributions in the chosen normalization
of the p-axis agree well up to
and for densities larger
than 10-2. Here the Vlasov code results are similar to the equivalent
PIC simulations in Dieckmann et al. (2004a). We find, however,
significant differences at lower densities, which are not well-represented
by the PIC code, and at the corresponding higher momenta. The parametric
instabilites and the sideband instabilities
have further accelerated electrons beyond the peak momenta measured in
Dieckmann et al. (2004a). This is particularly pronounced
for the beam speed
in which we find an electron density
plateau extending to a value of +10 for the normalised momentum, i.e.
twice as high as the corresponding value at negative momenta. For these
beam speeds the secondary waves have been most efficient as electron
accelerators. A further increase of
beyond 0.6c yields broadening
momentum distributions. The peak momentum in units of
is, for
positive momenta comparable for
and for
.
The peak momentum for
is about twice as high as for
.
The peak relativistic kinetic energies
in eV the electrons reach are
eV,
eV,
eV,
eV,
eV
and
eV. Note that all these peak
electron energies are comparable or above the threshold energy of
105 eV: the injection energy for Fermi acceleration at perpendicular shocks, given by Treumann & Terasawa (2001).
The observation of the emission of highly energetic cosmic ray particles by SNRs suggests the acceleration of particles from the thermal pool of the ISM plasma to highly relativistic energies by such objects. The acceleration site is apparently linked to the shock that develops as the supernova blast shell encounters the ambient plasma (Lazendic 2004). Such shocks are believed to accelerate electrons and ions to highly relativistic energies by means of Fermi acceleration (Fermi 1949, 1954). The Fermi acceleration of electrons is most efficient if the shock is quasi-perpendicular (Galeev 1984). For such shocks, however, Fermi acceleration works only if we find a relativistically hot electron population prior to the shock encounter, since slow electrons could not repeatedly cross the shock and pick up energy. Since the plasma, into which the SNR shock expands, has a thermal speed comparable to that of the ISM or the stellar wind of the progenitor star with temperatures of up to a few eV, if we take the solar wind as reference, no mildly relativistic electrons may exist. A mechanism is thus required that accelerates electrons up to speeds at which their Larmor radius exceeds the shock thickness. As Galeev (1984) proposed, electrons could be pre-accelerated from the initial thermal pool to mildly relativistic energies by their interaction with strong ESWs in the foreshock region which, in turn, are driven by shock-reflected beams of ions.
![]() |
Figure 18: The electron momentum distributions at the simulation's end times. |
Open with DEXTER |
We have examined in this work the growth, saturation and collapse of ESWs in a system dominated by the presence of two counter-propagating proton beams. The currents of both beams cancel, allowing the introduction of periodic boundary conditions. These initial conditions can be motivated as follows: the upstream protons, that have initially been reflected by the shock, are rotated by the global magnetic field oriented perpendicularly to the shock normal and return as a second counter propagating proton beam, with a slightly increased temperature. The system modelled in this work represents a small region ahead of a SNR shock. Here the magnetic field has been neglected in order to focus on the ESWs and nonlinear BGK modes. While this does not describe a complete model for the foreshock dynamics of high Mach number shocks, it is applicable to parts of the foreshock of perpendicular shocks where local variations magnetic field cause it to vanish, or become beam-aligned. These variations may, for example, be due to the presence of turbulent magnetic field structures in the foreshock of SNR shocks (Jun & Jones 1999; Lazendic et al. 2004).
The system is unstable to the relativistic Buneman instability (Thode & Sudan 1973) which saturates via the trapping of electrons to form BGK modes (Rosenzweig 1988). These trapped particle distributions are themselves unstable to the sideband instability and collapse after a period of stability.
Since our model assumes that the particle trajectories are not
affected by the local magnetic field and because we do not consider
here electromagnetic waves or waves in magnetised plasma, we are able to utilise a relativistic,
electrostatic Vlasov code. Previous work, for example (Thode & Sudan 1973; Dieckmann et al.
2000, 2004b; Shimada & Hoshino 2004) has made
extensive use of PIC codes for this problem and in particular Dieckmann et al. (2004b)
have compared Vlasov and PIC codes in certain conditions. We benefit from the Eulerian Vlasov code's
ability to resolve electron and ion phase space accurately, irrespective of the local particle density.
This allows us to identify a secondary acceleration mechanism which may not be immediately apparent
otherwise.
Overall the results of this work are in agreement with previous
studies (Dieckmann et al. 2000; Dieckmann et al. 2004b) showing the lifetimes
of the BGK modes to be dependent on the initial beam velocity. As
is increased, we observe a
reduced ESW stability up to
but with a significantly increased stability at the highest
beam speed.
At low beam velocities (
)
we observe long wavelength, high phase-velocity
modes. These are able to trap electrons, producing a population with kinetic energies above the injection
energy for Fermi acceleration, even at non-relativistic beam, and thus shock, velocities. We believe these secondary
(that is to say, not associated with the initial ESW saturation) trapped distributions to be the result
of trapping in ESWs produced by parametric coupling between low frequency oscillations and plasma waves
at
.
Above
the ESWs produced by this coupling have super-luminal phase velocities
and are unable to trap electrons. Hence we do not observe such BGK modes in the case of
or
0.9c. Our simulation box, at
in length, can only accommodate one wavemode with wavenumber
below that of the most unstable mode and this may have an influence on the appearance of this secondary
acceleration.
Future work has to examine how these parametric instabilities depend on a
magnetic field and on the introduction of a second spatial dimension.
We will need to consider significantly larger simulation boxes, capable of
resolving a broader spectrum below .
It may be the case that the availability of modes with
will result in the partition of ESW energy across a greater region of the spectrum, perhaps inhibiting
the trapping of electrons at high velocity. However, it may be the case that our observed coupling and
resultant electron trapping is the first step of a cascade, capable of accelerating electrons to high
energies for relatively modest shock velocities. This is since, even for
as low as 0.15c or
corresponding shock speeds of
which can be reached by the fastest SNR main shocks
(Kulkarni et al. 1998), electrons can reach energies of 105 eV. According to
Treumann & Terasawa (2001), this may increase the electron gyroradius beyond the shock thickness
by which they can repeatedly cross the shock front. These repeated shock crossings allow the electrons
to undergo Fermi acceleration to highly relativistic speeds. Even higher energies could be achieved if
we were to get shock precursors that outran the main shock as has been observed for the supernova SN1998bw
(Kulkarni et al. 1998). The numerical simulations in this work thus present strong
evidence for the ability of ESWs and processes driven by electrostatic turbulence to accelerate electrons beyond the threshold
energy at which they can undergo Fermi acceleration as it has previously been
proposed, for example
by Galeev (1984).
Acknowledgements
This work was supported in part by: the European Commission through the Grant No. HPRN-CT-2001-00314; the Engineering and Physical Sciences Research Council (EPSRC); the German Research Foundation (DFG); and the United Kingdom Atomic Energy Authority (UKAEA). The authors thank the Centre for Scientific Computing (CSC) at the University of Warwick, with support from Science Research Investment Fund grant (grant code TBA), for the provision of computer time. N J Sircombe would like to thank Padma Shukla and the rest of the Institut für Theoretische Physik IV at the Ruhr-Universität Bochum for their kind hospitality during his stay.