Download Theory of relativistic electron holes in hot plasmas

Physics Letters A 340 (2005) 237–242
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Theory of relativistic electron holes in hot plasmas
B. Eliasson, P.K. Shukla ∗
Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 31 January 2005; received in revised form 7 February 2005; accepted 7 February 2005
Available online 8 April 2005
Communicated by V.M. Agranovich
Abstract
We present a theory for finite-amplitude relativistic electron holes, which are localized Bernstein–Greene–Kruskal (BGK)
solutions characterized by a trapped population of electrons moving with the electron hole. We find that the relativistic effects
can drastically modify the electron hole, which becomes wider and is associated with a larger electrostatic potential, accelerating
the electrons to highly relativistic energies. The theory has relevance for understanding the properties of strong localized electric
fields in high-energy laser-plasma experiments and in supernovae remnants, where relativistic electron holes are a natural
product of streaming instabilities.
 2005 Elsevier B.V. All rights reserved.
PACS: 52.27.Ny; 52.25.Dg; 52.35.Sb; 52.65.Ff
In his pioneering papers, Schamel [1–5] developed
theories for electron and ion holes in a non-relativistic
plasma. The electron (ion) hole is characterized by
electron (ion) density depression in association with
positive (negative) potential distribution. In order for
the electron and ion holes to exist, the electron and ion
distribution functions have to be non-isothermal. The
modification of the distribution functions arises due to
the trapping of particles in the localized electrostatic
potential. In fact, these theoretical models of the electron and ion holes, which are exact solutions of the
* Corresponding author.
E-mail address: [email protected] (P.K. Shukla).
0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2005.02.047
Poisson and collisionless non-relativistic Vlasov equations, are governed by only two parameters, namely,
the speed of the holes and the temperature of the
trapped particle population. Far away from the holes,
the particles follow a non-relativistic Maxwellian distribution.
Schamel’s theories of electron and ion holes have
been spectacularly verified in small scale laboratory
experiments [6–8] as well as in numerical simulations [9,10] and in space [11–13]. Specifically, recent
observations from Wind and Cluster missions reveal
signatures of large amplitude electron holes due to
trapped electrons in the Earth’s magnetotail [12,13]
during magnetic reconnection. In laboratory experiments, the phase space electron and ion vortices are
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B. Eliasson, P.K. Shukla / Physics Letters A 340 (2005) 237–242
created on account of two-stream instabilities, and the
resulting electron and ion holes are fully non-linear kinetic structures. In non-relativistic Vlasov simulations
[9,10], exact hole solutions of Schamel have been used
as initial conditions for studying the complex interactions between electron/ion holes and their interactions
with the background plasma. Interesting dynamical
behaviors of interacting electron holes include the generation of energetic electrons and Langmuir waves as
well as the emergence of new features to the electron
holes in the presence of the ion dynamics. A theoretical investigation [14] reveals the trapping and interactions between large-amplitude Langmuir waves and
ion holes. The present status of the electron and ion
hole physics as well as pertinent simulations and observations are contained in Ref. [15].
In this Letter, we have, for the first time, developed a theory for the steady state electron holes in a
relativistic plasma. Relativistic effects play a very important role in high-energy laser-plasma experiments
[16,17], in plasma based electron and photon accelerators [18,19], in supernova remnants and gamma ray
bursts [20], where electrons can be accelerated to relativistic energies by strong electrostatic fields. We find
that the dynamics of the electron holes are dramatically changed by the relativistic mass increase of the
electrons. Our relativistic simulation studies show that
strongly relativistic electron holes are long-lived, and
that they survive head-on collisions. We expect that intense electron hole potentials can accelerate electrons
to extremely high energies.
The dimensionless relativistic Vlasov–Poisson system, where ions are assumed to form a neutralizing
background, is
p
∂φ ∂f
∂f
∂f
+
+
= 0,
2
2
∂t
∂x
∂x ∂p
1+α p
(1)
and
∂ 2φ
=
∂x 2
∞
f dp − 1,
(2)
−∞
where α = VTe /c, VTe = (Te /me )1/2 is the electron
thermal speed, Te is the electron temperature, me is the
rest mass of the electron, the distribution function f
has been normalized by n0 /mVTe , the time t by ωp−1 ,
the space x by rD , the momentum p by me VTe , and
the potential φ by Te /e. Here c is the speed of light
in vacuum, e is the magnitude of the electron charge,
the plasma frequency is ωp = (4πn0 e2 /me )1/2 , where
n0 is the background electron number density and
rD = VTe /ωp is the electron Debye length. In previous investigations of non-relativistic electron holes
[1,2,5], the system has been Galilei-transformed into
a frame moving with the electron hole, where stationary solutions of the Vlasov equation have been found.
The corresponding method in a relativistic description would be to Lorentz transform the system to a
frame moving with the electron hole, and then to find
stationary solutions in that frame. Here, however, we
shall follow a different path: we (the observer) will
remain in the frame of the bulk plasma, while the electron hole is moving with a constant speed v0 . Accordingly, we shall seek solutions of the form f (p, ξ ) and
φ(ξ ), where ξ = x − v0 t. With this ansatz, the Vlasov–
Poisson system (1)–(2) takes the form
∂f
dφ ∂f
p
+
= 0,
−v0 + (3)
dξ ∂p
1 + α 2 p 2 ∂ξ
and
d 2φ
=
dξ 2
∞
f dp − 1.
(4)
−∞
Integrating Eq. (3) along its trajectories, one finds
the general solution of the form f = f0 (E), where f0
is some function of one variable and
1
1
2
2
− φ,
1+α p −
E = −v0 p + 2
(5)
γ0
α
is a conserved quantity (energy) along the particle tra-
jectories. Here we have denoted γ0 = 1/ 1 − α02 v02 .
At the electron hole, a population of electrons are
trapped in a localized positive potential φ, which goes
to zero far away from the electron hole. The energy E
has been shifted so that the trapped electrons have a
negative energy, while the untrapped (free) electrons
have a positive energy.
A condition for the untrapped electrons is that far
away from the electron hole, the distribution should
connect smoothly to a Jüttner–Synge distribution [22],
which restricted to one momentum dimension takes
the form
1
f˜0 (p) = A exp − 2 1 + α 2 p 2 − 1 ,
(6)
α
B. Eliasson, P.K. Shukla / Physics Letters A 340 (2005) 237–242
where the normalization constant is
∞
−1
1
A=
exp − 2 1 + α 2 p 2 − 1 dp
α
viz.
γ0 − 1
− βE ,
ft = A exp −
α2
−∞
α exp(−α −2 )
,
=
2K1 (α −2 )
(7)
and K1 is a modified Bessel function of second kind.
The weakly relativistic limit √corresponds to α 1
so that A ≈ (1 + 3α 2 /8)−1 / 2π [23], and Eq. (6)
converges to a non-relativistic Maxwellian distribution
when a → 0. In order to impose the above mentioned
condition for the free electrons, we will use the solution
ff = f0 (E) = f˜0 p̃(E) ,
(8)
where p̃(E) is a function of the energy such that
p̃(E) → p when φ → 0. Such a function can be found
with the help of Eq. (5) by setting
1
1
2
2
−v0 p̃(E) + 2
(9)
= E.
1 + α p̃ (E) −
γ0
α
Solving for p̃(E), we have
1
p̃± (E) = γ02 v0 α 2 E +
γ0
γ02 1 2
1
2
α E+
±
− 2,
α
γ0
γ0
(10)
where p̃+ (E) and p̃− (E) correspond to a modified
momentum for free electrons on each side of the
trapped electron population in momentum space. Using p̃(E) = p̃+ (E) and p̃(E) = p̃− (E) in Eq. (8),
we obtain the distribution function for the free electrons. In the limit of vanishing energy, E = 0, we have
p̃± (0) = γ0 v0 , and the value of the distribution function is
ff |E =0 = f0 (0)
γ0 − 1
˜
,
= f0 (γ0 v0 ) = A exp −
α2
239
(11)
which should be matched with the distribution function for the trapped electrons with E = 0 in order
to obtain a continuous distribution function. For the
trapped electrons, we choose a relativistic Maxwell–
Boltzmann distribution with a negative “temperature”,
(12)
where β is a trapping parameter, leading to a vortex
distribution for β < 0, a flat top distribution for β = 0
and a shifted Jüttner–Synge distribution for β = 1.
Clearly, the separatrix between the free and trapped
electron distributions is found where E = 0 in Eq. (5).
Solving for p in Eq. (5) with E = 0, we have
1
p± = γ02 v0 α 2 φ +
γ0
γ2
1 2
1
± 0 α2 φ +
(13)
− 2,
α
γ0
γ0
where p− (φ) and p+ (φ) constitute the limits between
the trapped and free electron distributions in momentum space. Using these limits and combining Eqs. (8),
(10) and (12), we can now write the full electron distribution function as


2 (E) − 1) ,

A exp − α12 ( 1 + α 2 p̃+




p > p+ ,




A exp − γ0α−1
2 − βE ,
f=
(14)


 p− p p+ ,



2 (E) − 1) ,

A exp − α12 ( 1 + α 2 p̃−



p < p− ,
where E is given by Eq. (5). Integrating the distribution function over momentum space we obtain the total
electron density as a function of φ, which is calculated
self-consistently by means of Poisson’s equation (4).
In order to proceed further and to explore the impact of the relativistic effects on the electron holes for
different sets of parameters, we have solved Poisson’s
equation (4) numerically, where the integration of the
trapped and free electron populations have been done
with a sum representation of the integrals, and Eq. (4)
has been solved as a non-linear boundary value problem where the potential has been set to zero far away
from the electron hole, and the resulting non-linear
system of equations have been solved iteratively with
a Newton-like method. An alternative would be to use
the so-called potential method [1]. In Fig. 1 we display the electron hole speed v0 as a function of the
amplitude ψ of the potential for different values of the
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B. Eliasson, P.K. Shukla / Physics Letters A 340 (2005) 237–242
Fig. 1. The velocity of the electron hole as a function of the amplitude ψ of the electron hole potential φ, for a constant trapping
parameter β = −0.5 and for different values of α.
parameter α. We see that for larger values of α, the amplitude of the electron holes drastically increases. Note
that the electron hole potential has been normalized
by Te /e, so the curves in Fig. 1 show pure relativistic effects due to the relativistic mass increase of the
electrons. In the non-relativistic limit (α = 0), we recover the previous results of Bujarbarua and Schamel;
see Fig. 1 in Ref. [2]. The profiles of the electrostatic
potential and the electron density are shown in Fig. 2
for a few sets of parameters. To convert the potential to Volts, the values of φ should be multiplied by
the factor 5.1 × 105 α 2 ; accordingly, the amplitude of
the potential for the case (a) (ψ ≈ 33, α = 0.4) is
≈ 2.4 × 106 V, while for the case (b) (ψ ≈ 9, α = 0.2)
it is ≈ 0.18 × 106 V. The profiles of the electron holes
in phase space is shown in Fig. 3 for the same sets of
parameters as in Fig. 2. We have solved the Vlasov–
Poisson system (1)–(2) and have investigated interactions between two large-amplitude electron holes. The
results are shown in Fig. 4. The left hole has initially
the speed v0 = 0.7 and the right hole has the negative speed v0 = −0.2, while both holes have β = −0.5
and α = 0.4. During head-on collisions, the two holes
merge and form a new electron hole which survives
through the rest of the simulation time. As a comparison, shown in Fig. 5, we changed the relativistic factor
to α = 0.01, while keeping the same trapping parameter β = −0.5 and the initial speeds of the holes. In
this case, the electron holes are considerably smaller,
Fig. 2. The potential (left panel) and electron density (right panel)
of electron holes with (a): β = −0.5, α = 0.4 and v0 = 0.2;
(b): β = −0.5, α = 0.2 and v0 = 0.2; (c): β = −0.5, α = 0.4 and
v0 = 0.7, and (d): β = −0.7, α = 0.4 and v0 = 0.2.
Fig. 3. Phase space plot for an electron hole with (a): β = −0.5,
α = 0.4 and v0 = 0.2; (b): β = −0.5, α = 0.2 and v0 = 0.2;
(c): β = −0.5, α = 0.4 and v0 = 0.7, and (d): β = −0.7, α = 0.4
and v0 = 0.2, corresponding to the cases (a)–(d) in Fig. 2.
and do not merge as they did in the strongly relativistic case. The numerical method used in our simulations was a pseudo-spectral method to approximate
the x-derivatives, a fourth-order compact Padé scheme
to approximate the p derivative, and the fourth-order
Runge–Kutta method for the time stepping. The integral over p space in Eq. (2) was done with a simple
sum representation of the integral.
In conclusion, we have presented a theory for relativistic electron holes, taking into account the relativistic mass increase in a hot plasma. The relativistic
B. Eliasson, P.K. Shukla / Physics Letters A 340 (2005) 237–242
Fig. 4. Interactions between strongly relativistic electron holes. Initially, the parameters are α = 0.4 and β = −0.5 for both holes, and
the left and right holes have the speed v0 = 0.7 and v0 = −0.2, respectively, corresponding to the cases (a) and (c) in Figs. 2 and 3.
Time intervals are: t = 0 (upper left panel), t = 50 (upper right
panel), t = 100 (middle left panel), t = 150 (middle right panel),
t = 200 (lower left panel) and t = 250 (lower right panel).
241
plasmas. We find that relativistic electron holes interact in a complex fashion, and survive head on collisions either by merging and forming new electron
holes or by keeping their identities. Intense electrostatic potential associated with relativistic electron holes
can accelerate charged particles to extremely high energies that are relevant for laser plasma interactions
and astrophysical conditions (e.g., gamma-ray bursts
[21]). As an illustration, we take laser-plasma parameters (n0 ∼ 1020 cm−3 and Te in the hundreds of keV
range) relevant for supernova II, and find that relativistic solitary electrostatic pulses with potential humps
would have electric potential of the order of a megavolt, a typical width of 100 microns, and the pulse duration of the order of one tenth of a picosecond. These
numbers suggest the possibility of observing relativistic electron holes in forthcoming intense laser-plasma
experiments.
Acknowledgements
This work was partially supported by the Deutsche
Forschungsgemeinschaft through the Sonderforschungsbereich 591, as well as by the European Commission through contract No. HPRN-CT-2001-00314
for carrying out the task of the RTN “Turbulent
Boundary Layers in Geospace Plasmas”.
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Fig. 5. Interactions between weakly relativistic electron holes. Initially, the parameters are α = 0.01 and β = −0.5 for both holes,
and the left and right holes have the speed v0 = 0.7 and v0 = −0.2,
respectively. Times: t = 0 (upper left panel), t = 50 (upper right
panel), t = 100 (middle left panel), t = 150 (middle right panel),
t = 200 (lower left panel) and t = 250 (lower right panel).
effect on the electron hole is that the size of the electron hole as well as the amplitude of the electron hole
potential increases with increasing relativistic temperature. We have also performed simulation studies of
interactions between relativistic electron holes in hot
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