Nonlinear Processes in Geophysics, 12, 269–289, 2005
SRef-ID: 1607-7946/npg/2005-12-269
European Geosciences Union
© 2005 Author(s). This work is licensed
under a Creative Commons License.
Nonlinear Processes
in Geophysics
The dynamics of electron and ion holes in a collisionless plasma
B. Eliasson and P. K. Shukla
Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum,
Germany
Received: 2 November 2004 – Revised: 26 January 2005 – Accepted: 27 January 2005 – Published: 11 February 2005
Part of Special Issue “Nonlinear plasma waves-solitons, periodic waves and oscillations”
Abstract. We present a review of recent analytical and numerical studies of the dynamics of electron and ion holes in
a collisionless plasma. The new results are based on the
class of analytic solutions which were found by Schamel
more than three decades ago, and which here work as initial
conditions to numerical simulations of the dynamics of ion
and electron holes and their interaction with radiation and
the background plasma. Our analytic and numerical studies reveal that ion holes in an electron-ion plasma can trap
Langmuir waves, due the local electron density depletion associated with the negative ion hole potential. Since the scalelength of the ion holes are on a relatively small Debye scale,
the trapped Langmuir waves are Landau damped. We also
find that colliding ion holes accelerate electron streams by
the negative ion hole potentials, and that these streams of
electrons excite Langmuir waves due to a streaming instability. In our Vlasov simulation of two colliding ion holes, the
holes survive the collision and after the collision, the electron distribution becomes flat-topped between the two ion
holes due to the ion hole potentials which work as potential
barriers for low-energy electrons. Our study of the dynamics between electron holes and the ion background reveals
that standing electron holes can be accelerated by the selfcreated ion cavity owing to the positive electron hole potential. Vlasov simulations show that electron holes are repelled
by ion density minima and attracted by ion density maxima.
We also present an extension of Schamel’s theory to relativistically hot plasmas, where the relativistic mass increase of the
accelerated electrons have a dramatic effect on the electron
hole, with an increase in the electron hole potential and in
the width of the electron hole. A study of the interaction between electromagnetic waves with relativistic electron holes
shows that electromagnetic waves can be both linearly and
nonlinearly trapped in the electron hole, which widens further due to the relativistic mass increase and ponderomotive
force in the oscillating electromagnetic field. The results of
Correspondence to: B. Eliasson
(bengt@tp4.rub.de)
our simulations could be helpful to understand the nonlinear
dynamics of electron and ion holes in space and laboratory
plasmas.
1
Introduction
About a quarter century ago, Schamel (1971, 1972, 1979,
1986; Bujarbarua and Schamel, 1981) presented a theory for
ion and electron holes, where a vortex distribution is assigned
for the trapped particles, and where the integration over the
trapped and untrapped particle species in velocity space gives
the particle number density as a function of the electrostatic
potential. The potential is then calculated self-consistently
from Poisson’s equation. This model has been used in theoretical analyses of the existence criteria for the ion and electron holes (Bujarbarua and Schamel, 1981), and in the analysis of stability of phase space holes (Schamel, 1982, 1986).
Numerical and theoretical studies of the interaction between electron and ion holes have been performed by several authors. Newman et al. (2001) in their numerical
experiments studied the dynamics and instability of twodimensional phase-space tubes, and Daldorff et al. (2001)
investigated the formation and dynamics of ion holes in thee
dimensions. Krasovsky et al. (1999) showed theoretically
and by computer simulations that that electron holes perform
inelastic collisions, and Vetoulis (2001) studied the radiation generation due to bounce resonances in electron holes.
Guio et al. (2003, 2004) have studied numerically the dynamics of phase space vortices in a collisionless plasma as
well as the generation of phase space structures by an obstacle in a streaming plasma. Saeki and Genma (1998) studied
the disruption of electron holes in an electron-ion plasma.
The asymptotic nonlinear saturation of electrostatic waves
has been treated both theoretically (Medvedev et al., 1998;
Lancelotti and Dorning, 1998) and numerically (Manfredi,
1997). It has also been pointed out that plasma waves can
be undamped due to particle trapping effects in waves with
arbitrarily small amplitudes (Holloway and Dorning, 1991;
270
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
Schamel, 2000). Theoretical investigations of trapped particle effects in magnetized plasmas shows that trapped ions
influence strongly ion cyclotron waves (Abbasi et al., 1999).
In laboratory experiments, the formation and dynamics of
solitary electron holes (Saeki et al., 1979; Petraconi and Macel, 2003) and ion holes (Pécseli et al, 1981, 1984, 1987;
Nakamura et al., 1999) as well as accelerated ion holes
(Franck et al., 2001) have been observed. Observations
of broad electrostatic noise (BEN) by the GEOTAIL (Matsumoto et al., 1994) and FAST spacecrafts (Ergun et al.,
1998a, b, 2001) in the auroral acceleration regions have
revealed that BEN is connected to solitary electron BGK
modes/electron holes. Recent observations by the WIND
satellite in the Earth’s bow shock also reveal localized structures with bipolar electric fields typical for the electron BGK
modes/holes (Bale, 1998, 2002; Hoshino, 2003). The geomagnetic field-aligned bipolar electric field pulses associated
with electron phase space holes have also been observed by
the Polar and Cluster spacecrafts in the magnetosheath and
at the Earth’s magnetopause (Cattel et al., 2002; Dombeck
et al., 2001; Pickett et al., 2002, 2004; Drake et al., 2003),
while signatures of ion holes have been observed by the Fast
Auroral Snapshot (FAST) spacecraft (McFadden et al., 2003)
and by the Viking satellite (Boström, et al., 1988).
The interaction between low-frequency ion-acoustic
waves with high-frequency Langmuir turbulence and electromagnetic waves was described more than three decades
ago by Hasegawa (1970), Karpman (1971, 1975a, b) and Zakharov (1972), in which high-frequency photons and plasmons interact nonlinearly with low-frequency ion-acoustic
waves via the ponderomotive force arising due to the spatial gradient of the high-frequency wave intensity. This nonlinear interaction is typically described by the two-fluid and
Poisson-Maxwell equations, and the governing equations admit the localization of photon and plasmon wavepackets,
leading to the formation of envelope light and Langmuir
wave solitons (Rudakov, 1972; Varma and Rao, 1980; Rao
and Varma, 1982; Schamel, 1977). The latter are composed
of electron/ion density depression which traps photon and
Langmuir wave envelops. Schamel and Maslov (1999) studied theoretically the contraction of Langmuir waves trapped
in small-amplitude electron holes. Califano and Lontano
(1998) and Wang et al. (1997) studied numerically by Vlasov
simulations the dynamics of finite-amplitude high-frequency
Langmuir waves in an one-dimensional electron-ion plasma,
where wave collapse and heating of the electrons have been
observed. In two and three dimensions, one encounters photon self-focusing and Langmuir wave collapse (Zakharov,
1972; Shapiro and Shevchenko 1984; Goldman, 1984). The
formation of cavitons has been observed in the ionosphere
(Wong, 1974) as well as in several laboratory experiments
(Ikezi, 1976; Intrator, 1984; Wong, 1975). Yan’kov (1979)
studied the response of kinetic untrapped ions in the Langmuir envelope soliton theory, and predicted the formation of
sub ion thermal small-amplitude negative potential wells in
plasmas. On the other hand, Mokhov and Chukbar (1984)
found a Langmuir envelope soliton accompanied with small-
amplitude negative potential well created by localized Langmuir wave electric fields in a quasi-neutral plasma with nonisothermal ions whose temperature is much smaller than the
electron thermal temperature.
Relativistic effects play a very important role in highenergy laser-plasma experiments (Shukla, 1986; Montgomery, 2001), in plasma based electron and photon accelerators (Bingham, 2003; Bingham et al., 2004; Mendonça,
2001), in supernova remnants and in gamma ray bursts (Piran, 1999), where electrons can be accelerated to relativistic energies by strong electrostatic fields. For short laser
pulse intensities exceeding 1019 W/cm2 , electrons in the
laser beam oscillate relativistically. Interactions between intense short laser pulses and background plasma give rise to
a number of nonlinear effects (Mendonça, 2001; Shukla et
al., 1986; Bingham et al., 2003; Bingham, 2004) associated with relativistic electron mass increase in the electromagnetic fields and the plasma density modification due to
relativistic radiation ponderomotive force. In the past, several authors presented theoretical (Kaw et al., 1992; Esirkepov et al., 1998) and particle-in-cell simulation (Esirkepov et
al., 1998; Bulanov et al., 1999; Farina and Bulanov, 2001;
Naumova et al., 2001) studies of intense electromagnetic envelope solitons in a cold plasma where the plasma slow response to the electromagnetic waves is modeled by the electron continuity and relativistic momentum equations, supplemented by Poisson’s equation. Experimental observations
(Borghesi et al., 2002) have shown bubble-like structures in
proton images of laser-produced plasmas, which are interpreted as remnants of electromagnetic envelope solitons.
In this paper, we present a review of recent theoretical and
numerical results of the interaction between ion and electron
holes with the background plasma. Specifically, we find that
colliding ion holes may lead to non-Maxwellian electron distributions due to the acceleration of electrons in the negative
ion hole potential, which works as a barrier for the negatively
charged electrons (Eliasson and Shukla, 2004a). Streams of
electrons are accelerated by the colliding ion holes, and these
electron streams can excite high-frequency Langmuir waves
due to a streaming instability. On the other hand, Langmuir waves can be trapped in the density cavity of the ion
holes, as described by a nonlinear Schrödinger equation for
the Langmuir waves (Shukla and Eliasson, 2003). Numerical studies show that the trapped Langmuir waves are Landau damped due to the relatively small Debye-scale of the
ion holes (Eliasson and Shukla, 2004a). We have also studied the fully nonlinear interaction between electron holes and
oxygen ions by means of a Vlasov simulations (Eliasson and
Shukla, 2004b), where it is found that the large-amplitude
electron hole potential accelerates the ions locally and that
the self-created ion density cavity accelerates the electron
hole, which propagates away from the ion cavity with a constant speed close to half the electron thermal speed. Finally,
we have extended the idea of Schamel to relativistic plasma,
where the trapped and free particles are given by solutions
of the relativistic Vlasov equation. We have studied theoretically and numerically the properties of relativistic electron
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
holes, and their interaction with relativistically high amplitude electromagnetic waves.
tude electromagnetic waves.
The article is organized as follows. In Section 2, we reThe
article is organized as follows. In Sect. 2, we review the general theory of ion and electron holes based on
view
the
general
theory
and electron holes
based on
Schamel’s
solution
of of
theion
quasi-stationary
Vlasov-Poisson
Schamel’s
solution
of
the
quasi-stationary
Vlasov-Poisson
system. The dynamics of ion holes in plasmas and their insystem.
Thewith
dynamics
of ion holes
in and
plasmas
andelectrons
their in-is
teraction
high-frequency
waves
kinetic
teraction
with
high-frequency
waves
and
kinetic
electrons
is
discussed in Section 3. In Section 4, we study the dynamics
discussed
in
Sect.
3.
In
Sect.
4,
we
study
the
dynamics
of
of electron holes in a plasma with mobile ions. An extenelectron
in a plasma
mobile ions.
An extension
sion ofholes
Schamel’s
theorywith
is presented
in Section
5, where
of Schamel’s
theory isholes
presented
in Sect.
5, where
relativisrelativistic electron
and their
interactions
with
intense
tic electromagnetic
electron holes and
their interactions
intense electroradiation
are studied with
theoretically
and numagnetic
radiation
areconclusions
studied theoretically
and
numerically.
merically.
Finally,
are drawn in
Section
6.
Finally, conclusions are drawn in Sect. 6.
2 General theory of ion and electron holes
2 General theory of ion and electron holes
The dynamics of electrons and ions in an unmagnetized, colplasma
is governed
by the
system
Thelisionless
dynamics
of electrons
and ions
in Vlasov-Poisson
an unmagnetized,
collisionless plasma∂f
isj governed
by
the
Vlasov-Poisson
system
∂fj
qj ∂ϕ ∂fj
+v
−
= 0,
(1)
mj ∂x ∂v
∂fj
∂fj
q∂t
j ∂ϕ ∂f∂x
j
+v
−
= 0,
(1)
∂tand ∂x
mj ∂x ∂v
and
∂ 2ϕ
X
∂2ϕ
qj nj ,
X∂x2 = −4π
j
= −4π
qj n j ,
(2)
(2)
∂x 2
j
where the number density is
where the number density
Z ∞ is
Z ∞
nj =
fj dv.
(3)
nj =
fj dv.
(3)
−∞
−∞
Here fj is the distribution function of the particle species
Here
is thee distribution
the particle
species
j (jfjequals
for electronsfunction
and i forofions),
mj is the
mass,
j (jqe equals
e
for
electrons
and
i
for
ions),
m
is
the
mass,
j
= −e, qi = e, e is the magnitude of the electron charge,
qe E
==
−e,
qi = e, ise the
is the
magnitude
of the
charge,
−∂ϕ/∂x
electric
field, and
ϕ iselectron
the electrostatic
E =
−∂ϕ/∂x
the1,electric
field,
andnumerically
ϕ is the electrostatic
potential.
In isFig.
we have
solved
the Vlasov
potential.
Fig.
we havewhile
solved
the assumed
Vlasov
equationInfor
the1,electrons,
thenumerically
ions have been
equation
the electrons,
while
thenions
havethe
been
assumed
to havefor
a constant
density,
ni =
initial
condi0 . Here,
tion isa aconstant
large-amplitude,
to have
density, ndensity-modulated
initialwhich
condi-is
i = n0 . Here, the wave,
−1
at t = 0 ωpe
, and
which then evolves
in time,
where
tionreleased
is a large-amplitude,
density-modulated
wave,
which
is
−1
2
1/2
released
t = 00ω
, and
which
then
evolves
in
time,
where
ωpe =at (4πn
e pe
/m
)
is
the
electron
plasma
frequency
e
1/2 is theelectron
n0 isn0the
equilibrium
Weand
see
ωpeand
= (4π
e2 /m
electronnumber
plasmadensity.
frequency
e)
−1
at equilibrium
t = 7ωpe
, some
of the
electrons
have been
accelern0 that
is the
electron
number
density.
We see
that
by−1
the self-consistent electrostatic field and
have formed
at tated
= 7ω
accelerated
by
pe , some of the electrons have been
−1
in velocity
space. At
t =
ωpe
, the streams
thestreams
self-consistent
electrostatic
field
and70have
formed
streamsof
electronsspace.
have mixed
in−1
a complex fashion. Two elecin velocity
At t further
= 70 ω
pe , the streams of electrons
tron
holesfurther
have been
v ≈ ±3 V
associated
with
T e , electron
have
mixed
in a created
complexatfashion.
Two
holes
a depletion
of the
function
a pophave
been created
at electron
v ≈ ±3distribution
VT e , associated
withand
a depletrapped electrons
moving
with and
the electron
holes.
tionulation
of theofelectron
distribution
function
a population
1/2
Here,
V
=
(T
/m
)
is
the
electron
thermal
speed
and
e
e moving
e
of trappedTelectrons
with the electron holes. Here,
Te is the electron
temperature.
1/2
VT e = (Te /me ) is the electron thermal speed and Te is the
In order to describe ion and electron holes mathematically,
electron temperature.
we look for quasi-stationary structures moving with velocity
In order to describe ion and electron holes mathematically,
v0 . In this case, we make the ansatz fj = fj (ξ, v) and ϕ =
we look
for quasi-stationary structures moving with velocity
ϕ(ξ), where ξ = x − u0 t, which transforms the system of
v0 .equations
In this case,
we make the ansatz f = fj (ξ, v) and ϕ =
(1) and (2) to a new set of jequations
ϕ(ξ ), where ξ = x − u0 t, which transforms the system of
qj ∂ϕ ∂fj
∂fj
Eqs. (1) and (2)(−u
to a0 new
= 0,
(4)
+ v)set of−equations
∂ξ
mj ∂x ∂v
∂fj
qj ∂ϕ ∂fj
(−u0 + v)
−
= 0,
(4)
∂ξ
mj ∂x ∂v
271
3
v/VT e
−1
t = 0 ωpe
−1
t = 70 ωpe
−1
t = 7 ωpe
15
feVT e/n0
0.6
10
0.5
0.4
5
0.3
0
0.2
−5
0.1
−10
0
−15
x/rD
0
2π
4π
x/rD
x/rD
0
2π
4π
0
2π
4π
−1
Fig. 1. The electron distribution function at t = 0 ωpe
(left panel),
−1
−1
t = 7ωpe (middle panel) and t = 70 ωpe (lower panel). The initial
−1
Fig. 1. The electron distribution −1
function at t = 0 ωpe
(left panel),
distribution
f = n0 (me /2πTe )1/2 [1 +
−1 function at t = 0ωpe is −1
t = 7ωpe (middle panel) and 2t = 70 ωpe (lower panel). The initial
0.5 cos(0.5x/rD )] exp(−me v /2T
−1e ).
distribution function at t = 0ωpe
is f = n0 (me /2πTe )1/2 [1 +
0.5 cos(0.5x/rD )] exp(−me v 2 /2Te ).
and
2ϕ
dand
Z ∞
=
−4π
e
(fi − fZe ) ∞
dv.
(5)
dξ 2
−∞
d2 ϕ
= −4πe
(fi − fe ) dv.
(5)
2
dξbe
Equation 4 can
integrated−∞
along its particle trajectories, so
that
the distribution
functionsalong
fj are
of the particle
Equation
4 can be integrated
itsfunctions
particle trajectories,
so
energy
that the distribution functions fj are functions of the particle
energy
mi (v − u0 )2
Ei =
+ eϕmi (v − u0 )2
+ eϕ
Ei =
2
2
(6)
(6)
and
and
me (v − u0 )2
− eϕ
(7)
me (v − u0 )2Ee =
2
Ee =
− eϕ
(7)
2 electrons, respectively. Here, E > 0 correfor ions and
j
sponds
untrapped
particles,
while EjHere,
< 0 corresponds
to
for
ions toand
electrons,
respectively.
Ej > 0 corretrappedtoparticles,
when
ϕ → 0 while
at |ξ| =Ej∞.
sponds
untrapped
particles,
< 0 corresponds to
Electron
holes are
characterized
by| a=localized
positive potrapped
particles,
when
ϕ → 0 at |ξ
∞.
tential,
in which
population
of theby
negatively
charged
elecElectron
holes aare
characterized
a localized
positive
potrons can
be trapped.
A special
classnegatively
of electron
hole solutential,
in which
a population
of the
charged
elections can
can be
be trapped.
found byAprescribing
the of
distribution
function
trons
special class
electron hole
solufor the
trapped
and free
electrons (Schamel
1971, 1972),
tions
can
be found
by prescribing
the distribution
function
( free·electrons (Schamel 1971,
)
for the
trapped and
³
´1/2 ¸2 1972),

2


 ae exp − 1" ±Ee1/2 + me u0
2 1/2 #2  , Ee > 0,
 Te
fe =
2

m´i

h 1 ³ 1/2
e u0

±E +
, E > 0,
−

m u2
aae eexp
exp −TT1ee βEee + e2 0 2 ,
 Eee < 0,
(8)
fe =

(8)
2

m
u

e
0
1/2
expn0 (m
− T1ee/2πT
βEee)+
, the trapping
Ee param< 0,
aaee =
where
, 2and
β is
eter describing the “temperatures” of the trapped electrons.
1/2we
Negative
ofe /2πT
β, which
are βinterested
in here,paramleads
where
ae values
= n0 (m
, and
is the trapping
e)
to
a
vortex
distributions
of
the trapped
electrons,
the
eter describing the “temperatures”
of the
trappedwhere
electrons.
distribution
function
is
excavated
locally.
This
is
illustrated
Negative values of β, which we are interested in here, leads
to a vortex distributions of the trapped electrons, where the
4
B.B.
Eliasson
The
dynamicsof
ofelectron
electron
and
ion
holes
Eliasson
and
K.
Shukla:
and
ion
holes
B.
Eliassonand
andP.P.
P.K.
K.Shukla:
Shukla: The
The dynamics
dynamics
of
electron
and
ion
holes
4272
Fig. 2. The electron distribution function in velocity space (from
Schamel, 1986). In a frame moving with
√ speed of the the electron
hole, electrons with a velocity |v| < 2φ are trapped. Negative
Fig.
2. The electron
distributionvortex
function
in velocity
space
β corresponds
to an excavated
distribution,
while
β (from
= 0
Fig. 2.Schamel,
The electron
distribution
function
in
velocity
space
(from
1986).
In a(constant)
frame moving
with
speed
of
the
the
electron
corresponds
to
a
flat
distribution.
The
velocity
v
is
scaled
√
√
hole,
velocity
< speed
2φ are
trapped.
Negative
Schamel,
awith
frame
moving
with
of the
the electron
by 1986).
Telectrons
, and
the apotential
φ|v|
by
e /meIn
√Te /e.
β corresponds
an excavated
β = 0
hole, electrons
with to
a velocity
|v| <vortex
2φdistribution,
are trapped.while
Negative
corresponds
to
a
flat
(constant)
distribution.
The
velocity
v
is
β corresponds
to
an
excavated
vortex
distribution,
while
β
=scaled
0
p
by
Tto
and
the potential
φ by Tlocally.
e /m
e ,function
e /e.
distribution
is distribution.
excavated
This isv illustrated
corresponds
a flat
(constant)
The velocity
is scaled
p
Figs.
2 and
(taken from
Tin
the3potential
φ bySchamel,
Te /e. 1986), where electrons
e /m
e , and
with negative energies are trapped in an excavated region of
in
2 anddistribution
3 (taken from
Schamel,
where
theFigs.
electron
function.
In1976),
the study
of electrons
electron
with
negative
energies
are
trapped
in
an
excavated
region
of
holes
which
are moving
much faster
than the
ion sound
speed
in Figs.
2 and
3 (taken
from Schamel,
1976),
where
electrons
the
electron
distribution
function.
In
the
study
of
electron
1/2
(Te /mi ) energies
, one can
theexcavated
ions formregion
a constant
with negative
areassume
trappedthat
in an
of
holes
which are
moving much
faster
than
the ion sound
speed
neutralizing
background,
n
=
n
.
We
discuss
a
deviation
i
0
the electron distribution
function. In the study of electron
1/2
(T
, one canfor
assume
thatofthestanding
ions form
a constant
e /m
i ) treatment
from
this
the
case
electron
hole
holes neutralizing
which
are moving
much
faster
than
the discuss
ion sound
speed
background,
n
=
n
.
We
a
deviation
i
0
in
Sect.
1/2 4, where the ion dynamics becomes important and
(Te /mfrom
, one
can assume
that
the
ions form
a constant
i)
for theare
case
of standing
hole in
wherethis
thetreatment
electron holes
accelerated
by electron
the self-created
neutralizing
background,
n
=
n
.
We
discuss
a
deviation
i dynamics
0
section
4,
where
the
ion
becomes
important
and
ion density. Integration of the electron distribution function
from this
treatment
for the
case
of
standing electron
hole in
where
the
electron
holes
are
accelerated
by
the
self-created
over velocity space gives the electron density as a function of
section
where Integration
the
ionwhich
dynamics
becomes
important
and
ion
density.
of inserted
the electron
distribution
function
the4,potential,
ne (ϕ),
into the
Poisson equation
whereover
the
electron
holes
are
accelerated
by
the
self-created
velocity
space
gives
the
electron
density
as
a
function
of
gives an equation for the self-consistent potential (Schamel,
the
potential,
n
(ϕ),
which
inserted
into
the
Poisson
equation
ion density.
Integration
of
the
electron
distribution
function
1971, 2000). e
gives
equation
forthe
theelectron
self-consistent
potential
(Schamel,
over velocity
space
as negative
a function
of
Ionan
holes
aregives
characterized
by adensity
localized
poten1971,
2000).
the potential,
ne (ϕ),
which
inserted into
theions.
Poisson
equation
tial, which
can trap
a population
of the
Similarly
as for
holes are
by aoflocalized
negative
gives the
anIon
equation
forcharacterized
thea special
self-consistent
potential
(Schamel,
electron
holes,
class
solutions
can
be potenfound
which can the
trapion
a population
the ions.for
Similarly
by
prescribing
distributionoffunction
free (Ei as
> for
0)
1971, tial,
2000).
the
electron
holes,
a
special
class
of
solutions
can
be
found
trapped
(Ei < 0) ions as
Schamel,
1981)
Ionand
holes
are characterized
by(Bujarbarua
a localizedand
negative
potenby prescribing
ion" distribution function#for
free (Ei > 0)

tial, which
can trap
athepopulation
of the ions.
as for
2
1/2Similarly


2
and trapped
(E
<
0)
ions
as
(Bujarbarua
and
Schamel,
1981)

i
m
u

1/2
i
1
0
the electron
a−
special
class+of solutions
can, be
found
holes,
Ei >
0,
i
ai exp (
Ti ±E
2
)
·
¸2 (E > 0)
³
´
by prescribing
the
ion
distribution
function
for
free
1/2
(9)
fi = 
i

1/2
−
mi u20
1

ai i<
exp
±E
+
,
E
>
0,
2

i
and trapped
(E
0)
ions
as
(Bujarbarua
and
Schamel,
1981)
i
m
u
T
2

1 i
f =
αE + i2 0 ´i
,
E < 0,
 ai exp −
i 
( h T·i 1 ³ i
) i

¸
mi u20 ´
2

³
αEi + m
ai exp − Ti 1/2
,1/2
Ei < 0,

2 u2

1
0

ai exp
±ETii )1/2+, Ti isi2the
, Ei > and
0, (9)
where
ai = n−
ion temperature
α
0 (m
Ti i /2π
fi = is the trapping
parameter
which
describes
the
“temperature”
1/2 ´i
³
h
where
ai = n01(mi /2πTi )mi u,2Ti is the ion temperature and


0
the
of
trapped
in negative
values
αEWe
exp
− Tions.
< 0, on
i +are 2interested
α
isaithe
trapping
parameter
which, describes
theEi“temperai
α,
giving
an
excavated
vortex
distribution
of
the
ions.
In(9)
the
ture” of the trapped ions. We are interested in negative values
study
of
ion
holes,
the
electrons
are
usually
assumed
to
1/2 vortex distribution of the ions. be
excavated
whereon
aiα,=giving
n0 (man
, Ti is the ion temperature andIn
i /2πT
i)
Maxwell-Boltzmann
distributed,
the
study
of
ion
holes,
the
electrons
are usually
assumed to
α is the trapping parameter which
describes
the “tempera
1/2 distributed, 2 !
be
Maxwell-Boltzmann
ture” of the trapped
me ions. We areeϕinterested
me v in negative values
fe = n0
exp ¶
− µ
,
µ
¶ions. (10)
on α, giving
an2πT
excavated
vortex
distribution
of the
In
1/2
T
2Teϕ
e
e
e
me
me v 2
,
(10)
fe =
n0 the electrons
expare usually
−
the study of ion
holes,
assumed
to
2πTmathematics
Te
2Tecompared to
which
simplifies
the
somewhat
e
be Maxwell-Boltzmann distributed,
using an exact solution of the Vlasov equation for elecwhich simplifies
the¶1/2
mathematics somewhat¶compared to
µ see,
trons. We will
however, µ
in the following
2 section that
e
ev
using an exact msolution
of the eϕ
Vlasovmequation
for elec-
by
fe = n0
2πTe
exp
Te
−
2Te
,
(10)
which simplifies the mathematics somewhat compared to
Fig. 3.
Particle trajectories in the vicinity of an electron hole
(from Schamel, 1986), for different normalized (by Te ) energies
Fig. 3. Particle trajectories in the vicinity of an electron hole
Ee . Trapped electrons have negative energies (Ee < 0), while free
(from Schamel, 1986), for different normalized (by Te ) energies
Fig.
3. Particle
trajectories
vicinity
an electron
electrons
have positive
energiesin
(Eethe
> 0).
Here, σofdenotes
the signhole
Ee . Trapped electrons have negative
energies (Ee < 0), while free
of theSchamel,
velocities 1986),
of the free
electrons,
in
a frame moving
with
the
(from
for
different
normalized
(by
T
) energies
e
electrons have positive energies (Ee > 0). Here, σ denotes the sign
of the electron
hole.
Espeed
.
Trapped
electrons
have
negative
energies
(E
<
0),
while
free
e
e
of the velocities of the free electrons, in a frame moving with the
electrons
have
positive
energies
(E
>
0).
Here,
σ
denotes
the
sign
e
speed of the electron hole.
of the velocities of the free electrons, in a frame moving with the
the dynamical
interaction
speed
of the electron
hole. between ion holes may change
the
electron
distribution
so that in
it deviates
strongly
from the
trons. We will see, however,
the following
section
that
Maxwellian
distribution.
By
integrating
the
distributions
for
the dynamical interaction between ion holes may change
the
free
and
trapped
ions
and
the
electrons,
both
the
ion
and
trons.
We will
see, however,
the following
that
the electron
distribution
so that in
it deviates
stronglysection
from the
electron
densities
can
be
expressed
as
functions
of
the
pothe
dynamical
interaction
between
ion
holes
may
change
Maxwellian distribution. By integrating the distributions for
tential,
and neions
(ϕ), and
Inserting
theion
partii (ϕ)
theelectron
freenand
trapped
theitelectrons,
both
the
andthe
the
distribution
sorespectively.
that
deviates
strongly
from
cle
number
densities
into
the
Poisson
equation,
one
obtains
electron densities
can be By
expressed
as functions
of the po-for
Maxwellian
distribution.
integrating
the distributions
an
equation
forand
the nself-consistent
potential
of the ion
tential,
ni (ϕ)
respectively.
Inserting
the hole,
partie (ϕ),and
the
free may
and
trapped
ions
the electrons,
both
the
ion
and
which
be
solved
analytically
by
means
the
Sagdeevcle number
densities
into
the
Poissonas
equation,
oneofobtains
electron
densities
can
be
expressed
functions
the
potential method (Schamel, 1971, 2000), or numerically aspoan equation
for
thenself-consistent
potential
of the ion
tential,
ni (ϕ)boundary
and
Inserting
thehole,
partie (ϕ),
a nonlinear
valuerespectively.
problem (Eliasson
and Shukla,
which
may
be
solved
analytically
by
means
the
Sagdeevcle
number densities into the Poisson equation, one obtains
2004a).
potential method (Schamel, 1971, 2000), or numerically as
an equation for the self-consistent potential of the ion hole,
a nonlinear boundary value problem (Eliasson and Shukla,
which may be solved analytically by means the Sagdeev2004a).
potential
method (Schamel,
1971,
2000), or numerically as
3 The dynamics
of ion holes
in plasmas
a nonlinear boundary value problem (Eliasson and Shukla,
In the present section, we will study some kinetic effects as2004a).
3 The dynamics of ion holes in plasmas
sociated with the interaction between ion holes and the background electron-ion plasma. Ion holes are associated with
In the present section, we will study some kinetic effects asnegative
electrostatic
potential,
which
repels the electrons.
3asociated
The dynamics
of ion
holes
in
plasmas
with the interaction
between
ion holes and the backTherefore, low-energy electrons can be trapped between two
ground electron-ion plasma. Ion holes are associated with
ion holes, and during collisions between the ion holes, the
negative
electrostatic
potential,
which
repels
the electrons.
Ina the
present
section, we
will study
some
kinetic
effects aslow-energy electrons are accelerated to high energies by the
Therefore,
low-energy
electrons
can beion
trapped
between
two
sociated
with
the
interaction
between
holes
and
thetwo
backion hole potential, forming electron streams escaping the
ion holes,
and duringplasma.
collisions
between
the associated
ion holes, the
ground
electron-ion
Ion
holes
are
with
colliding ion holes. Streams of electrons can excite highlow-energy
electrons
are
accelerated
to
high
energies
by
the
a frequency
negative electrostatic
potential,
repels
the electrons.
Langmuir waves
due to awhich
streaming
instability,
and
ion hole potential,
forming
electron streams
escaping
the two
Therefore,
low-energy
electrons
be trapped
between
thus the processes
on the
slower can
ion timescale
gives
rise totwo
colliding ion holes. Streams of electrons can excite highion
holes, and during
collisions
between
the On
ionthe
holes,
high-frequency
waves on
an electron
timescale.
otherthe
frequency Langmuir waves due to a streaming instability, and
low-energy
electrons
are accelerated
high energies
by the
hand, ion holes
are characterized
by a to
depletion
of the electhus the processes on the slower ion timescale gives rise to
trons,
which
gives forming
the possibility
thatstreams
high-frequency
Langion
hole
potential,
electron
escaping
the
two
high-frequency waves on an electron timescale. On the other
colliding ion holes. Streams of electrons can excite highfrequency Langmuir waves due to a streaming instability, and
thus the processes on the slower ion timescale gives rise to
B. Eliasson and P. K. Shukla: The dynamics of electron and5ion holes
on and ion holes
6
0.4
he elecy Langy, if the
a modrce actction of
se elecvia the
on holes
of comwill extion.
s of ion
mploy a
erically
and (2).
l distri, which
(9). We
= −1.0,
ron disann dise initial
em into
chamel,
ential ϕ
q. (5) is
is then
nditions
ing ion
/ωpe =
the inially are
ween the
on holes
for the
s of the
ding ion
e propa/mi )1/2
anding.
ed with
−1
= 0 ωpi
and afThe left
rgo coln in the
ly nonthe ion
ve plott v/VT e
wellian
on with
y before
ma after
273
B. Eliasson and P
f V
e Te
n
maintai
t=0.0 ω−1
pi
0.2
3.1 Tr
0
0
−4
0.4
−3
−2
−1
0
1
2
3
4
v/vTe
−2
−1
0
1
2
3
4
v/vTe
−2
−1
0
1
2
3
4
v/vTe
t=35.9 ω−1
pi
feVTe
n0
0.2
0
−4
0.4
f V
e Te
n
−3
t=133 ω−1
pi
0.2
0
0
−4
−3
Fig. 5. The electron velocity distribution at x = 8 rD , for t =
−1
−1
−1
0 ωpi
(upper panel), t = 35.9 ωpi
(middle panel) and t = 133 ωpi
Fig.
5.
The
electron
velocity
distribution
at
x
=
8
r
,
for
D
(lower panel).
−1
−1
t = 0 ωpi
(upper panel), t = 35.9 ωpi
(middle panel) and
−1
t = 133 ωpi
(lower panel).
tribution, given by Eq. (10). The potential ϕ for the initial
n T ) (10) is obtained by
Power inserting
spectrum of E (decibel)
conditions (9)E /(4πand
them into
40
Eq.0.25(5), which is then integrated30 (Bujarbarua and Schamel,
0.2
1981;
Shukla and Eliasson, 2003)
to obtain the potential ϕ
0.15
20
as 0.1
a function of x; in practice the
integration of Eq. (5) is
10
0.05
performed
numerically.
The
potential
thus obtained is then
0
0
−0.05
inserted
into Eqs. (9) and (10) to−10obtain the initial conditions
−0.1
for
the ion and electron distribution
functions.
−0.15
−20
−0.2
First, we investigate the process
of two colliding ion
−30
−0.25
holes,
as shown in Figs. 4–6. −40Here, rD = VT e /ωpe =
0
100
150
0
0.5
1
1.5
2
2.5
3
2 501/2 is the
(Te /4πn
electron
Debye
length.
0e )
ω/ωFor the initial
ω t
conditions, we considered that the ion holes initially are well
separated in space, and that the interaction between the ion
holes is weak, so that the solutions for the single ion holes
can
in field
spaceat to
form
condition
for
the
Fig.be
6. matched
The electric
x=
8.0 an
rD initial
as a function
of ωpi
t (left
panel)
andtwo
the power
spectrum
of the
electric field
as a function
of
case
with
ion holes.
Figure
4 displays
the features
of the
ω/ω
panel).
pe (right
ion
and
electron
distribution functions for two colliding ion
holes, where initially (upper panels) the left ion hole propagates with the speed u0 = 0.9 VT i , where VT i = (Ti /mi )1/2
−1
.and
Thethe
large-amplitude
elecat time
t ≈ 10
ωpi
iscited
the ion
thermal
speed,
right ion holebipolar
is standing.
tric ion
fields
the ion distribution
hole appear functions
as it crosses
x = 8.0with
rD
The
andofelectron
associated
−1
−1
at time
t ≈ 50–80
ωpi
.before
By examining
frequency
the
ion holes
are shown
collision the
at times
t = 0specωpi
trum of
the electric
(theω−1
right
panel),panels),
we notice
(upper
panels)
and t field
= 35.9
(middle
andthat
afthe high-frequency oscillations pi
have broad frequency spec−1
ter
collision
at
time
t
=
133
ω
(lower
panels).
The
left
pi to the electron plasma fretrum which is upshifted compared
4 exhibit
the downshifted
two ion holes
undergoband
colpanels
of We
Fig.also
quency.
have a that
slightly
frequency
lisions
without
being
destroyed,
and
as
can
be
seen
in
around ω/ωpe = 0.9, which we may be the frequenciesthe
of
right
panelswaves
of Fig.
4, theinelectrons
have as
a strongly
Langmuir
trapped
the ion holes,
describednonbeMaxwellian
flat top distribution in the region between the ion
low.
holes
after
that
the solutions
collision has
taken
place. We have
plotThe numerical
of the
Vlasov-Poisson
system
ted
the
electron
velocity
distribution
function
against
v/V
Te
were performed by means of a Fourier transform method
at(Eliasson,
x = 8.0 r2001).
We500
seeintervals
that the in
initial
Maxwellian
D in Fig.
We 5.
used
x space
with the
distribution
(the
changes
to a distribution
with
domain −40
≤ upper
x/rD panel)
≤ 40 with
periodic
boundary condibeams
at
v
≈
±0.6
V
(the
middle
panel)
slightly
before
Te
tions. We used 300 intervals
in velocity space. The eleccollision,
a flat-top
distribution
maxima
tron and and
ion to
speed
intervals
were inwith
the two
ranges
−15.7after
≤
collision
(the
lower
panel).
The
reason
for
the
creation
of the
v/VT e ≤ 15.7 and −0.118 ≤ v/VT e ≤ 0.118, respectively.
−1 is that the two ion holes are
flat-topped
velocity
The timestep
∆t ≈distribution
0.013 ωpe
was adapted dynamically to
associated with negative electrostatic potentials, and the elec1/2
0
Fig. 4. The scaled ion distribution function fi VT i /n0 (left panels)
and electron distribution function fe VT e /n0 (right panels) of two
−1
Fig. 4. The
distribution
function
fi VTt i /n
(left
colliding
ionscaled
holes, ion
before
the collision
at times
= 00 ω
(upper
pi panels)
and
electron
distribution
−1 function fe VT e /n0 (right panels) of two
panel) and t = 35.9 ωpi (middle panel), and after the collision
at
−1
colliding ion
holes, before the collision at times t = 0 ωpi
(upper
−1
tpanel)
= 133
ωpi
The color bar goes from dark blue
−1
and
t =(lower
35.9 ωpanel).
pi (middle panel), and after the collision at
(small
values)
−1to dark red (large values).
t = 133 ωpi (lower panel). The color bar goes from dark blue
(small values) to dark red (large values).
muir waves can be trapped in the ion hole. Clearly, if the
Langmuir
waves
have
a largeThe
amplitude,
there
be a of
modcollision (the
lower
panel).
reason for
the will
creation
the
ification
of
the
ion
hole
due
to
the
ponderomotive
force
flat-topped velocity distribution is that the two ion holesactare
ing
on the with
electrons,
which
are repelled
in the and
direction
of
associated
negative
electrostatic
potentials,
the elecdecreasing
Langmuir
electric
field
amplitudes.
These
electrons entering the region between the ion holes after collision
trons
the ion hole
potential,
ascross
described
via the
must then
havechange
a large enough
kinetic
energy to
the potential
Poisson
equation.
Thus,
the
interaction
between
the
ion
holes
barriers that are set up by the ion holes. The flattening of
the
with
the
background
plasma
gives
rise
to
a
multitude
of
comlow-velocity region then occurs probably due to repeated
replex
phenomena
on different
we will
exflections
and slowing
down oftimescales,
the trappedwhich
electrons
against
plore
theoretically
and numerically
thethe
present
the two
moving potential
barriers.inOn
othersection.
hand, we
In
order
to
investigate
the
time-dependent
dynamics
of ion
also observe that before collision, the low-energy population
holes
as
well
as
kinetic
effects
for
electrons,
we
employ
a
of electrons are compressed between the ion holes (see the
newly
developed
code
(Eliasson,
2001),
which
numerically
middle right panel of Fig. 4) and released during collision;
solves
the Vlasov-Poisson
system oftoEqs.
(1)beams
and (2).
The
these electrons
are then accelerated
form
of elecinitial
conditions
for
f
is
taken
as
the
Schamel
distribution
trons (see the middle panel
of Fig. 5) which escape the ion
i
(Schamel,
and trapped
ions,
which
in the
holes. The1986)
energyforofthe
thefree
electron
beams are
then
released
in
rest
frame
of
the
bulk
plasma
is
given
by
Eq.
(9).
We
use
a beam-plasma instability triggering high-frequency plasma
(Shukla
and as
Eliasson,
2003)
mi /m
1836,
α where
= −1.0,
oscillations,
illustrated
in Fig.
6 (the
panel)
the
e =left
and
T
/T
=
10.
The
initial
condition
for
the
electron
electric
at x = 8.0 rD is plotted as a function of ωdise field
i
pi t.
tribution
functionplasma
is takenoscillations
to be the Maxwell-Boltzmann
High-frequency
are beginning to bedisex-
pi
i
pe
Ion hol
and elec
Clearly,
in this
overden
cavity.
Schame
waves i
tion, we
holes, w
magnet
We h
in the p
holes. A
is the u
(j equa
muir wa
is the w
acting n
envelop
ison wi
nonline
2iωpe
= 0,
µ
where v
muir wa
bining t
well as
retainin
tion ne
Langmu
phase s
thermal
tron equ
presenc
The res
where τ
Langmu
trostatic
term in
ear term
2π/ωpe
electron
Eh = (
If the
ist, in g
We here consider an unmagnetized electron-ion plasma
in the presence of Langmuir waves and large-amplitude ion
Fig. 5. The electron velocity distribution at x = 8 rD , for
holes. At equilibrium, we have ne0 = ni0 = n0 , where nj0
−1
−1
t = 0 ωpi
(upper panel), t = 35.9 ωpi
(middle panel) and
is the
number
density ofofthe
particleand
species
j
−1
274
B. Eliasson
andunperturbed
P. K. Shukla:
The dynamics
electron
ion holes
t = 133 ωpi
(lower panel).
(j equals e for electrons and i for ions). The linear Lang¡ 2
¢1/2
muir
wave frequency
ω0 = are
ωpeprevented
+ 3k02 VT2from
,escaping
where k0the
e
overdense
so that theiswaves
E /(4π n T )
Power spectrum of E (decibel)
40
is the wavenumber. Large-amplitude Langmuir waves intercavity. A similar scenario was investigated theoretically by
0.25
30
acting nonlinearly with ion holes generate Langmuir wave
0.2
Schamel and Maslov (1999) for the trapping of Langmuir
0.15
20
envelope whose electric field E evolves slowly (in comparwaves in small-amplitude electron holes. In the present sec0.1
10
ison with the electron plasma wave period) according to a
0.05
tion, we investigate the trapping of Langmuir waves in ion
nonlinear Schrödinger equation
0
0
holes, while in Sect. 5 we consider the trapping of electro−0.05
−10
−0.1
magnetic
holes.
¶
µ
¶
µ waves in electron
−0.15
−20
2
ne plasma
∂ an unmagnetized
∂ consider
We here
electron-ion
2 ∂ E
2
−0.2
E
+
3V
1
−
+
v
+
ω
E
2iω
−30
g
pe
e
pe
−0.25
∂t
∂xLangmuir Twaves
∂x2 and large-amplitude
n0
in the presence
of
ion
−40
0
50
100
150
0
0.5
1
1.5
2
2.5
3
holes.
At
equilibrium,
we
have
n
=
n
=
n
,
where
n
e0
i0
0
j0
= 0,
ω/ω
ω t
is the unperturbed number density of the particle species
(11) j
(j equals e for electrons and i for ions). The linear Lang
1/2
Fig. 6. The electric field at x = 8.0 rD as a function of ωpi t (left
group
velocity
Lang-k
where
vg = frequency
3k0 VT2e /ωis
2 2 of the
2
peωis the
muir wave
, where
panel) and the power spectrum of the electric field as a function of
0 = ωpe + 3k0 VT e
0
muir
waves.
We
note
that
Eq.
(11)
has
been
derived
by
comFig.
6.
The
electric
field
at
x
=
8.0
r
as
a
function
of
ω
t
(left
D
pi
ω/ωpe (right panel).
is the wavenumber.
Large-amplitude
Langmuir
waves interbining
the electron continuity
and momentum
equations
as
panel) and the power spectrum of the electric field as a function of
acting
nonlinearly
with
ion
holes
generate
Langmuir
well as by using Poisson’s equation with fixed ions, and wave
by
ω/ωpe (right panel).
envelopethe
whose
electric
field
E evolves
slowly
(in comparretaining
arbitrary
large
electron
number
density
variatrons entering the region between the ion holes after collision
isonnwith
the
electron
plasma
wave
period)
according
to a
tion
associated
with
the
ion
holes
in
the
presence
of
the
e
must have a large enough kinetic energy to cross the potential
nonlinear wave
Schrödinger
equation force. Assuming that the
−1
Langmuir
ponderomotive
Theion
large-amplitude
bipolar of
eleccited at that
timeare
t ≈set10upωpi
barriers
by. the
holes. The flattening
the
electron
than the
phase speed
of ionholes is much
smaller
2
tric
fields
of
the
ion
hole
appear
as
it
crosses
x
=
8.0
D
low-velocity region then occurs probably due to repeatedrre∂
∂
ne
2 ∂ E from
2 the inertialess
−1
thermal
speed,
we
readily
obtain
2iω
+
v
E
+
3V
+
ω
1
−
E = elec0, (11)
pe
g
at time tand
≈ 50–80
ωpidown
. Byof
examining
the electrons
frequencyagainst
specpe
flections
slowing
the trapped
∂t
∂xmotion theT eelectron
n0
∂x 2
tron
equation
of
number
density
in
the
trum
of moving
the electric
field barriers.
(the right On
panel),
we notice
the
two
potential
the other
hand,that
we
presence
of
the
ponderomotive
force
of
the
Langmuir
waves.
2
the high-frequency
oscillations
have
frequency
specwhere vg = 3k0 VT e /ωpe is the group velocity of the Langalso
observe that before
collision,
the broad
low-energy
population
The result is
trum
which
is
upshifted
compared
to
the
electron
plasma
fremuir
waves. We note that Eq. (11) has been derived by comof electrons are compressed between the ion holes (see the
quency.right
We also
have
a slightly
downshifted
frequency
band
bining the electron continuity and momentum
equations as
middle
panel
of Fig.
4) and
released during
collision;
(12)
n = n0 exp[τ (φ̃ − W 2 )],
around
ω/ωpe =
we maytobe
the beams
frequencies
of
well as by using ePoisson’s
equation with fixed ions, and by
these
electrons
are0.9,
thenwhich
accelerated
form
of elec1/2
Langmuir
waves
trapped
in
the
ion
holes,
as
described
bewhere
τ =theTiarbitrary
/Te , W large
= |E|/(16πn
Ti )
isdensity
the scaled
retaining
electron 0number
variatrons (see the middle panel of Fig. 5) which escape the ion
low. The energy of the electron beams are then released in
Langmuir
wave envelope,
andion
φ̃ =holes
eϕ/Tini is
scaled election ne associated
with the
thethepresence
of the
holes.
The numerical
solutions
of the Vlasov-Poisson
trostatic
potential
of the ion hole.force.
We note
that thethat
W 2 -the
Langmuir
wave ponderomotive
Assuming
a beam-plasma
instability
triggering
high-frequency system
plasma
were performed
by means
method
term
Eq. (12)
comes
from
the averaging
of the
phaseinspeed
of ion
holes
is much
smaller than
thenonlinelectron
oscillations,
as illustrated
in of
Fig.a 6Fourier
(the lefttransform
panel) where
the
(Eliasson,
2001).
We8.0
used
intervals
x space with
ear
term
m
v
∂v
/∂x
over
the
Langmuir
wave
period
e
he
he
thermal
speed,
we
readily
obtain
from
the
inertialess
elecelectric
field
at x =
rD 500
is plotted
as in
a function
of ωthe
t.
pi
domain −40 ≤ plasma
x/rD ≤oscillations
40 with periodic
boundary
2π/ω
, where of
vhemotion
≈ −eE
the high-frequency
e ωpe isnumber
tron pe
equation
theh /m
electron
density in the
High-frequency
are beginning
to condibe ex−1
tions.at We
in velocity space. The elecelectron
in the Langmuir
wave
electric waves.
field
presencequiver
of thevelocity
ponderomotive
force of the
Langmuir
cited
timeused
t ≈ 300
10 ωintervals
pi . The large-amplitude bipolar electron
and
ion
speed
intervals
were
in
the
ranges
−15.7
≤
E
=
(1/2)E
exp(ik
x
−
iw
t)+complex
conjugate.
h result is
0
0
The
tric fields of the ion hole appear as it crosses x = 8.0 rD at
v/VTt e≈≤50–80
15.7 and
If the potential has a maximum φ̃max > 0, then there ex−1−0.118 ≤ v/VT e ≤ 0.118, respectively.
time
ωpi
. By examining
the
frequency
spectrum
−1
The
∆t ≈(the
0.013
ωpe
was adapted
dynamically
to
ist,
ne in
= general,
n0 exp[τ trapped
(φ̃ − W 2ions
)], where φ̃ < φ̃max , while at the
(12)
of
thetimestep
electric field
right
panel),
we notice
that the highfrequency oscillations have broad frequency spectrum which
where τ = Ti /Te , W = |E|/(16π n0 Ti )1/2 is the scaled
is up-shifted compared to the electron plasma frequency. We
Langmuir wave envelope, and φ̃ = eϕ/Ti is the scaled
also have a slightly “downshifted” frequency band around
electrostatic
potential of the ion hole. We note that the
ω/ωpe = 0.9, which we may be the frequencies of Lang2 -term in Eq. (12) comes from the averaging of the nonW
muir waves trapped in the ion holes, as described below.
linear term me vhe ∂vhe /∂x over the Langmuir wave period
The numerical solutions of the Vlasov-Poisson system
2π/ω
pe , where vhe ≈ −eEh /me ωpe is the high-frequency
were performed by means of a Fourier transform method
electron
quiver velocity in the Langmuir wave electric field
(Eliasson, 2001). We used 500 intervals in x space with the
E
=
(1/2)E
exp(ik0 x − iw0 t)+complex conjugate.
h
domain −40 ≤ x/rD ≤ 40 with periodic boundary condiIf
the
potential
has a maximum φ̃max > 0, then there extions. We used 300 intervals in velocity space. The electron
ist,
in
general,
trapped
ions where φ̃ < φ̃max , while at the
and ion speed intervals were in the ranges −15.7 ≤ v/VT e ≤
point
where
φ̃
=
φ̃
max there are no trapped ions. Similar
15.7 and −0.118 ≤ v/VT e ≤ 0.118, respectively. The time−1
to
Schamel
(1971),
we
chose at this point a Maxwellian disstep 1t ≈ 0.013 ωpe was adapted dynamically to maintain
tribution
for
the
free
ions.
The ion distribution function asnumerical stability.
sociated with the ion holes can then be obtained by solving
the ion Vlasov equation for free and trapped ions, which have
3.1 Trapping of Langmuir waves in ion holes
speeds larger and smaller than [2(φ̃max − φ̃)]1/2 , respectively.
Ion holes are associated with a local depletion of both ions
The electric potential will turn out to be essentially negative,
and electrons, the latter due to the negative ion hole potential.
with only a small-amplitude positive maximum φ̃max comClearly, there is a possibility that waves could be trapped
pared to the large-amplitude negative potential well with a
in this plasma density cavity if the surrounding plasma is
minimum at φ̃min ≡ −ψ. Thus, the potential is restricted by
1/2
0
pi
i
pe
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
−ψ ≤ φ̃ ≤ φ̃max , where ψ plays the role of the amplitude.
Integrating the sum of the free and trapped ion distribution
functions over velocity space, we obtain the ion number density (Schamel, 1971)
M2
ni = n0 b exp − i
I (φ̃max − φ̃)
2
2
q
Mi
2
, φ̃max − φ̃ + √
WD [ α(φ̃ − φ̃max )] , (13)
+κ
2
π|α|
where Mi = u0 /VT i is the Mach number, and u0 is the ion
hole speed. The normalization constant
2 Mi
b = exp
I (φ̃max )
2
2
−1
q
Mi
2
+κ
WD [ −α φ̃max ]
, φ̃max + √
,
2
π|α|
is chosen so that when φ̃ = 0, the total density of ions is
n0 . The special functions are defined as (Bujarbarua and
Schamel, 1981)
√
I (8) = exp(8)[1 − erf( 8)],
Z π/2 √
2
κ(X, 8) = √
X cos θ exp(−8 tan2 θ
(14)
π 0
√
+X cos2 θ )erf( X cos θ ) dθ,
and the Dawson integral
Z Z
WD (Z) = exp(−Z 2 )
exp(t 2 ) dt.
(15)
0
We are here mostly interested in negative values of α; however, for positive
√ 1972)
√ α, we use (Abramowitz and Stegun,
(where
i
=
−1) and
WD (iZ) = i( π/2) exp(Z 2 )erf(Z)
q
√
replace the term (2/ π |α|)WD [ α(φ̃ − φ̃max )] in Eq. (13)
q
√
by (1/ α) exp[−α(φ̃ − φ̃max )]erf[ −α(φ̃ − φ̃max )]; we
note especially that Mi = 0, α = 1 leads to a Boltzmann distribution ni = n0 exp(−φ̃) for the ion number density. The
Langmuir wave ponderomotive force acting on the ions is
weaker by the electron to ion mass ratio in comparison with
the one acting on the electrons, and therefore it is ignored in
Eq. (13). The electron ponderomotive force is transmitted to
ions via the ambipolar potential φ̃, which is determined from
Poisson’s equation
∂ 2 φ̃
= Ne − Ni ,
(16)
∂x 2
where Ne = ne /n0 and Ni = ni /n0 .
We are interested in quasi-steady state solutions of
Eqs. (11)–(16), which are fully nonlinear. We insert
E(x, t) = W (ξ ) exp {i[Kx − 2t]} and φ̃ = φ̃(ξ ) where
ξ = (x − u0 t)/rD , and where W (ξ ), K and 2 are assumed
to be real, into Eqs. (11)–(16) and obtain a coupled set of
nonlinear equations
2
τ rD
3
∂ 2W
− (λ − 1)W − W exp[τ (φ̃ − W 2 )] = 0,
∂ξ 2
(17)
275
and
Mi2
∂ 2 φ̃
2
−
exp[τ
(
φ̃
−
W
)]
+
b
exp
−
×
2
∂ξ 2
2
Mi
I (φ̃max − φ̃) + K
, φ̃max − φ̃
2
q
2
+√
WD [ α(φ̃ − φ̃max )] = 0,
π|α|
τ
(18)
−1 2 − 3k 2 r 2 (1 − u2 /v 2 ) = −2ω−1 2 −
where λ = −2ωpe
pe
D
0 g
2
2
2
2
3k rD +u0 /3VT e represents a nonlinear frequency shift. The
system of Eqs. (17) and (19) admits the first integral in the
form of a Hamiltonian
!2
∂W 2 τ ∂ φ̃
H (W, φ̃; Mi , τ, α, λ) = 3
−
∂ξ
2 ∂ξ
1
−(λ − 1)W 2 + {exp[τ (φ̃ − W 2 )] − 1}
τ
Mi2
+b exp −
P (φ̃max − φ̃, α)
2
2
Mi
, 0, φ̃max − φ̃ − 1 − H0 = 0,
+h
2
(19)
where in the unperturbed state (|ξ | = ∞) we have used the
boundary conditions W = 0, φ̃ = 0, ∂W/∂ξ = 0, ∂ φ̃/∂ξ =
0. The constant
2
Mi2
Mi
H0 =b exp −
P (φ̃max , α)+h
, 0, φ̃max −1
2
2
is chosen so that H = 0 at |ξ | = ∞. The auxiliary functions
are defined as
r
8
P (8, α) = I (8) + 2
(1 − α −1 )
π
√
2
+ √
WD ( −8α),
(20)
α π|y|
and
b
Z
h(X, a, b) =
κ(X, 8) d8.
(21)
a
Because we are interested in symmetric solutions defined
by W (ξ ) = W (−ξ ) and φ̃(ξ ) = φ̃(−ξ ), the appropriate
boundary conditions at ξ = 0 are W = W0 , φ̃ = −ψ,
∂W/∂ξ = 0, and ∂ φ̃/∂ξ = 0. Hence, from Eq. (19) we
have
1
(λ − 1)W02 − {exp[τ (−ψ − W02 )] − 1}
τ
Mi2
−b exp −
P (φ̃max + ψ, α)
2
2
Mi
+h
, 0, φ̃max + ψ − 1 − H0 = 0,
2
(22)
which shows how the maximum values of W0 and ψ are
related to Mi , φ̃max and λ for given values of τ and α. A
276 and P. K. Shukla: The dynamics of electron and ion holes
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
B. Eliasson
1
(22)
Ni
defined
propriate
= −ψ,
(20) we
0.6
0.4
−10
i
−5
0
5
φ
(24)
=0
¸¾
−1 .
(25)
) in the
ermines
ded that
−ψ. The
tween ψ
on holes
negative
quations
aves for
mplitude
odify the
electron
Eq. (18)
(26)
ξ
M =1.4
i
M =0.9
i
M =0.7
i
M =0
−2
−4
i
−6
(23)
10
0
~
ψ are red α. A
heck the
Eqs. (18)
imes for
s.
are gov-
max
M =1.4
i
M =0.9
i
M =0.7
i
M =0
0.8
−10
−5
0
5
10
ξ
Fig. 7. The ion density (upper panel) and the potential (lower
panel) of ion holes without Langmuir waves (W = 0) for different
Fig. 7.numbers
The ionM
density
panel)
potential (lower panel)
Mach
τ = 0.1
andand
α =the
−1.0.
i , with(upper
of ion holes without Langmuir waves (W = 0) for different Mach
numbers Mi , with τ = 0.1 and α = −1.0.
practical application of the Hamiltonian (19) is to check the
correctness of any numerical scheme used to solve Eqs. (17)
with(19),
the eigenvalue
λ anddepicts
the corresponding
eigenfunction
and
while Eq. (23)
the parameter
regimes for
W ,existence
and where
is obtained
by assuming
= 0holes.
in the soluthe
ofφ̃trapped
Langmuir
waves W
in ion
tion
of Eq.
(19); in
and lower
of Fig.
we
In the
absence
of the
the upper
Langmuir
waves,panels
ion holes
are7,govdisplay
density
profiles
and the1966)
associated ambipoerned
by the
the ion
energy
integral
(Sagdeev,
lar potentials for different sets of parameters, where the ob!2
tained negative
potentials are used in the eigenvalue problem
1 ∂ φ̃
(26). The+eigenvalue
VS (φ̃; Mi ,problem
τ, α) = (26)
0, has a continuous spec(23)
2trum∂ξfor λ < 0, associated with non-localized, propagating Langmuir wave envelopes W , and a point spectrum for
where
thecorresponding
Sagdeev potential
for our purposes
with φ̃maxwave
=0
λ > 0,
to localized,
trapped Langmuir
isenvelopes.
(Bujarbarua
1981)
Weand
haveSchamel,
investigated
numerically the cases corresponding to four different
Mach numbers displayed in Fig. 7,
1 1
and
found
the
corresponding
positive
eigenvalues listed in
VS (φ̃; Mi , τ, α) = −
[exp(τ
φ̃) − 1]
τ τ 1, where each eigenvalue λ is asthe left columns of Table
2 W . Using
to a 2bell-shaped
sociated
eigenfunction
k0 = 0
Mi
Mi
+inexp
−
P
(−
φ̃,
α)
+
h
,
0,
−
φ̃
−
1
. (24)
the definition
of
λ,
we
obtain
the
oscillation
frequency
2
2
of the trapped Langmuir waves as ω/ωpe = 1 + Θ =
2
1 − λ/2 +(23),
(u0 /V
1 − λ/2 from
+ Mi2 τEq.
me /6m
Equation
which
is=obtained
(19) i .inOnly
the
T e ) /6
one positive
eigenvalue
was wave
foundelectric
for each
case,determines
admitting
limit
of vanishing
Langmuir
fields,
onlyprofile
the ground
states
for theThe
Langmuir
to be linearly
the
of ion
holes.
latter waves
exist provided
that
trapped.
VS (φ̃; Mi , τ, α) is negative for φ̃ between zero and −ψ. The
Next, we
examine
of finite-amplitude
Langcondition
VS (−ψ;
Mithe
, τ, influence
α) = 0 gives
a relation between
ψ
muir
on the
ionofhole,
theout
fully
and
Miwaves
for given
values
α andinτwhich
. It turns
thatnonlinear
ion holes
system large-amplitude
of equations (18)Langmuir
and (19) waves
has tohave
be solved
numerwithout
only negative
ically, andas
where
we use the
same parameters as in the linear
potentials,
pre-assumed
earlier.
treatment
The
solutions
the ion
We haveabove.
carried
outnumerical
numerical
studiesreveal
of thethat
equations
hole deepened
and widened,
the eigenvalue
to
governing
ion holes
with and admitting
without Langmuir
wavesλ for
become
larger.
We
have
investigated
the
special
case
with
a
τ = 0.1 and α = −1.0. First, we consider small-amplitude
nonlinear
shift
of
0.1
of
λ,
listed
in
the
column
with
nonlinear
Langmuir waves which are not strong enough to modify the
eigenvalues
Table can
1, and
found
solutions
for all
cases
ion
hole, butinwhich
behave
linearly
trapped
in the
electron
2
except
for
M
=
1.4;
the
numerical
solutions
are
depicted
density well ofi the hole. Accordingly, for W 1 Eq. (17)
in Fig.
with Fig.
7, weofsee
the presence
turns
into8.a Comparing
linear eigenvalue
problem
thethat
form
of trapped finite-amplitude Langmuir waves makes the ion
density
d 2 W depletion both deeper and wider, and the same holds
3 2 + [1 − exp(τ φ̃) − λ]W = 0,
(25)
dξ
with the eigenvalue λ and the corresponding eigenfunction
W , and where φ̃ is obtained by assuming W = 0 in the solution of Eq. (19); in the upper and lower panels of Fig. 7, we
display the ion density profiles and the associated ambipolar
potentials for different sets of parameters, where the obtained
negative potentials are used in the eigenvalue problem (25).
The eigenvalue problem (25) has a continuous spectrum
for λ < 0, associated with non-localized, propagating Langmuir wave envelopes W , and a point spectrum for λ > 0, corresponding to localized, trapped Langmuir wave envelopes.
We have investigated numerically the cases corresponding
to four different Mach numbers displayed in Fig. 7, and
found the corresponding positive eigenvalues listed in the
left columns of Table 1, where each eigenvalue λ is associated to a bell-shaped eigenfunction W . Using k0 = 0
in the definition of λ, we obtain the oscillation frequency
of the trapped Langmuir waves as ω/ωpe = 1 + 2 =
1 − λ/2 + (u0 /VT e )2 /6 = 1 − λ/2 + Mi2 τ me /6mi . Only
one positive eigenvalue was found for each case, admitting
only the ground states for the Langmuir waves to be linearly
trapped.
Next, we examine the influence of finite-amplitude Langmuir waves on the ion hole, in which the fully nonlinear system of Eqs. (17) and (18) has to be solved numerically, and
where we use the same parameters as in the linear treatment
above. The numerical solutions reveal that the ion hole deepened and widened, admitting the eigenvalue λ to become
larger. We have investigated the special case with a nonlinear shift of 0.1 of λ, listed in the column with nonlinear
eigenvalues in Table 1, and have found solutions for all cases
except for Mi = 1.4; the numerical solutions are depicted
in Fig. 8. Comparing with Fig. 7, we see that the presence
of trapped finite-amplitude Langmuir waves makes the ion
density depletion both deeper and wider, and the same holds
for the ambipolar potential well. The deepening of the ambipolar negative potential well is a feature closely related to
the strongly non-isothermal trapped ion distribution function.
For this case, the electrostatic potential had small-amplitude
maxima φ̃max of the order ≈ 10−3 on each side of the ion
hole, and this maximum of the potential increased with increasing Mi .
Physically, the broadening of the ion hole and the enhancement of negative ambipolar potential occur because the ponderomotive force of the Langmuir waves locally expels electrons, which pull ions along to maintain the local charge
neutrality. The deficit of the ions in plasmas, in turn, produces more negative potential within the ion hole that is now
widened and enlarged to trap the localized Langmuir wave
electric field envelope.
In order to investigate the conditions for the existence of
ion holes in the presence of strong Langmuir fields, we numerically solved Eq. (22) for ψ as a function of Mi ; see
Fig. 9. We used the same parameters τ = 0.1 and α = −1.0
as above. Here, we assumed the Langmuir field to be given
as an external parameter (say W0 = 0.8) and with a nonlinear shift that approximately follows the relation λ(Mi ) =
0.3 − 0.14 Mi obtained from the fourth column of Table 1.
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
277
B.B.Eliasson
and
Shukla:
dynamics
ofofelectron
and
99
Eliasson
andP.P.K.K.
Shukla:The
The
dynamics
electron
andion
ionholes
holes
Table
1. Theoretical
eigenvalues
λ and
normalized
frequencies
ω/ω
for trapped Langmuir waves inside ion holes propagating with
pe
different Mach numbers Mi . Theoretical linear values are compared with numerical values obtained from direct Vlasov simulation (left and
Mach
Linear
eigenvalues
Frequencies
from
Nonlinear
eigenvalues
Mach
Linear
eigenvalues
Frequenciesused
fromin the the nonlinear
Nonlinear
eigenvalues
middle
columns).
Eigenvalues
(and frequencies)
theoretical
treatment of large-amplitude trapped Langmuir wave
number
Vlasov
and
number
andfrequencies
frequencies
Vlasovsimulation
simulation
andfrequencies
frequencies
envelopes
are and
shown
in the right columns.
MM
λλ
ω/ω
ω/ω
λλ
ω/ω
ω/ω
ω/ω
(ω
iγ)/ω
ω/ω
ii
pepe
pepe==(ω
r r++iγ)/ω
pepe
pepe
1.4
0.0013
0.999
—
—
1.4
0.0013 Mach
0.999 Linear eigenvalues
—
—
Frequencies from
Nonlinear eigenvalues
0.9
0.0463
0.977
0.1463
0.9
0.0463 number
0.977 and0.98
0.98−−0.00i
0.00i
0.1463 0.927
0.927
frequencies
Vlasov simulation
and frequencies
0.7
0.0772
0.7
0.0772 0.961
0.961
0.1772
0.911
λ
ω/ωpe
ω/ωpe =0.1772
(ωr + iγ 0.911
)/ω
M
λ
ω/ωpe
pe
i
0.0
0.1906
0.93
0.2906
0.0
0.1906 0.905
0.93−−0.03i
0.03i
0.2906 0.855
0.905
0.855
1.4
0.0013
0.999
—
—
0.9
0.0463 0.977
0.98 − 0.00i
0.1463 0.927
Table
Table1.1. Theoretical
Theoreticaleigenvalues
eigenvaluesλλand
andnormalized
normalizedfrequencies
frequenciesω/ω
ω/ω
fortrapped
trappedLangmuir
Langmuirwaves
wavesinside
insideion
ionholes
holespropagating
propagatingwith
with
pepefor
0.7
0.0772 0.961
0.1772 0.911
different
differentMach
Machnumbers
numbersMM
Theoreticallinear
linearvalues
valuesare
arecompared
comparedwith
withnumerical
numericalvalues
valuesobtained
obtainedfrom
fromdirect
directVlasov
Vlasovsimulation
simulation(left
(leftand
and
i .i .Theoretical
0.0
0.1906 0.905
0.93 − 0.03i
0.2906 0.855
middle
middlecolumns).
columns).Eigenvalues
Eigenvalues(and
(andfrequencies)
frequencies)used
usedininthe
thethe
thenonlinear
nonlineartheoretical
theoreticaltreatment
treatmentofoflarge-amplitude
large-amplitudetrapped
trappedLangmuir
Langmuirwave
wave
envelopes
envelopesare
areshown
shownininthe
theright
rightcolumns.
columns.
11
1.6
1.6
MM=0.9
=0.9
i i
MM=0.7
=0.7
i i
MM
=0
=0
i i
WW 0.5
0.5
00
−10
−10
−5
−5
00
55
1010
ξξ
MM=0.9
=0.9
i i
MM=0.7
=0.7
i i
MM
=0
=0
i i
0.8
0.8
i
0.6
0.6
0.4
0.4
−10
−10
1.2
1.2
WW
=0.8
=0.8
00
11
WW
=0.0
=0.0
00
MM0.8
i 0.8
i
0.6
0.6
11
NN
i
1.4
1.4
−5
−5
00
55
1010
ξξ
0.4
0.4
0.2
0.2
00
00
55
10
10
ψψ
15
15
20
20
00
~~
φφ
−2
−2
MM
=0.9
=0.9
i i
MM=0.7
=0.7
i i
MM=0
=0
i
−4
−4
−6
−6
−10
−10
i
−5
−5
00
55
1010
ξξ
Fig. 8. The electric field (upper panel), the ion density (middle
panel) and the potential (lower panel) of ion holes in the presence of
Fig.
field
(upper
panel),
the
(middle
Fig.8.8. The
Theelectric
electric
fieldwaves
(upper
panel),
theion
iondensity
density
(middle
large-amplitude
Langmuir
for
different
Mach
numbers
Mi ,
panel)
and
potential
(lower
panel)
and
the
potential
(lowerpanel)
panel)ofofion
ionholes
holesininthe
thepresence
presenceofof
with
τ=
0.1the
and
α = −1.0.
large-amplitude
large-amplitudeLangmuir
Langmuirwaves
wavesfor
fordifferent
differentMach
Machnumbers
numbersMM
i ,i ,
with
withτ τ==0.1
0.1and
andαα==−1.0.
−1.0.
This relation overestimates slightly the Langmuir field W0
for
Mi and potential
underestimates
slightly
the field
for
the
for
the
ofofthe
forsmall
theambipolar
ambipolar
potentialwell.
well.The
Thedeepening
deepening
theamamhighest
M
;
see
the
upper
panel
in
Fig.
8.
We
assumed
a
maxbipolar
potential
bipolarnegative
negative
potentialwell
wellisisa afeature
featureclosely
closelyrelated
relatedtoto
i
imum
potential
of φ̃max = 0.003.
Weion
found
that for this
set of
the
non-isothermal
trapped
function.
thestrongly
strongly
non-isothermal
trapped
iondistribution
distribution
function.
parameters,
the
solution
had
an
upper
bound
M
=
1.25
for
For
this
case,
the
electrostatic
potential
had
small-amplitude
For this case, the electrostatic potential had small-amplitude
i
−3
−3
the
existence
of
localized
solutions,
which
is
clearly
smaller
maxima
φ̃φ̃
of
the
order
≈
10
on
each
side
of
the
ion
maxima
of
the
order
≈
10
on
each
side
of
the
ion
max
max
hole,
this
ofofthe
increased
with
inthan
theand
existence
in the absence
ofpotential
the Langmuir
fields.
a
hole,
and
thismaximum
maximum
thepotential
increased
withIn
increasing
.
creasing
.
more
exactMM
mapping
of
the
existence
of
ion
holes,
one
needs
ii
to explore
more
carefully
theofof
relationships
between
differPhysically,
the
broadening
the
Physically,
the
broadening
theion
ionhole
holeand
andthe
theenhanceenhanceent
parameters
in
Eq.
(22),
possibly
by
solving
the
system
ment
mentofofnegative
negativeambipolar
ambipolarpotential
potentialoccur
occurbecause
becausethe
theponponofderomotive
Eqs. (17) force
and
(18)
forLangmuir
differentwaves
cases.
Furthermore,
the
ofofthe
expels
deromotive
force
the
Langmuir
waveslocally
locally
expelselecelecdynamics
of
the
time-dependent
system
is
not
explored
here,
trons,
which
pull
ions
along
to
maintain
the
local
charge
trons, which pull ions along to maintain the local charge
neutrality.
neutrality. The
Thedeficit
deficitofofthe
theions
ionsininplasmas,
plasmas,ininturn,
turn,propro-
Fig. 9. Numerical solutions of Eq. (22), depicting ψ (= −φ̃min )
versus Mi for W0 = 0.0 and W0 = 0.8, with τ = 0.1 and α =
Fig.
(23),
φ̃φ̃
))
Fig.9.9.Numerical
Numerical
solutions
ofEq.
Eq.
(23),depicting
depicting
ψ(=
(=−−
min
−1.0.
We
see thatsolutions
the ion of
hole
loaded
with
the ψLangmuir
wave
min
versus
WW
αα==is
versusMM
forW
W
0.0and
and
=0.8,
0.8,
withτ τ=
=0.1
0.1and
and
i ifor
0 0=
0 0=on
electric
fields
has
an=0.0
upper
bound
thewith
Mach
number,
which
−1.0.
loaded
the
Langmuir
−1.0. We
Wesee
see
that
theion
ionhole
holeLangmuir
loadedwith
with
thefields.
Langmuirwave
wave
smaller
than
thethat
onethe
without
the
wave
electric
electricfields
fieldshas
hasananupper
upperbound
boundononthe
theMach
Machnumber,
number,which
whichisis
smaller
smallerthan
thanthe
theone
onewithout
withoutthe
theLangmuir
Langmuirwave
wavefields.
fields.
but is studied by direct simulations of the Vlasov-Poisson
system below.
duces
ducesmore
morenegative
negativepotential
potentialwithin
withinthe
theion
ionhole
holethat
thatisisnow
now
It should be stressed that the properties of the present
widened
widenedand
andenlarged
enlargedtototrap
trapthe
thelocalized
localizedLangmuir
Langmuirwave
wave
Langmuir envelope solitons significantly differ from those
electric
electricfield
fieldenvelope.
envelope.
based
on Zakharov’s model
(1972) which
utilizes
the fluid
InInorder
ordertotoinvestigate
investigatethe
theconditions
conditionsfor
forthe
theexistence
existenceofof
ion
response
for
driven
(by
the
Langmuir
wave
ponderomoion
ionholes
holesininthe
thepresence
presenceofofstrong
strongLangmuir
Langmuirfields,
fields,we
wenunutive
force)
ion-acoustic
perturbations
and
yield
subsonic
denmerically
solved
Eq.
(23)
for
ψ
as
a
function
of
M
merically solved Eq. (23) for ψ as a function of M
see
i ;i ;see
sity
depression
accompanied
with aτ τpositive
localized
amFig.
==0.1
αα==−1.0
Fig.9.
9.We
Weused
usedthe
thesame
sameparameters
parameters
0.1and
and
−1.0
bipolar
potential
structure.
of a
as
Here,
assumed
the
field
asabove.
above.
Here,we
we
assumedFurthermore,
theLangmuir
Langmuirconsideration
fieldtotobebegiven
given
Boltzmann
ion
density
distribution,
viz.
N
=
exp(−
φ̃),
i witha anonlinasasananexternal
externalparameter
parameter(say
(sayWW
0.8)and
andwith
nonlin0 0==0.8)
would
correspond
to
the
case
M
=
0
and
α
=
1
in
Eq.
(18).
i
ear
the
earshift
shiftthat
thatapproximately
approximatelyfollows
follows
therelation
relationλ(M
λ(M
i )i ) ==
Here,
as0.14
shown
in
Fig. 10,
wethe
have
a localized
Langmuir
0.3
MM
from
fourth
column
0.3−−0.14
obtained
from
the
fourth
columnof
ofTable
Table
i iobtained
wave
electric
field
envelope trapped
inthe
athe
standing
ionfield
den1.
relation
overestimates
slightly
Langmuir
1. This
This
relation
overestimates
slightly
Langmuir
field
sity
cavity.
The
corresponding
slow
ambipolar
potential
WW
forsmall
smallMM
andunderestimates
underestimatesslightly
slightlythe
thefield
fieldfor
foris
0 0for
i iand
positive
and
localized.
the
thehighest
highestMM
seethe
theupper
upperpanel
panelininFig.
Fig.8.8.We
Weassumed
assumed
i ;i ;see
In
the numerical
solutions
of Eqs.
(17) and
(18)
, thethat
sec== 0.003.
We
found
a amaximum
potential
ofofφ̃φ̃
0.003.
We
found
that
maximum
potential
max
max
ond
derivatives
were
approximated
by
a
second-order
cenfor
forthis
thisset
setofofparameters,
parameters,the
thesolution
solutionhad
hadananupper
upperbound
bound
tered
difference
scheme
(Isaacson
and Keller,
1994),
and is
the
M
1.25
existence
ofoflocalized
solutions,
which
M
1.25for
forthe
the
existence
localized
solutions,
which
is
i i==
values
of
W
and
φ̃
were
set
to
zero
at
the
boundaries
of
the
clearly
smaller
than
the
existence
in
the
absence
of
the
Langclearly smaller than the existence in the absence of the Langmuir
muirfields.
fields.InIna amore
moreexact
exactmapping
mappingofofthe
theexistence
existenceofofion
ion
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
B.B.
Eliasson
and
Eliasson
andP. P.K.K.Shukla:
Shukla:The
Thedynamics
dynamicsofofelectron
electronand
andion
ionholes
holes
ωpeωtpet Electric
field,
MM=0M
0 0
Trapped
Langmuir
wave,
M=0
Electric
field,
i =
Trapped
Langmuir
wave,
i =
100100
100100
2 2
90
1.51.5
WW 1
1
70
50
40404040
30
−5−5
0 0
5 5
1010
ξ ξ
30
20202020
10
0.80.8
Ni Ni
0.60.6
0.40.4
−10
−10
−5−5
0 0
5 5
1010
ξ ξ
0.40.4
0.30.3
~ ~
φ φ0.20.2
0.10.1
0 0
−10
−10
−5−5
0 0
5 5
1010
ξ ξ
Fig. 10. The Langmuir electric field (upper panel), the ion density
(middle panel) and the potential (lower panel) of a Langmuir caviFig.
The
Langmuir
electric
field
(upper
the
denFig.
10.10.
The
Langmuir
electric
field
(upper
ion
ton
with
a Boltzmann
ion distribution
for
M
0,panel),
λ =the
0.1,
τion
=den0.1
i =panel),
panel)
potential
(lower
panel)
a Langmuir
sitysity
andand
thethe
potential
(lower
panel)
of of
a Langmuir
and
α(middle
=(middle
1.0. panel)
caviton
with
a Boltzmann
distribution
0.1,
i =
caviton
with
a Boltzmann
ionion
distribution
forfor
MM
0, 0,
λ λ= =0.1,
i =
τ =
α=
1.0.
τ =
0.10.1
andand
α=
1.0.
computational domain at ξ = ±40. The resulting nonlinear
system
of equations
was
solved
iteratively.
We relationships
used 2000
holes,
needs
explore
more
carefully
holes,
oneone
needs
to to
explore
more
carefully
thethe
relationships
sampling
points
to
resolve
the
solution.
between
different
parameters
(23),possibly
possiblybybysolvsolvbetween
different
parameters
in in
Eq.Eq.
(23),
system
equations
(18)
and
(19)
different
cases.
inging
thethe
system
of of
equations
(18)
and
(19)
forfor
different
cases.
3.2
Vlasov simulations
of trapped
Langmuir
waves
Furthermore,
dynamics
time-dependent
systemis is
Furthermore,
thethe
dynamics
of of
thethe
time-dependent
system
explored
here,
studied
direct
simulations
notnot
explored
here,
butbut
is is
studied
byby
direct
simulations
ofof
thethe
InVlasov-Poisson
order
to explore
the dynamics
Vlasov-Poisson
system
below. of trapped Langmuir waves
system
below.
in an
hole,be
we
have
performed
aproperties
new
set ofofsimulations
to
It
should
bestressed
stressed
thatthethe
properties
ofthethepresent
present
It ion
should
that
study
the
time-dependence
of
trapped
Langmuir
wave
elecLangmuir
envelope
solitons
significantly
differ
from
those
Langmuir envelope solitons significantly differ from those
tric
fields.
The
results are
displayed
in
Fig. 11.
based
Zakharov’s
model
(1972)
which
utilizesthethefluid
fluid
based
onon
Zakharov’s
model
(1972)
which
utilizes
ion
response
for
driven
(by
the
Langmuir
wave
ponderomoIn
these
simulations,
the
electron
density
is
initially
perion response for driven (by the Langmuir wave ponderomoforce)
ion-acoustic
perturbations
and
yield
subsonic
denturbed
so
that
Langmuir
waves are and
excited
around
thedenion
tivetive
force)
ion-acoustic
perturbations
yield
subsonic
sity
depression
accompanied
with
a
positive
localized
amhole
centered
at
x/r
=
0.
We
examine
the
above
discussed
sity depression accompanied
with a positive localized amD
bipolar
potential
structure.
Furthermore,
consideration
of
ion
hole
with
zero-speed
(the
left
panel)
and
the
ion
hole
havbipolar potential structure. Furthermore, consideration of a a
Boltzmann
ion
density
distribution,
viz.N
N = 11
exp(−
ing
the speedion
M
=
0.9 distribution,
(the
right panel).
Figure
shows
Boltzmann
viz.
φ̃),φ̃),
idensity
i i= exp(−
would
correspond
case
=
0ion
and
α
=
1 in
Eq.
(19).
i the
would
correspond
to to
thethe
case
MM
=
0
and
α
=
1
in
Eq.
(19).
that
Langmuir
waves
are
trapped
in
hole,
with
a
maxi
Here,
shown
Fig.
wehave
have
a localized
Langmuir
Here,
as as
shown
in in
Fig.
10,10,
we
a
localized
Langmuir
imum
wave
amplitude
at x/r
=
0.
We
find
that
the
trapped
D
wave
electric
field
envelope
trapped
in
a standingion
iondendenwave
electric
field
envelope
trapped
a standing
Langmuir
waves
are
oscillating
with
a in
real
frequency
slightly
sity
cavity.
The
corresponding
slow
ambipolar
potential
sity cavity.
The
corresponding
slow ambipolar
is is
lower
than the
electron
plasma frequency,
ωr /ωpotential
pe ≈ 0.93
positive
and
localized.
positive
and localized.
for
the standing
ion hole case, and ωr /ωpe ≈ 0.98 for for
In
numerical solutions
of Eqs.
(18)
and
(19),thethesecsecthethe
numerical
(18)
and
(19),
the In
moving
ion hole solutions
case withof
MEqs.
which
is consistent
i = 0.9,
ond
derivatives
were
approximated
by
a
second-order
cenond the
derivatives
were listed
approximated
by For
a second-order
with
linear results
in Table 1.
the standingcenhole
tered
difference
scheme
(Isaacson
and
Keller, 1994),
and
tered
difference
scheme
and
Keller,
and
thethe
case,
the
Langmuir
waves(Isaacson
are strongly
Landau1994),
damped,
with
values
of
W
and
φ̃
were
set
to
zero
at
the
boundaries
of
of W
andγ φ̃≈were
to, zero
at the
boundaries
thethe
avalues
damping
rate
0.03set
ωpe
which
is an
effect notofcovcomputational
domain
at
ξ
=
±40.
The
resulting
nonlinear
computational
domain model
at ξ = which
±40. is
The
resulting
ered
by ouroftheoretical
based
onWe
thenonlinear
electron
system
equations
was
solved
iteratively.
used
2000
system
of equations
solved
iteratively.
used
2000
hydrodynamic
model was
for the
Langmuir
wave We
packets.
In
the
moving ion hole case, the damping is significantly weaker
90
0.10.1
0.1 0.1
10
10
0.05
0.05
0.05 0.05
0
00 0
-0.05
-0.05
−0.05−0.05
-0.1
-0.1
−0.1 −0.1
10
−0.1 −0.1
0 −4000 −400−30 −30−20 −20−10 −100 0 10 1020 2030 30
-0.1
-0.1 0 −4000 −400 −30 −30−20 −20−10 −10 0 0 10 10 20 20 30 30
-40-40 -20-20 x/r0 x/r0 2020 4040
-40-40 -20-20 x/r0 x/r0 2020 4040
x/r
x/r
x/r
x/r
D D
DD
D
1 1
90
ωpt
ωpt
ωpt
50
ωpeωtpet Electric
field,
M
0.90.9
Trapped
Langmuir
wave,
M=0.9
Electric
field,
M
i =
Trapped
Langmuir
wave,
M=0.9
i =
100 100
0.15 0.15
100
100
0.15
0.15
80808080
0.05
0.05
0.05 70 70
0.05
60606060
50 50
00 00
40
404040
30 30
−0.05
−0.05
-0.05
-0.05 20 20
2020
60606060
0.50.5
0 0
−10
−10
90
80808080
70
0.10.1
0.1 0.1
ωpt
278
1010
D
D
D
p
Fig. 11. Normalized Langmuir wave Electric field E/ 4πn0 Ti
(the bipolar electric field has been removed from the
data) as a
√√
Fig.11.11.Normalized
Langmuir
wave
Electric
field
E/
4πn
Fig.
Langmuir
wave
field
E/
4πn
T0iTi
0Langfunction
ofNormalized
the normalized
space
x/rElectric
and
time
ω
t.
The
pe
D
(the
bipolar
electric
field
has
been
removed
from
the
data)
as
(the
bipolar
electric
field
has
been
removed
from
the
data)
as
a aat
muir wave is trapped in the ion hole which is initially centered
function
of
the
normalized
space
x/r
and
time
ω
t.
The
Langfunction
of
the
normalized
space
x/r
and
time
ω
t.
The
LangD
pe
D
pe
x/rD = 0, and moving with the speed Mi = 0 (left panel) and
waveis istrapped
trappedininthetheion
ionhole
holewhich
whichis isinitially
initiallycentered
centeredatat
muir
wave
Mmuir
i = 0.9 (right panel).
x/r
andmoving
movingwith
withthethespeed
speedMM
(leftpanel)
panel)and
and
x/r
0, 0,and
(left
D D= =
i i==0 0
(right
panel).
MM
0.90.9
(right
panel).
i =
i =
than in the standing hole case. This can be understood in
that the frequency shift is smaller for the moving ion hole
samplingpoints
pointstotoresolve
resolvethe
thesolution.
solution.
sampling
case, making the scalelength larger for the Langmuir wave
envelope, resulting in a smaller Landau damping; some disVlasovsimulations
simulationsofoftrapped
trappedLangmuir
Langmuirwaves
waves
3.23.2 Vlasov
tance away from the ion hole density minimum the potential
φ̃ vanishes, and there the Langmuir wave envelope
decreases
√
order
explorethe
thedynamics
dynamicsofoftrapped
trappedLangmuir
Langmuirwaves
waves
InIn
order
totoexplore
exponentially with ξ with the scalelength 3/λ. A smaller
in
an
ion
hole,
we
have
performed
a
new
set
of
simulations
in an ion hole, we have performed a new set of simulations toto
λstudy
leadsthe
to time-dependence
a larger length-scale,
resulting
in a weaker
Lantrapped
Langmuir
waveelecelecstudy
the time-dependence
ofoftrapped
Langmuir
wave
dau
damping.
In
numerical
simulations
of
large-amplitude
tricfields.
fields.The
Theresults
resultsare
aredisplayed
displayedininFig.
Fig.11.
11. InInthese
these
tric
Langmuir
waves,
one has
to apply
an external
pump
field
simulations,
electron
density
initially
perturbed
that
simulations,
thetheelectron
density
isisinitially
perturbed
sosothat
toLangmuir
sustain the
Langmuir
field
at
constant
amplitude,
since
wavesare
areexcited
excitedaround
aroundthe
theion
ionhole
holecentered
centeredatat
Langmuir waves
the
trapped Langmuir
waves
are
Landau
damped.
We
have
x/r
Weexamine
examine
theabove
above
discussed
ionhole
hole
with
x/r
the
discussed
ion
with
D D==0.0.We
observed
the
following
phenomena
in
Vlasov
simulations:
zero-speed(the
(theleft
leftpanel)
panel)and
andthe
theion
ionhole
holehaving
havingthe
thespeed
speed
zero-speed
High-intensity
Langmuir
waves
are
trapped
inside
M
0.9(the
(theright
rightpanel).
panel).
Figure
showsthat
that
Langmuir
i =
M
0.9
Figure
1111initially
shows
Langmuir
i =
the
ion
hole.
The
ion
density
well
associated
with
the
waves
are
trapped
in
the
ion
hole,
with
a
maximum
wave
waves are trapped in the ion hole, with a maximum waveion
hole
deepened,
and =
the0.ion
surrounding
ion hole
amplitude
x/r
Wedensity
find that
the trappedthe
Langmuir
amplitude
atatx/r
D D= 0. We find that the trapped Langmuir
is
depleted.
The
Langmuir
field
trapped
in
the
ion
acwaves
are
oscillating
with
a
real
frequency
slightly
lower
waves are oscillating with a real frequency slightly hole
lower
celerated
electrons
which
formed
propagating
electron
BGK
than
the
electron
plasma
frequency,
ω
/ω
≈
0.93
for
the
r
pe
than the electron plasma frequency, ωr /ωpe ≈ 0.93 for the
standing
ionhole
hole
case,and
and
ωr /ω
0.98 forafor
thewidenmovwaves
(electron
holes).
Weωcould
not≈observe
clear
standing
ion
case,
r /ω
pepe≈ 0.98 for for the moving
ion
hole
case
with
=
0.9,
which
is
consistent
with
the
ing
of
the
ion
hole
predicted
by
theory.
Possible
explanai
ing
ion
hole
case
with
MM
=
0.9,
which
is
consistent
with
the
i
linear
listedininTable
Table
For
the
standing
hole
case,
the
tions
ofresults
this discrepancy
can
that
the
electrons
are
strongly
linear
results
listed
1.1.be
For
the
standing
hole
case,
the
Langmuir
waves
arestrongly
strongly
Landau
damped,
witha adampdampheated
in waves
the
simulation,
which
increases
the with
temperature
raLangmuir
are
Landau
damped,
ingrate
0.03
ωscenario
, whichisinisanwhich
aneffect
effect
not
covered
byour
ourto
pe
tio
τ rate
, leading
toωape
the
ion
hole by
tends
ing
γ γ≈≈0.03
, which
not
covered
theoretical
modelif
which
isbased
basedononthe
theelectron
electron
hydrodybecome
smaller
otherisparameters
are
kept
constant
(Butheoretical
model
which
hydrodynamic
model
for
the
Langmuir
wave
packets.
In
the
moving
jarbarua
andfor
Schamel,
1981).wave
Further,
the In
theory
does
not
namic
model
the Langmuir
packets.
the moving
ion
hole
case,
dampingisissignificantly
weaker
than
the
ion
hole
case,
weaker
than
ininthe
predict
how
thethedamping
parameters
ofsignificantly
an ion hole
changes
dynamistanding
hole
case.
Thiscan
canbe
beunderstood
understood
thatthe
thefrefrestanding
hole
case.
This
ininthat
cally
when
the
amplitude
of
the
trapped
Langmuir
envelope
quency
shift
is
smaller
for
the
moving
ion
hole
case,
making
quency
shift
is smaller for the moving ion hole case, making
changes
in time.
scalelengthlarger
largerforforthe
theLangmuir
Langmuirwave
waveenvelope,
envelope,rerethethescalelength
sultinginina asmaller
smallerLandau
Landaudamping;
damping;some
somedistance
distanceaway
away
sulting
ionhole
holedensity
densityminimum
minimumthe
thepotential
potentialφ̃φ̃vanishes,
vanishes,
from
thetheion
4from
The
dynamics
of nonrelativistic
electron
holes
in plasand
there
the
Langmuir
wave
envelope
decreases
exponenp decreases exponenand there
mas the Langmuir wave envelope
p
tiallywith
withξ ξwith
withthe
thescalelength
scalelength 3/λ.
3/λ.AAsmaller
smallerλλleads
leads
tially
to
a
larger
length-scale,
resulting
in
a
weaker
Landau
damptoInathis
larger
length-scale,
in a weaker
Landau dampsection,
we studyresulting
the dynamics
of nonrelativistic
elecing. Innumerical
numericalsimulations
simulationsofoflarge-amplitude
large-amplitude Langmuir
ing.
tron In
holes in an electron-ion
plasma. The shapeLangmuir
of the elecwaves,one
onehas
has toapply
applyananexternal
externalpump
pumpfield
field to sustain
waves,
tron hole
is not to
unique but
is strongly
dependenttoonsustain
the histhe
Langmuir
field
at
constant
amplitude,
since
thetrapped
trapped
the
Langmuir
field at constant
amplitude,
since the
the
tory
of
its
creation.
In
order
to
investigate
dynamics
Langmuirwaves
wavesare
areLandau
Landaudamped.
damped.We
Wehave
haveobserved
observedthe
the
Langmuir
of electron
holes numerically
in a controlled
manner, we
use Schamel’s solution of the stationary Vlasov-Poisson sys-
ters of an ion hole changes dynamically when the amplitude
of the trapped Langmuir envelope changes in time.
panel), associated with a standing electron hole (Me = 0) with
β = −0.7 (solid lines) and β = −0.5 (dash-dotted lines), and a
moving electron hole with Me = 0.5 and β = −0.7 (dashed lines)
in plasmas with fixed ion background (Ni = 1).
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
279
ron and ion holes
11
4 The dynamics of nonrelativistic electron holes in plas-
mas
intensity
ole. The
ned, and
ed. The
electrons
(electron
the ion
this dised in the
, leading
maller if
Schamel,
paramemplitude
in plas-
stic electhe electhe hisynamics
nner, we
sson sysons. The
t the sond that it
from the
Followsolutions
ributions
here use
ntrapped
e, we ob-
φ)1/2 ]
/2
¸
,
(27)
speed of
tial. The
4)–(16),
(28)
onlinear
ar away
ence apPoisson’s
s, which
Newton’s
and elecase of a
Ne
10
In this section, we study the dynamics of nonrelativistic elec5
tron holes in an electron-ion plasma. The shape of the electron hole is not unique but is strongly dependent on the his0
−20of its
−15creation.
−10
−5 order0 to investigate
5
10 the15dynamics
20
tory
In
of electron holes numericallyξ in a controlled manner, we
1.2Schamel’s solution of the stationary Vlasov-Poisson sysuse
N
1 to construct initial conditions for our simulations. The
e
tem
0.8
Schamel solution is well-behaved in the sense that the solution
for the distribution function is continuous and that it
0.6
−20
−15
−10
−5
0
5
10
15
20
goes to a shifted Maxwellian distribution
far away from the
ξ
electron hole where the electric potential vanishes. Following12.
Bujarbarua
and(upper
Schamel
(1981),
prescribe
solutions
Fig.
12.
The potential
potential
(upper
panel)
and
electron
density
(lower
Fig.
The
panel)
and the
thewe
electron
density
(lower
to
the
electron
Vlasov
equation
in
the
form
of
distributions
panel),
associated
with
a
standing
electron
hole
(M
=
e
panel), associated with a standing electron hole (Me = 0)
0) with
with
−0.7
(solid
lines)electrons,
and ββ =
= as
−0.5
(dash-dotted
and
aa
and
trapped
given
in Eq. (8)lines),
We here
ββof
==free
−0.7
(solid
lines)
and
−0.5
(dash-dotted
lines),
anduse
moving
electron
holefor
withoxygen
Mee =
= 0.5
0.5
andIntegrating
β
lines)
mi /melectron
ions.
the untrapped
moving
hole
with
M
and
β=
= −0.7
−0.7 (dashed
(dashed
lines)
e = 29500
plasmas
withelectron
fixedion
iondistributions
background (N
(N
1).
i = 1).
ininand
plasmas
with
fixed
background
trapped
over
space, we obi =velocity
tain the electron density
Ni
φ
Ne ·
Ni
¸
µ 2 ¶
1/2
]
Me for our2W
tem to construct
simulations.
D [(−βφ) The
−Me2 /2 initial conditions
,φ +
,
I(φ) + κ
Ne = e
Schamel solution is well-behaved
that1/2the so2 in the sense
(π|β|)
lution
for the distribution function is continuous and that(27)
it
ωpe t
where
=
u
/V
is
the
Mach
number,
u
is
the
speed
of
goes
to aMshifted
Maxwellian
distribution
far
away
from
the
e
0
Te
0
the electron
hole, and
= eϕ/Tpotential
is
the
scaled
potential.
The
electron
hole where
theφelectric
vanishes.
Followe
functions
κ and W(1981),
in Eqs. (14)–(16),
ingspecial
Bujarbarua
andI,Schamel
we prescribe
solutions
D are given
Poisson’s
equation
√
torespectively.
the electron Vlasov
equation
in the form of distributions of
E/ 4πn0 Te
φ
free and trapped electrons,2 as given in Eq. (8). We here use
∂
φ
2
mi /me = 29500 for oxygen
ions.
the untrapped
= NeIntegrating
− Ni ,
(28)
rD
∂x2
and trapped electron distributions
over velocity space, we obωpe tthe electron density
tain
where Ne is given by Eq. (27), is solved as a nonlinear
"
! φ is set to zero far
# away
boundary value
problem 2where
1/2 ]
M
2W
[(−βφ)
2
D
e
−Me /2
sideI (φ)+κ
of the electron
difference
apNeon
=eeach
, φ hole;
+ a central1/2
, (26)
(π |β|)
proximation x/r
is Dused for2the second derivative
in
Poisson’s
x/rD
equation, leading to a system of nonlinear equations, which
where
Me =iteratively
u0 /VT e is with
the Mach
number,modified)
u0 is the Newton’s
speed of
is solved
(a slightly
Fig.
13. The hole,
electron
density
(upper
left
panel),
the ion
density (upthe
electron
and
φ
=
eϕ/T
is
the
scaled
potential.
The
e
method.
per
right panel), the electric field (lower left panel) and the potential
special
functions
I
,
κ
and
W
are
given
in
Eqs.
(14)–(15),
D
In right
Fig. 12,
weofhave
plotted
the electric
potential
elec(lower
panel)
an initially
standing
electron
hole. Theand
chosen
respectively.
Poisson’s
tron number
density
of−0.7.
the electron hole for the case of a
trapping
parameter
is β =equation
2
2 ∂ φ
rD
= Ne − Ni ,
(27)
∂x 2
fixed ion background, viz. Ni = 1. We note that larger valwhere
bygive
Eq. (26),
is solved
as aψnonlinear
boundues ofNM
and |β|
smaller
maxima
of the potential
e is
e given
ary
value
problem
where
φ
is
set
to
zero
far
away
on
each
and less deep electron density minima, in agreement with
Fig.of3(a)
Bujarbarua
Schamel
(1981). The
potentials
side
theofelectron
hole;and
a central
difference
approximation
Fig.
12 are
used to construct
the equation,
numericalleadiniisobtained
used forinthe
second
derivative
in Poisson’s
tialtoconditions
fornonlinear
the electron
distribution
function
of elecing
a system of
equations,
which
is solved
iteratron holes
and Schamel,
1981),method.
to be used in our
tively
with (Bujarbarua
(a slightly modified)
Newton’s
Vlasov
including
dynamics.
In Fig.simulations
12, we have
plotted the ion
electric
potential and elecFig. 13density
we show
the electron
time development
of case
an electronInnumber
of the
hole for the
of a
tron ion
holebackground,
initially at viz.
rest.Ni =
As1.the
condition
for
fixed
Weinitial
note that
larger valtheofelectron
function,
weψuse
thepotential
parameters
ues
Me and distribution
|β| give smaller
maxima
of the
and
Medeep
= 0electron
and β =
−0.7minima,
(the solid
lines in Fig.with
12),Fig.
with
less
density
in agreement
3a
small local and
perturbation
the plasma
near the obtained
electron
ofa Bujarbarua
Schamel of
(1981).
The potentials
perturbation
consisted
a Maxwellian
inhole.
Fig. The
12 are
used to construct
theofnumerical
initialpopulacondition of
added
to the initial
condition
for the elections
forelectrons
the electron
distribution
function
of electron
holes
tron hole, with
the same 1981),
temperature
as theinbackground
(Bujarbarua
and Schamel,
to be used
our Vlasov
simulations including the ion dynamics.
ωpe t
√
E/ 4πn0 Te
φ
ωpe t
x/rD
x/rD
Fig.
density (up(upFig.13.
13.The
Theelectron
electrondensity
density (upper
(upper left panel), the ion density
per
potential
perright
rightpanel),
panel),the
theelectric
electric field
field (lower
(lower left panel) and the potential
(lower
chosen
(lowerright
rightpanel)
panel)of
ofan
aninitially
initially standing
standing electron hole. The chosen
trapping
trappingparameter
parameterisis ββ =
= −0.7.
−0.7.
fixed ion background, viz. Ni = 1. We note that larger valIn of
Fig.
timemaxima
development
of potential
an elecues
Me13andwe|β|show
give the
smaller
ψ of the
tron
hole deep
initially
at rest.
the initial
conditionwith
for
and less
electron
densityAsminima,
in agreement
the
distributionand
function,
use the
Fig.electron
3(a) of Bujarbarua
Schamelwe
(1981).
Theparameters
potentials
M
β 12
= are
−0.7
(thetosolid
lines the
in Fig.
12), with
e = 0 and
obtained
in Fig.
used
construct
numerical
iniatial
small
local perturbation
of the
plasma near
the electron
conditions
for the electron
distribution
function
of elechole.
The (Bujarbarua
perturbationand
consisted
of 1981),
a Maxwellian
tron holes
Schamel,
to be usedpopulain our
tion
of electrons
added
to thethe
initial
condition for the elecVlasov
simulations
including
ion dynamics.
tronInhole,
the same
as the background
elecFig.with
13 we
show temperature
the time development
of an electrons
the density
of thecondition
form δNe for
=
tron and
holewith
initially
at rest.perturbation
As the initial
2
−0.008
sinh(x/2r
(x/2rD ).we
Theuse
most
feaD )/ coshfunction,
the electron
distribution
thestriking
parameters
−1
ture,
time
130 ω(the
that lines
the electron
Me seen
= 0 atand
β t=≈ −0.7
solid
in Fig. hole
12), “sudwith
pe , is
denly”
in the negative
x direction
a Mach
a smallstarts
localmoving
perturbation
of the plasma
near with
the electron
number
Me perturbation
≈ 0.55. (The
direction
the propagation
dehole. The
consisted
of of
a Maxwellian
populapends
the perturbation
theinitial
initialcondition
condition.)
trantion ofonelectrons
added tointhe
for The
the elecsition
seemswith
to happen
when
the ion density
formed a
tron hole,
the same
temperature
as thehas
background
deep enough cavity. When the electron hole has escaped
the ion cavity, the ion density cavity continues to deepen
and an electron density cavity is created at the same place,
neutralizing the plasma. In Fig. 14, we have repeated the
same numerical experiment as in Fig. 13, including the local perturbation of the plasma, but with β = −0.5, making
the electron hole larger. In this case, the electron hole starts
−1 , also with
moving in the positive x direction at t ≈ 100 ωpe
a Mach number Me ≈ 0.55. For both the β = −0.7 and
β = −0.5 cases, the potential maximum decreases slightly
during the transition from standing to moving electron holes,
from ψ = 4.5 to ψ = 3.8 and from ψ = 7.8 to ψ = 7.0, respectively. For Me = 0.55, the diagram in Fig. 14a of Bujarbarua and Schamel (1981) predicts that the trapping parameter has changed to β ≈ −0.67 and β ≈ −0.45, respectively,
for the two cases. We next analyze the distribution of energy in the system. The total energy is conserved exactly in
the continuous system and to a high degree in our numerical
simulations (Eliasson, 2001), and is distributed between the
12
280
12
B.Eliasson
Eliassonand
andP.P.
P.K.
K.Shukla:
Shukla:The
electron
and
ion
holes
B.
Eliasson
and
K.
Shukla:
Thedynamics
dynamicsofofelectron
electron
and
ion
holes
B.
and
ion
holes
NN
ee
NN
i i
2.5
2.5
6.5
6.5
β=−0.7
β=−0.7
β=−0.5
β=−0.5
66
WW
pot pot
W
pot
Wpot
4πn
4πn
0 T0eTe
ωωpepet t
W
W
5.55.5
Wpot
pot
Wpot
pot
4πn
4πn
0 Te
0 Te
22
55
1.5
1.5
00
√√
E/ 4πn
4πn
E/
0T
0T
ee
−4 −4
x 10
x 10
55
β=−0.7
β=−0.7
00
φφ
φ φmin
min
φmin
φmin
ωωpepet t
x/r
x/r
DD
Fig.
14.
electron
ion
density
(upFig.
14.
The
electron
density
(upper
Fig.
14.The
The
electrondensity
density(upper
(upperleft
leftpanel),
panel),the
theion
iondensity
density(up(upper
right
panel),
and
the
potential
per
right
panel),
the
electric
field
(lower
per
right
panel),the
theelectric
electricfield
field(lower
(lowerleft
leftpanel)
panel)and
andthe
thepotential
potential
(lower
right
panel)
hole.
The
cho(lower
right
panel)
ofofan
an
initially
stationary
(lower
right
panel)of
aninitially
initiallystationary
stationaryelectron
electronhole.
hole.The
Thechochosen
trapping
parameter
sen
trapping
parameter
==−0.5.
−0.5.
sen
trapping
parameterisisisβββ=
−0.5.
kinetic energy of the particles and the potential energy stored
electrons and
with
and
with the
the density
density perturbation
perturbation ofof the
the form
form
inelectrons
the electric
field.
22
δN
−0.008
sinh(x/2r
cosh
(x/2r
).). The
most
=
−0.008
sinh(x/2r
)/
cosh
(x/2r
The
most
δN
ee =
DD)/
DD
In the upper panel of Fig.
we have−1
plotted the potenR 15,
striking
t t≈≈130
thatthe
theelecelecstrikingfeature,
feature,seen
seenatattime
time
130ωω −1, ,isisthat
tial energy Wpot = (1/2) E 2 dx of thepepe
system for the two
tron
tronhole
hole“suddenly”
“suddenly”starts
startsmoving
movingininthe
thenegative
negativexxdirecdirecsimulation runs, as a function of time. Here, most of the
tion
0.55.(The
(Thedirection
directionofofthe
the
tionwith
withaaMach
Machnumber
numberM
Me ≈≈0.55.
potential energy is stored in thee large-amplitude bipolar elecpropagation
propagationdepends
dependson
onthe
theperturbation
perturbationininthe
theinitial
initialcondiconditric
field of transition
the electron hole,
while some
potential
energy
tion.)
tion.) The
The transitionseems
seemstotohappen
happenwhen
whenthe
theion
iondensity
density
ishas
released
in
high-frequency
Langmuir
waves.
We
observe
hasformed
formedaadeep
deepenough
enoughcavity.
cavity. When
Whenthe
theelectron
electronhole
hole
ahas
gradual decrease
of the potential
energy till the
transition
hasescaped
escapedthe
theion
ioncavity,
cavity,the
theion
iondensity
densitycavity
cavitycontinues
continuestoto
time
when the
electron hole
leaves the is
ion cavity at
and the
podeepen
same
deepenand
andan
anelectron
electrondensity
densitycavity
cavity iscreated
created atthe
the−1
same
tential
energy
performs
large
fluctuations
(t
≈
130
ω
pe for
place,
place,neutralizing
neutralizingthe
theplasma.
plasma.InInFig.
Fig.14,
14,we
wehave
haverepeated
repeated
−1
the
= −0.7
case and
t ≈ 100 ωas
for
the13,
β
−0.5 case),
the
numerical
experiment
Fig.
including
the
pe
theβsame
same
numerical
experiment
asinin
Fig.
13,=
including
the
whereafter
the potential
performs
oslocal
ofofthe
plasma,
but
ββ==−0.5,
localperturbation
perturbation
theenergy
plasma,
butwith
withhigh-frequency
−0.5,making
making
cillations
attributed
to Langmuir
waves
released
thestarts
tranthe
hole
InInthis
the
electron
hole
theelectron
electron
holelarger.
larger.
thiscase,
case,
the
electronin
hole
starts
−1
−1
sition,
around
somewhat
smalleratconstant
W
attributed
,
also
with
moving
ininthe
xxdirection
ωωpe
pot
,
also
with
moving
theapositive
positive
direction
att t≈≈100
100
pe
Mach
number
both
−0.7
toaathe
propagating
electron
holeFor
bipolar
electric
The
ee ≈
≈ 0.55.
0.55.
For
boththe
theββ ==field.
−0.7and
and
Mach
numberM
M
ββ ==decrease
−0.5
the
potential
maximum
decreases
slightly
slow
of the
potential
energy
in the
initial phase
in−0.5cases,
cases,
the
potential
maximum
decreases
slightly
during
the
from
standing
moving
holes,
dicates
the
positive
electrostatic
potential
ofelectron
the electron
duringthat
thetransition
transition
from
standingtoto
movingelectron
holes,
from
ψψ == 4.5
==3.8
from
ψψ=
==elec7.0,
hole
accelerates
the
ions
which
leave
vicinity
ofψψ
the
from
4.5to
toψψ
3.8and
and
fromthe
=7.8
7.8toto
7.0,
respectively.
For
M
=
0.55,
the
diagram
in
Fig.
14(a)
tron
hole.
In
a
homogeneous
ion
background,
the
single,
respectively. For Mee = 0.55, the diagram in Fig. 14(a)ofof
Bujarbarua
and
predicts
the
steady-state
hole is(1981)
associated
withthat
athat
positive
potenBujarbaruaelectron
and Schamel
Schamel
(1981)
predicts
thetrapping
trapping
parameter
has
changed
to
β
≈
−0.67
and
β
≈
−0.45,
tial
which
traps
electrons.
Since
the
positive
potential
of rethe
parameter has changed to β ≈ −0.67 and β ≈ −0.45,
respectively,
cases.
We
electron
holefor
andthe
thetwo
negative
ofanalyze
the ion the
cavity
are
spectively,
for
the
two
cases.potential
Wenext
next
analyze
thedistridistribution
The
total
isisconbution ofof energy
energy ininthe
thesystem.
system.
The
totalenergy
energy
concompeting
processes,
there
could be
a problem
of the
exisserved
ininthe
continuous
system
totoaahigh
served
the
continuous
system
and
highdedetence
of exactly
aexactly
stationary
electron
hole if
the ionand
density
becomes
gree
ininour
simulations
(Eliasson,
2001),
and
isis
greeenough.
ournumerical
numerical
simulations
(Eliasson,
2001),
and
deep
In
our case,
the electron
hole remains
stable
but
distributed
between
energy
and
distributed
between
the
kinetic
energy
theparticles
particles
and
escapes
the ion
cavity.the
In kinetic
the
lower
panelofofthe
Fig.
15, we have
the
energy
ininthe
electric
field.
thepotential
potential
energystored
stored
thein
electric
field. In
Inthe
theupuptaken
the ion density
N
our Vlasov
simulation
i obtained
per
panel
of
Fig.
15,
we
have
plotted
the
potential
energy
panel to
of Poisson’s
Fig.
have plotted
RR 215, we
asper
an input
equation,
which the
we potential
then solveenergy
with
W
= (1/2)
EE 2given
dx
the
system
the
two
dxofofby
theEq.
system
for
thepotential
twosimulation
simulation
Wpot
pot = (1/2)
the
electron
density
(26).for
The
of the
runs,
as
a
function
of
time.
Here,
most
of
the
potential
enruns, ashole
a function
of time.
Here,
most of the
potential
enelectron
is normally
positive
everywhere
when
the ions
ergy
is
stored
in
the
large-amplitude
bipolar
electric
field
is stored
in the large-amplitude
electric
fieldofof
areergy
fixed
background.
In the presence bipolar
of a local
ion density
the
hole,
some
energy isisreleased
theelectron
electron
hole,while
while
somepotential
releasedinin
cavity,
the potential
may have
apotential
slightly energy
negative minimum,
high-frequency
Langmuir
waves.
We
observe
a
gradual
high-frequency
Langmuir
waves.
We observe
a gradualdedeand
at this point we
prescribe
a Maxwellian
distribution
for
the untrapped electrons in the same manner as often done by
200
200
300
300
4.54.5
0 0
ωtpe t
ωpe
100
100
−3 −3
x 10
x 10
11
β=−0.5
β=−0.5
00
−5−5
φ φmin −1−1
−10
−10
−2−2
−15
−15
00
x/r
x/r
DD
100
100
200
200
300
300
ωpeωtpe t
φmin
φmin
min
100
100
200
200
300
300
−3−3
0 0
100
100
200
200
300
300
t ωtωt
ωt pe
ωt tpe t
pe pe t
R RR 2 2
2dx
Fig.
asas
a aa
Fig.15.
15. The
Thetotal
potentialenergy
energy
(1/2)
Fig.
15.
The
totalpotential
energyWW
W
=(1/2)
(1/2) EE
Edx
dx
as
pot
pot
pot==
function
ofof
the
electrostatic
functionofof
oftime
time(upper
(upperpanels)
andthe
theminimum
minimum
the
electrostatic
function
time
(upper
panels)and
minimum
of
the
electrostatic
potential
panels)
and
β ββ
===
−0.5
potential(lower
(lowerpanels)
panels)for
forβ β==−0.7
−0.7
(left
panels)
and
−0.5
potential
(lower
−0.7(left
(left
panels)
and
−0.5
(right
electron
(rightpanels),
panels),obtained
obtainedfrom
themodel
modelforfor
stationary
electron
(right
panels),
obtained
fromthe
fora aastationary
stationary
electron
hole
where
the
ion
density
holeinin
inthe
thepresence
presenceofofa avarying
iondensity,
density,
where
the
ion
density
hole
the
presence
varyingion
density,
where
the
ion
density
isisisobtained
obtainedfrom
fromthe
theVlasov
Vlasovsimulations.
simulations.
obtained
from
the
crease
energy
time
creaseofofthe
thepotential
potential
energy
tillthe
thetransition
transition
timewhen
whenthe
the
Schamel
(1971,
1986).
Wetill
plotted
the potential
minimum
electron
hole
leaves
the
ion
cavity
and
the
potential
energy
electron
hole
leaves
the
ion
cavity
and
the
potential
energy
φmin as a function of time, and found
that at t ≈ 170 ω−1
−1
−1for the β = −0.7pe
performs
large
fluctuations
(t
≈
130
ω
the β =that
−0.7
performs
large
fluctuations
(t
≈
130
ω
pe
pe
a part of the potential
becomes negative. for
It seems
the
−1
−1for the β = −0.5 case), whereafter
case
and
t
≈
100
ω
for
the
β
=
−0.5
case),
whereafter
case
and
t
≈
100
ω
pe
pe
acceleration of the electron hole occurs at approximately the
the
energy performs
high-frequency
oscillations
at-atthepotential
potential
performs
high-frequency
oscillations
same
time as energy
a part of
the potential
for the theoretical
model
tributed
to
Langmuir
waves
released
in
the
transition,
around
tributed tonegative.
LangmuirAn
waves
releasedway
in the
becomes
alternative
of transition,
explainingaround
accela asomewhat
smaller
constant
WW
attributed
totothe
attributed
thepropapropasomewhat
smaller
constant
pot
pot
erating electron holes is as follows: The positive
electron
gating
electron
electric
field.
slow decrease
gating
electronhole
holebipolar
bipolar
electric
field.The
The
decrease
hole
potential
starts
to
reflect
low energy
ionsslow
giving
rise to
ofofthe
the
thepotential
potentialenergy
energyininthe
theinitial
initialphase
phaseindicates
indicatesthat
a local reduction of Ni . After a while, at the centerthat
of the
the
positive
the
electron
hole
accelerpositiveelectrostatic
electrostaticpotential
potentialof
of
the
electron
hole
accelerelectron hole, the condition φ 00 (x) = Ne (x) − Ni (x) < 0
ates
atesthe
theions
ionswhich
whichleave
leavethe
thevicinity
vicinityofofthe
theelectron
electronhole.
hole.
fora xhomogeneous
near zero is no longer
met and the
standing structure
In
In a homogeneousion
ionbackground,
background,the
thesingle,
single,steady-state
steady-state
ceases tohole
exist. Then the with
electron hole ispotential
accelerated such
electron
electron holeisisassociated
associated witha apositive
positive potentialwhich
which
that
a
free
ion
component
exists,
giving
rise
to
a free
ion dentraps
trapselectrons.
electrons. Since
Sincethe
thepositive
positivepotential
potentialofofthe
theelectron
electron
sity and
Nif (x)
necessarypotential
for the maintenance
of the competinequality
hole
the
hole and
thenegative
negative potentialofofthe
theion
ioncavity
cavityare
are competnearprocesses,
the electron
hole
center.
Figure
13 of
indicates
that indeed
ing
there
could
be
a
problem
the
existence
ing processes, there could be a problem of the existenceofof
(x) = Nifelectron
(x) ≈ 1 at the
location
of the propagating
eleci stationary
aN
astationary
electronhole
holeififthe
theion
iondensity
densitybecomes
becomesdeep
deep
tron
hole.
enough.
In
our
case,
the
electron
hole
remains
stable
but
esenough. In our case, the electron hole remains stable but esFinally,
wecavity.
have In
studied
interactions
two
eleccapes
the
panel
Fig.
capes
theion
ion
cavity.
Inthe
thelower
lower
panelofofbetween
Fig.15,
15,we
wehave
have
tron
holes
with
each
other
and
with
ions
in
a
longer
simutaken
the
ion
density
N
obtained
in
our
Vlasov
simulation
i
taken the ion density Ni obtained in our Vlasov simulation
16 and
17. Thewhich
electron
holes
withwith
β =
lation;
see toFigs.
as
equation,
we
asananinput
input
toPoisson’s
Poisson’s
equation,
which
wethen
thensolve
solve
with
the
given
bybyinitially
Eq.
The
−0.5
and βdensity
= −0.7
were
placed
at
x = of
−40
rD
theelectron
electron
density
given
Eq.(27).
(27).
Thepotential
potential
ofthe
the
electron
isisrnormally
positive
when
ions
and
x hole
=hole
40
, respectively.
A local electron
density
D
electron
normally
positiveeverywhere
everywhere
whenthe
the
ions
are
InInthe
presence
ofofa alocal
perturbation
was taken
tothe
be
Maxwellian
withion
thedensity
density
arefixed
fixedbackground.
background.
presence
local
ion
density
2
cavity,
a aslightly
negative
δN
−0.08{sinh[(x/r
+40)/2]/
cosh
[(x/rDminimum,
+40)/2]+
cavity,
thepotential
potentialmay
mayhave
slightly
negative
minimum,
e =the
Dhave
and
a aMaxwellian
distribution
for
sinh[(x/r
−40)/2]/
cosh2 [(x/r
i.e. the same
andatatthis
this
pointwe
weprescribe
prescribe
Maxwellian
distribution
for
Dpoint
D +−40)/2]},
the
untrapped
electrons
asasoften
perturbations
in the in
single-hole
cases,
centered
atdone
theby
two
the untrappedas
electrons
inthe
thesame
samemanner
manner
oftendone
by
Schamel
We
plotted
minimum
electron
holes.
Here,
holes
also creates
local
Schamel(1971,
(1971,1986).
1986).the
Weelectron
plottedthe
thepotential
potential
minimum
−1
−1
φion
asasa afunction
time,
and
found
that
atatt t≈≈the
170
ωpe
min
density
cavities,ofof
and
after
some
time
escapes
density
function
time,
and
found
that
170
ωpe
φ
min
acavities
part
of
the
potential
becomes
negative.
It
seems
that
the
the potential
same times
as in thenegative.
single electron
hole
cases.
a part ofat the
becomes
It seems
that
the
acceleration
ofofthe
electron
atatapproximately
the
acceleration
the
electron
hole
occurs
approximately
the
This
can clearly
be
seen inhole
Fig.occurs
16, where
the two electron
−1
−1
same
a apart
for
model
sametime
time
partofat
ofthe
the
forthe
thetheoretical
model
holes
startasas
moving
t potential
≈potential
100 ωpe
and
ttheoretical
≈ 130 ω
,
pe rebecomes
An alternative
way
accelbecomesnegative.
negative.
wayof
ofexplaining
explaining
accelspectively.
At t ≈ An
170alternative
ω−1 , the two
electron
holes collide
pe
inelastically and merge into a new electron hole, in accor-
B.
Eliasson
and
P.P.P.
K.
Shukla:
electron
and
ion
holes
B.B.
Eliasson
and
K.K.
Shukla:
Eliasson
and
Shukla:The
Thedynamics
dynamicsofofelectron
electronand
andion
ionholes
holes
NeN
e
281
13
13
−1
t = 0 ωpe
−1
t = 0 ωpe
NiN
i
−1
t = 155 ωpe
−1
t = 155 ωpe
v/V
Te
v/V
Te
ωpeωt t
pe
√√
E/E/4πn
0 TeT
4πn
0 e
φφ
ωpeωt t
pe
−1
t = 175 ωpe
−1
t = 175 ωpe
−1
t = 251 ωpe
−1
t = 251 ωpe
−1
t = 461 ωpe
−1
t = 461 ωpe
−1
t = 576 ωpe
−1
t = 576 ωpe
v/V
Te
v/V
Te
v/V
Te
v/V
Te
x/r
D
x/r
D
x/r
D
x/r
D
Fig.
16.16.The
electron
density
(upper
left panel), the ion
density
Fig.
The
Fig.16.
Theelectron
electrondensity
density(upper
(upper left panel),thetheion
iondensity
density
(upper
right
panel),
the
electric
field
(lower
left
panel),
and
the
po(upper
(upperright
rightpanel),
panel),thetheelectric
electricfield
field (lower leftpanel),
panel),and
andthe
thepopotential
(lower
right
panel)
of
two
electron
initially
placed
tential
(lower
right
panel)
of
two
holes
initially
placed
tential (lower right panel) of two electron holes initially placedatatat
x/r
= −0.5
−0.5 for the
the left
left elecelecx/r
=±40.
±40. The
Thetrapping
trapping parameter
parameter β =
DD=
x/r
D = ±40. The trapping parameter β = −0.5 for the left electron
hole
initially
placed
at
==−0.7
−0.7
for
the
tron
hole
initially
placed
atatx/r
x/r
==−40,
−40,
D D=
D
tron
hole
initially
placed
x/r
−40,and
andβ β=
−0.7for
forthe
the
right
electron
hole
placed
at
x/r
=
40.
right
electron
hole
placed
at
x/r
=
40.
D
D D = 40.
right electron hole placed at x/r
dance
with
the results
of
(1994),
whereafter
the
erating
electron
holes
isisMatsumoto
asasfollows:
The
positive
electron
erating
electron
holes
follows:
The
positive
electron
single
electron
hole
propagates
slightly
in
the
positive
x
dihole
holepotential
potentialstarts
startstotoreflect
reflectlow
lowenergy
energyions
ionsgiving
givingrise
risetoto
rection,
and
becomes
trapped
at
a
local
ion
density
maximum
a alocal
i . i .After
localreduction
reductionofofNN
Aftera awhile,
while,atatthe
thecenter
centerofofthe
the
00 00 panel of Fig. 16 for the ion
atelectron
x ≈ 30hole,
rD ; see
the
upper
right
the
condition
φ
(x)
=
N
(x)
−
N
< 0
e
i (x)
electron hole, the condition φ (x) = Ne (x) − N
i (x) < 0
density
and zero
the lower right panel
for the
the electron structure
hole pofor
met
forx xnear
near zeroisisnonolonger
longer
metand
and thestanding
standing structure
−1 , a new
tential.
After
t
≈
400
ω
ion
density
cavity
is crepe
ceases
ceasestotoexist.
exist.Then
Thenthe
theelectron
electronhole
holeisisaccelerated
acceleratedsuch
such
ated
where
the
electron
hole
is
centered,
and
at
this
time
the
that
thata afree
freeion
ioncomponent
componentexists,
exists,giving
givingrise
risetotoa afree
freeion
iondendenelectron
hole
is
again
accelerated
in
the
negative
x
direction.
sity
(x)
ifif
sityNN
(x)necessary
necessaryfor
forthe
themaintenance
maintenanceofofthe
theinequality
inequality
−1
At
t ≈the
480
ωpe
, the
moving
hole
again that
encounters
near
electron
hole
center.
Figure
indicates
near
the
electron
hole
center.electron
Figure1313
indicates
thatindeed
indeed
an
density
maximum
located
at xofof
≈
−30
rD , where
the
NN
(x)
= Nifif
(x)
the
propagating
eleciion
(x)≈≈1 1atatthe
thelocation
location
the
propagating
eleci (x) = N
electron
hole
is
trapped,
performing
large
oscillations.
The
tron
hole.
tron hole.
electron
phase space
density interactions
is depicted in
Fig. 17:two
The iniFinally,
have
between
Finally,we
we
havestudied
studied
interactions
betweenholes
twoelecelectial
condition
(upper
left
panel),
the
two
electron
havtron
tronholes
holeswith
witheach
eachother
otherand
andwith
withions
ionsinina alonger
longersimsiming
started moving
(upper right
panel),electron
collisions between
ulation;
ulation;see
seeFigs.
Figs.1616and
and17.
17. The
The electronholes
holeswith
with
the
electron
holes (middlewere
left panel), theplaced
newly created
β βtwo
==−0.5
−0.5and
andβ β==−0.7
−0.7 wereinitially
initially placedatatx x==
electron
hole
trapped
at
x
=
30
r
(middle
right
panel), and
D
−40
, respectively.
−40rDrDand
andx x==4040rDrD
, respectively.AAlocal
localelectron
electrondendenthe
electron
hole
trapped
at
x
=
−30
r
(lower
panels).
We
D
sity
the
densityperturbation
perturbationwas
wastaken
takentotobebeMaxwellian
Maxwellianwith
with
the
den2 2accelerasee
that
the
electron
holes
remain
stable
during
the
sity
e e==−0.08{sinh[(x/r
DD++40)/2]/
DD++
sityδN
δN
−0.08{sinh[(x/r
40)/2]/cosh
cosh[(x/r
[(x/r
22
tion
by ion density cavities.
40)/2]
++40)/2]/
++40)/2]},
DD
DD
40)/2]++sinh[(x/r
sinh[(x/r
40)/2]/cosh
cosh[(x/r
[(x/r
40)/2]},i.e.
i.e.
The
numerical
solutions
of
the
Vlasov-Poisson
system
the
perturbations
asasinin
the
single-hole
cases,
thesame
same
perturbations
the
single-hole
cases,centered
centered
were
performed
with
a Fourier
method
(Eliasson,
2001).
atatthe
two
holes.
Here,
holes
crethe
twoelectron
electron
holes.
Here,the
theelectron
electron
holesalso
also
creWe
used
500
intervals
in
x
space
with
the
domain
−80
≤
ates
local
ion
density
cavities,
and
after
some
time
escapes
ates local ion density cavities, and after some time escapes
x/r
≤
80,
and
300
intervals
in
velocity
space.
The
electron
the
density
cavities
at
the
same
times
as
in
the
single
electron
D
the density cavities at the same times as in the single electron
hole
This
can
bebeseen
Fig.
velocity
interval
set
to −15.7
≤inin
v/V
≤where
15.7 the
and
the
T16,
e 16,
holecases.
cases.
Thiswas
canclearly
clearly
seen
Fig.
where
thetwo
two
−1
−1
−1
−1
and
t
≈
130
ω
, ,
electron
holes
start
moving
at
t
≈
100
ω
ion
velocity
interval
was
set
to
−0.118
≤
v/V
≤
0.118.
T
e
ωpe
electron holes start moving at t ≈ 100 pe
ωpe and t ≈ 130 pe
−1
−1
−1
respectively.
At
t
≈
170
ω
,
the
two
electron
holes
collide
The
time-step
1t
≈
0.013
ω
was
adapted
dynamically
to
pe
respectively. At t ≈ 170 ωpe
pe , the two electron holes collide
inelastically
and
into
maintain
numerical
stability.
inelastically
andmerge
merge
intoa anew
newelectron
electronhole,
hole,ininaccoraccordance
dancewith
withthe
theresults
resultsofofMatsumoto
Matsumoto(1994),
(1994),whereafter
whereafterthe
the
single
singleelectron
electronhole
holepropagates
propagatesslightly
slightlyininthe
thepositive
positivex xdi-diand
trapped
rection,
andbecomes
becomes
trapped
a localion
iondensity
densitymaximum
maximum
5rection,
Relativistic
electron
holesatata local
atatx x≈≈3030rDrD
; see
; seethe
theupper
upperright
rightpanel
panelofofFig.
Fig.1616for
forthe
theion
ion
density
and
the
lower
right
panel
for
the
electron
hole
poand the
lowerwe
right
panel afortheory
the electron
poIn density
the present
section,
develop
for thehole
steady
state electron holes in a relativistic plasma. We find that the
x/r
D
x/r
D
x/r
D
x/r
D
Fig.
17.
distribution
forfor
two
electron
holes
atatat
t t=
Fig.
17.
Theelectron
electron
distribution
for
two
electron
holes
t=
=
Fig.
17.The
The
electron
distribution
two
electron
holes
−1
−1
−1
−1
−1(upper
−1(upper
00ω
left
panel),
t tt==
155
t t=
pepe
(upper
left
panel),
155ωω
(upperright
rightpanel),
panel),
t=
=
0ωpe
ωpe
(upper
left
panel),
=
right
panel),
pe
−1−1
−1−1
(middle
left
panel),
t=
ωpe
(middle
right
panel), t =
175
ωpe
−1
−1
(middle
left
panel),
=251
251
ωpe
(middle
right
175
ωpe
pe −1
pe (middle
175
ω−1
left
panel),
tt =
251
ω
(middle
right panel),
panel),tt =
=
−1
−1
(lower
left
panel)
and
t t==576
ωpe
(lower
right
461
ωpe
(lower
left
panel)
and
576
ω
(lower
rightpanel).
panel).
461
ω−1
−1
pe
pe
461the
ωpe
(lower left
panel)etc.
and tFig.
= 16.
576Initially,
ωpe (lower
right
panel).
See
associated
densities
thethe
left
electron
See the
associated
densities
etc.inin
Fig.
16. Initially,
left
electron
See (placed
the associated
densities
etc.
ina Fig.
16. Initially,
theβleft
electron
hole
at
x/r
=
−40)
has
trapping
parameter
=
−0.5,
D
hole (placed at x/rD = −40) has a trapping parameter β = −0.5,
hole (placed
at x/rD =
−40) has aattrapping
parameter
β = −0.5,
while
==
40)
DD
whilethetheright
rightelectron
electronhole
hole(placed
(placed atx/r
x/r
40)has
hasβ β==−0.7.
−0.7.
while the right electron hole (placed at x/rD
= 40) has β = −0.7.
−1
−1
, a, anew
tential.
newion
iondensity
densitycavity
cavityisiscrecretential.After
Aftert t≈≈400
400ωpe
ωpe
properties
of the
electron
holes
are dramatically
changed
by
ated
where
the
electron
hole
is
centered,
and
at
this
ated where the electron hole is centered, and at thistime
timethe
the
the relativistic
mass accelerated
increase of in
thethe
electrons.
Our
relativiselectron
hole
is
again
negative
x
direction.
electron hole is again accelerated in the negative x direction.
−1
tic
simulation
studies
show that
strongly
relativistic
electron
−1
At
, the
Att t≈≈480
480ωpe
ωpe
, themoving
movingelectron
electronhole
holeagain
againencounters
encounters
holes
are
long-lived,
and
that
they
survive
head-on
collisions.
an
ion
density
maximum
located
at
x
≈
−30
r
,
where
D
an ion density maximum located at x ≈ −30 rD , wherethe
the
We
expect
that
electron
hole large
potentials
can accelerate
electron
hole
isisintense
trapped,
performing
The
electron
hole
trapped,
performing
largeoscillations.
oscillations.
The
electrons
to extremely
high energies.
electron
space
isisdepicted
electronphase
phase
spacedensity
density
depictedininFig.
Fig.17:
17:The
Theiniinitial
condition
(upper
left
panel),
the
two
electron
holes
havThe
dimensionless
relativistic
Vlasov-Poisson
system,
tial condition (upper left panel), the two electron holes
having
(upper
collisions
between
where
ionsmoving
are
assumed
toright
form
apanel),
neutralizing
background,
ingstarted
started
moving
(upper
rightpanel),
collisions
between
the
isthetwo
twoelectron
electronholes
holes(middle
(middleleft
leftpanel),
panel),the
thenewly
newlycreated
created
electron
electronhole
holetrapped
trappedatatx x==3030rDrD(middle
(middleright
rightpanel),
panel),and
and
∂feelectron hole
p trapped
∂φ=∂f−30
∂fe at x
e rD (lower panels). We
the
the +
electron
hole
trapped
at
x
=
−30
r
(lower
panels).
We
p
+
= 0,D
(28)
∂t
∂x
∂x ∂pstable
see
1 electron
+electron
µ2 p 2 holes
seethat
thatthe
the
holesremain
remain
stableduring
duringthe
theacceleraacceleration
tionbybyion
iondensity
densitycavities.
cavities.
and
The
Thenumerical
numericalsolutions
solutionsofofthe
theVlasov-Poisson
Vlasov-Poissonsystem
system
were
performed
with
a aFourier
method
(Eliasson,
2001).
Z
were
performed
with
Fourier
method
(Eliasson,
2001).
∞
∂ 2 φused 500
We
intervals
ininx xspace
with
the
domain
−40
≤≤
We used
500f
intervals
space
with
the
domain
−40
=
dp
−
1,
(29)
2 ≤ 40, ande 300 intervals in velocity space. The electron
∂x
x/r
DD ≤ −∞
x/r
40, and 300 intervals in velocity space. The electron
velocity
T eT e≤≤15.7
velocityinterval
intervalwas
wassetsettoto−15.7
−15.7≤≤v/V
v/V
15.7and
andthe
the
where
µ =interval
VT e /c,was
thesetdistribution
function
f≤
been
e has
ion
velocity
to
−0.118
≤
v/V
0.118.
T
e
ion velocity interval was set to −0.118
≤
v/V
≤
0.118.
T
e
−1 , x by r , the momennormalized
by n0≈/m
, −1
t −1
bywas
ωpe
e VT eω
D
adapted
The
was
adapteddynamically
dynamicallytoto
Thetimestep
timestep∆t
∆t ≈0.013
0.013 pe
ωpe
tum
p
by
m
V
,
and
φ
by
T
/e.
Here, c is the speed of
e
T
e
e
maintain
numerical
stability.
maintain numerical stability.
light in vacuum and the nonrelativistic plasma frequency is
ωpe = (4πn0 e2 /me )1/2 . We (the observer) remain in the
of the bulk plasma
while the electron hole is moving
5frame
5 Relativistic
Relativisticelectron
electronholes
holes
with the speed u0 . Accordingly, we look for solutions of the
form
fpresent
(p,
ξ ) and
φ(ξ ),wewe
where
ξ =axatheory
−theory
Me t,forfor
Mthe
= steady
usteady
e the
0 /VT e
In
section,
develop
Inthe
the
present
section,
develop
is
the
Mach
number
and
u
is
the
constant
propagation
speed
state
plasma.
the
0
stateelectron
electronholes
holesinina arelativistic
relativistic
plasma.We
Wefind
findthat
that
the
of
the
electron
hole.
With
this
ansatz,
the
Vlasov-Poisson
properties
propertiesofofthe
theelectron
electronholes
holesare
aredramatically
dramaticallychanged
changedbyby
system
(28) and
(29)increase
takes the
the
mass
ofofform
the
therelativistic
relativistic
mass
increase
theelectrons.
electrons.Our
Ourrelativisrelativisticticsimulation
studies
show
that
simulation studies show
stronglyrelativistic
relativisticelectron
electron
! thatstrongly
holes
survive
∂fethey
p and
∂fe head-on
holesare
arelong-lived,
long-lived,
andthat
that
theydφ
survive
head-oncollisions.
collisions.
+
= 0,
(30)
−Me + p
dξ ∂p
1 + µ2 p 2 ∂ξ
282
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
and
e± (0) = γ0 Me , and the value of the distriE = 0+ , we have p
bution function is
γ0 − 1
e
fe |E =0+ = f0 (0) = f0 (γ0 Me ) = a0 exp −
, (37)
µ2
d 2φ
=
dξ 2
Z
∞
fe dp − 1.
(31)
−∞
Integrating Eq. (30) along its trajectories, one finds the general solution of the form fe = f0 (E), where f0 is some function of one variable and
q
1
1
2
2
1+µ p −
−φ
(32)
E = −Me p + 2
γ0
µ
is a conserved quantity (energy) alongq
the particle trajectories. Here, we have denoted γ0 = 1/ 1 − µ20 Me2 . At the
electron hole, a population of electrons is 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 (de Groot et al.,
1980), which restricted to one momentum dimension takes
the form
q
1
2
2
e
f0 (p) = a0 exp − 2 ( 1 + µ p − 1) ,
(33)
µ
where the normalization constant is
nR
h
i o−1
p
∞
a0 = −∞ exp − µ12 ( 1 + µ2 p2 − 1) dp
=
µ exp(−µ−2 )
,
2K1 (µ−2 )
and K1 is a modified Bessel function of second kind. The
weakly relativistic
√ limit corresponds to µ 1 so that a0 ≈
(1 + 3µ2 /8)−1 / 2π (Gradshteyn and Ryzhik, 1965), and
Eq. (33) converges to a non-relativistic Maxwellian distribution when µ → 0. In order to impose the above mentioned
condition for the free electrons, we use the solution
fe = f0 (E) = fe0 [e
p(E)],
E > 0,
(34)
e(E) is a function of the energy such that p
e(E) → p
where p
when φ → 0. Such a function can be found with the help of
Eq. (32) by setting
q
1
1
2
2
e (E) −
e(E) + 2
1+µ p
= E.
(35)
− Me p
γ0
µ
e(E), we have
Solving for p
v
u
2
γ0 u
1
1 2 1
2
2
t
2
e± (E)=γ0 Me µ E+
p
±
µ E+
− 2 , (36)
γ0
µ
γ0
γ0
e+ (E) and p
e− (E) correspond to a modified momenwhere p
tum for free electrons on each side of the trapped electron
e(E) = p
e+ (E) and
population in momentum space. Using p
e(E) = p
e− (E) in Eq. (34), we obtain the distribution funcp
tion for the free electrons. In the limit of vanishing energy,
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”, viz.
γ0 − 1
fe = a0 exp −
−
βE
, E < 0,
(38)
µ2
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. (32). Solving for p in Eq. (32) with E = 0, we have
v
u
2
γ0 u
1
1 2 1
2
2
t
2
p± = γ0 Me µ φ+
±
µ φ+
− 2 , (39)
γ0
µ
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. (34), (36) and (38),
we can now write the full electron distribution function as

q

1
2
2

e+ (E)−1) , p>p+ ,

 a0 exp − µ2 ( 1+µ p


h
i
fe = a0 exp − γ0µ−1
p− ≤p≤p+ , (40)
2 −βE ,

q



2 (E)−1) , p<p ,

e−
 a0 exp − µ12 ( 1+µ2 p
−
where E is given by Eq. (32). Integrating the distribution
function over momentum space we obtain the total electron density as a function of φ, which is calculated selfconsistently by means of Poisson’s equation (31).
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 (31) numerically, where the integration of the trapped and free electron
populations has been done with a sum representation of the
integrals, and Eq. (31) has been solved as a nonlinear boundary value problem where the potential has been set to zero far
away from the electron hole, and the resulting nonlinear system of equations have been solved iteratively with a Newtonlike method. An alternative would be to use the so-called potential method (Schamel, 1971). In Fig. 18 we display the
electron hole Mach number Me as a function of the maximum amplitude ψ of the potential for different values of the
parameter µ. We see that for larger values of µ, 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. 18 show pure relativistic effects due to the relativistic mass increase of the electrons. In the non-relativistic
limit (µ = 0), we recover the previous results of Bujarbarua
and Schamel (1981); see their Fig. 1. The profiles of the
B.
Eliasson
ofofelectron
electron
and
ion
holes
B.
Eliassonand
andP.P.P.P.K.
K.Shukla:
Shukla:The
Thedynamics
dynamicsof
electronand
andion
ionholes
holes
B.
Eliasson
and
K.
Shukla:
The
dynamics
B.
Eliasson
and
K.
Shukla:
The
dynamics
of electron
and
ion
holes
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
283
15
15
15
15
1.4
1.4
1.4
1.4
1.2
1.2
1.2
1.2
111
1
M
MM
ee
Me
e
0.8
0.8
0.8
0.8
µ=0.4
µ=0.4
µ=0.4
µ=0.4
µ=0.3
µ=0.3
µ=0.3
µ=0.3
µ=0.2
µ=0.2
µ=0.2
µ=0.2
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4 µ=0.1
µ=0.1
µ=0.1
µ=0.1
0.2
0.2
0.2
0.2 µ=0
µ=0
µ=0
µ=0
00
55
00000
0
55
1010
10
10
1515
15
15
2020
20
20
ψψ
ψψ
2525
25
25
3030
3030
3535
3535
4040
4040
Fig.
18.
The
Mach
number
MM
ofofthe
Fig.
18.
asasaa afunction
function
Fig.
18.The
TheMach
Machnumber
numberM
theelectron
electronhole
holeas
function
Fig.
18.
The
Mach
number
Meeeeeof
ofthe
the
electron
hole
function
Fig.
18.
The
Mach
number
M
of
electron
hole
asasa afunction
the
maximum
ψ
of
the
electron
hole
potential
φ,φ,
forfor
aa
of
the
maximum
amplitude
ψ
of
the
electron
potential
ofof
the
maximum
amplitude
ψ
of
the
electron
hole
φ,
for
of the
the maximum
maximum amplitude
amplitudeψψof
ofthe
theelectron
electronhole
holepotential
potentialφ,φ,
for
of
for
aa
trapping
parameter
βββ==
−0.5
and
for
different
values
of
constant
trapping
parameter
=
−0.5
and
for
different
values
of
aconstant
constant
trapping
parameter
−0.5
and
for
different
values
of
constant trapping
trapping parameter
parameterββ == −0.5
−0.5and
andfor
fordifferent
differentvalues
valuesofof
constant
µ.µ.
µ.
µ.
µ.
4040
40
40
a)a)
a)
φ φ 2020 a)
φφ 20
20
00
−50
00−50
−50
−50
10
10
10 b)
10
b)
φ φ 5 b)
b)
φφ 555
00
−50
00−50
−50
−50
20
20
20
20 c)
c)
φ φ 1010 c)
c)
φφ 10
10
00
−50
00−50
−50
−50
1010
10
10 d)
d)
φ φ 5 d)
5 d)
φφ
55
00
−50
00−50
−50
−50
00
00
00
00
00
00
00
ξ00ξ
ξξ
1.2
1.2a)a)
1.2
1.2
1 a)a)
N N 111
e e0.8
N
N e 0.8
e 0.8
0.8
0.6
0.6
5050
−50
0.6
0.6
−50
50
−50
50
−50
1.2
1.2 b)b)
1.2 b)b)
1.2
NeN 1 1
11
NNe e0.8
0.8
e 0.8
0.8
0.6
0.6
5050
−50
0.6
0.6
−50
50
−50
50
−50
1.2
1.2 c)c)
1.2
1.2
1 c)c)
NN 1 1
1
NNe e0.8
0.8
ee 0.8
0.8
0.6
0.6
5050
−50
0.6
0.6
−50
50
−50
50
−50
1.2
1.2 d)d)
1.2
1.2 d)d)
NN 1 1
e 0.8
11
NN e 0.8
ee 0.8
0.8
0.6
0.6
5050
−50
0.6
−50
0.6
50
−50
50
−50
00
00
5050
5050
00
00
5050
5050
00
00
5050
5050
00
ξ0 0ξ
ξξ
5050
5050
Fig.
Fig.19.
19.The
Thepotential
potential(left
(leftpanel)
panel)and
andelectron
electrondensity
density(right
(rightpanel)
panel)
Fig.
19.
The
potential
(left
panel)
and
electron
density
(right
panel)
Fig.
19.
potential
(left
and
density
(right
panel)
Fig.
19.The
The
potential
(leftβpanel)
panel)
andelectron
electron
density
(right
panel)
of
electron
holes
with
a):
=
−0.5,
µ
=
0.4
and
M
==0.2,
b):b):
e
ofelectron
electronholes
holeswith
witha):
a):ββ==−0.5,
−0.5,µµ==0.4
0.4and
andMM
0.2,
e
of
=
0.2,
b):
ofβofelectron
holes
a):
ββe===−0.5,
µµ =
and
==0.4
0.2,
b):
electron
holes
withand
a):M
−0.5,
= 0.4
0.4−0.5,
andM
M
0.2,
b):
e=
=
−0.5,
µµ==with
0.2
0.2,
c):
β
=
µeeµ
and
β
=
−0.5,
0.2
and
M
=
0.2,
c):
β
=
−0.5,
=
0.4
and
e= 0.2, c): β = −0.5, µ = 0.4 and
β
=
−0.5,
µ
=
0.2
and
M
e
βM
−0.5,
µ
0.2
and
MMe e=
0.2,
c):
ββM
==e−0.5,
µµ == 0.4
β=e==
−0.5,
µ==
0.2
and
==
0.2,
c):
−0.5,
0.4and
and
0.7,
and
d):
β
=
−0.7,
µ
0.4
and
=
0.2.
Me e==0.7,
0.7,and
andd):
d): β =−0.7,
−0.7, µ =0.4
0.4and
andMM
0.2.
e==
0.2.
MM
0.2.
M
0.7,and
andd):
d):βββ==
=−0.7,
−0.7,µµµ===0.4
0.4and
andM
M
0.2.
e e==0.7,
ee=
e =
where
whereEEis
isgiven
givenby
byEq.
Eq.(33).
(33). Integrating
Integratingthe
thedistribution
distribution
where
is
given
by
Eq.
(33).
Integrating
the
distribution
where EE over
is potential
given
by and
Eq. the
(33).
Integrating
the
distribution
electrostatic
electron
density
are
shown in
function
momentum
space
we
obtain
the
function over
overmomentum
momentumspace
spacewe
weobtain
obtainthe
thetotal
totalelecelecfunction
total
elecfunction
over
momentum
space
we
obtain
the
total
elecFig.
19
for
a
few
sets
of
parameters.
To
convert
the
potential
tron
trondensity
densityas
asaaafunction
functionof
ofφ,
φ,which
whichis
iscalculated
calculatedselfselftron
density
as
function
of
φ,
which
is
calculated
selftron
density
asmeans
a function
of φ,
which
is(32).
calculated
selftoconsistently
Volts,
the by
values
of of
φofshould
beequation
multiplied
by the factor
Poisson’s
consistently
by
means
Poisson’s
equation
(32).
consistently
by
means
of
Poisson’s
equation
(32).
2 ;by
consistently
means of Poisson’s
equation
(32).
5.1
× order
105 µto
accordingly,
the amplitude
of the
potential
for
In
impact
Inorder
ordertotoproceed
proceedfurther
furtherand
andto
toexplore
explorethe
the
impactof
ofthe
the
In
proceed
further
and
toto
explore
the
of
the
6impact
the
case
a)
(ψ
≈
33,
µ
=
0.4)
is
≈
2.4
×
10
Volts,
while
In
order
to
proceed
further
and
explore
the
impact
of
the
relativistic
sets
relativisticeffects
effectson
onthe
theelectron
electronholes
holesfor
fordifferent
different
setsof
of
relativistic
effects
on
the
electron
holes
for
different
sets
of
6
for
the case effects
b) (ψ
≈
µ electron
= 0.2) itholes
is ≈for
0.18
× 10numeriVolts.
relativistic
on9,the
different
sets
of
parameters,
parameters,we
wehave
havesolved
solvedPoisson’s
Poisson’sequation
equation(32)
(32)numerinumeriparameters,
we
have
solved
Poisson’s
equation
(32)
The
profiles of
electron
in phase
spacefree
is shown
in
parameters,
wethe
have
solvedholes
Poisson’s
equation
(32)
numerically,
the
cally,where
wherethe
theintegration
integrationof
thetrapped
trappedand
andfree
freeelectron
electron
cally,
where
the
integration
ofofthe
trapped
and
electron
Fig.
20
for
the
same
sets
of
parameters
as
in
Fig.
19.
We
cally,
where
the
integration
of
the
trapped
and
free
electron
populations
has
been
done
with
a
sum
representation
of
the
populationshas
hasbeen
beendone
donewith
witha asum
sumrepresentation
representationofofthe
the
populations
populations
has
been
withsolved
a sumasas
representation
of
the
integrals,
and
Eq.
(32)
has
a nonlinear
boundhave
solved
the
Vlasov-Poisson
system
(28)–(29)
and
have
integrals,
and
Eq.
(32)done
hasbeen
been
solved
nonlinear
boundintegrals,
and
Eq.
(32)
has
been
solved
as a anonlinear
boundary
problem
where
the
has
zero
integrals,
Eq. (32)
has
been
solved
ashas
a been
nonlinear
boundinvestigated
interactions
between
two large-amplitude
elecaryvalue
valueand
problem
where
thepotential
potential
beenset
setto
zero
ary
value
problem
where
the
potential
has been
set
totozero
far
away
from
the
electron
hole,
and
the
resulting
nonlinary
value
problem
where
the
potential
has
been
set
to
zero
tron
holes.
The
results
are
shown
in
Fig.
21.
The
left
hole
far
away
from
the
electron
hole,
and
the
resulting
nonlinfar away from the electron hole, and the resulting nonlinfar initially
away from
the electron
and0.7
theand
resulting
nonlinhas
the Mach
numberhole,
Me =
the right
hole
Fig.
electron
holes
with
a):a):β β
==
−0.5,
Fig.20.
20.Phase
Phasespace
spaceplots
plotsfor
for
electron
−0.5,
Fig.
20.
Phase
space
plots
for
electron
holes
with
β
=
−0.5,
Fig.
20.
Phase
space
plots
for
electron
holes
with
a):
β
=
−0.5,
Fig.
20.
Phase
space
plots
for
electron
holes
with
a):
β=
=0.2,
−0.5,
µµµ===
0.4
and
M
=
0.2,
b):
β
=
−0.5,
µ
=
0.2
and
M
c):c):
e
e
0.4
and
M
=
0.2,
b):
β
=
−0.5,
µ
=
0.2
and
M
=
0.2,
e=
e =
0.4
and
M
0.2,
b):
β
=
−0.5,
µ
=
0.2
M
0.2,
e
e
µ
=
0.4
and
M
=
0.2,
b):
β
=
−0.5,
µ
=
0.2
and
M
=
0.2,
e
e
µ
0.4
andµM=e =
0.2,
b):
β=
−0.5,
µ=
0.2
and
Me =µ0.2,
c): c):
βββ=
=
−0.5,
0.4
and
MM
==
0.7,
and
d):d):
β=β
=−0.7,
−0.7,
=0.4
0.4
e=
=
−0.5,
µ
=
0.4
and
0.7,
and
=
−0.7,
µ
=
0.4
e
=
−0.5,
µ
=
0.4
and
M
0.7,
and
d):
β
µ
=
ee =
β==−0.5,
−0.5,µ µ==0.4
0.4and
and
0.7,
and
d):
β
−0.7,
µ=
0.4
e =
β
MM
0.7,
and
d):
=in=
−0.7,
µ=
0.4and
and
M
=
0.2,
corresponding
to to
the
cases
a)d)
–βin
d)
Fig.
19.
e0.2,
and
M
=
0.2,
corresponding
the
cases
a)
–
d)
in
Fig.
19.
e
M
=
corresponding
to
the
cases
a)
–
Fig.
19.
e MM
and
0.2,corresponding
corresponding
cases
– d)
Fig.
and
==0.2,
to to
thethe
cases
a) a)
– d)
in in
Fig.
19.19.
e e
Fig.
Interactions
between
strongly
relativistic
electron
holes.
Fig.21.21.Interactions
Interactionsbetween
betweenstrongly
strongly
relativistic
electron
holes.
Fig.
relativistic
electron
holes.
Fig.
21.
Interactions
between
strongly
relativistic
electron
holes.
Fig.21.
21.the
Interactions
between
strongly
relativistic
electron
holes.
Initially,
parameters
are
µ
=
0.4
and
β
=
−0.5
for
both
holes,
Initially,thethe
parameters
are
µ
=
0.4
and
β
=
−0.5
for
both
holes,
Initially,
parameters
are
µ
=
0.4
and
β
=
−0.5
for
both
holes,
Initially,
the
parameters
are
µ
=
0.4
and
β
=
−0.5
for
both
holes,
Initially,
the
parameters
are
µ
=
0.4
and
β
=
−0.5
for
both
holes,
and
thetheleft
right
holes
have
thethe
Mach
numbers
Me = =
0.7 and
andthe
leftand
and
right
holes
have
Mach
numbers
e 0.70.7
and
left
right
holes
have
thethe
Mach
numbers
MM
=
andand
e
and
the
left
and
right
holes
have
Mach
numbers
M
= 0.7
and
the
leftand
and
right
holes
have
the
M
0.7
eeand
M
=
−0.2,
respectively,
corresponding
to
the
cases
a)
c)
inand
e
M
= −0.2,respectively,
respectively,
corresponding
to
the
cases
a)
and
c)
in
M
corresponding
toto
thethe
cases
a) a)
and
c) c)
in
e e=
M
−0.2,
respectively,
corresponding
cases
and
M
=−0.2,
−0.2,
respectively,
corresponding
to
the
cases
a)
and
c) in
e=
e
Figs.
19
and
20.
Time
intervals
are:
t
=
0
(upper
left
panel),
t
=
50
Figs.
19
and
20.
Time
intervals
are:
t=
0 (upper
left
panel),
t=
50
Figs.
19
and
20.
Time
intervals
are:
t
=
0
(upper
left
panel),
t
=
50
Figs.19
19
and20.
20. Time
Time
intervals
are: ttleft
= 00
(uppert left
left
Figs.
and
intervals
are:
=
(upper
panel),
tt = 50
(upper
right
t=
100
(middle
panel),
=
150
(middle
(upperright
rightpanel),
panel),
t =100
100
(middle
left
panel),
t=
150
(middle
(upper
t t=
(middle
left
panel),
t=
150
(middle
(upper
rightpanel),
panel),
t(lower
= 100
100
(middle
left
panel),
=
150
(middle
(upper
right
panel),
=
(middle
left
panel),
tt =
150
(middle
right
panel),
t
=
200
left
panel),
and
t
=
250
(lower
right
right panel),t =
t =200
200 (lower
left
panel),
and
t=
250
(lower
right
right
left
panel),
and
t=
250
(lower
right
rightpanel),
panel), tt =
= 200
200(lower
(lower
left
panel),
and
=
250
(lower
right
right
panel),
(lower
left
panel),
and
tt =
250
(lower
right
panel).
panel).
panel).
panel).
panel).
ear
ofofequations
have
solved
iteratively
with
a
earsystem
systemof
equationshave
havebeen
beensolved
solved
iteratively
with
ear
iteratively
with
a aa
has
asystem
negative
Mach
number
M
−0.2,
while
both
holes
earsystem
ofequations
equations
havebeen
been
solved
iteratively
with
Newton-like
method.
An
alternative
be
to
use
the
soe =would
Newton-likemethod.
method.AnAnalternative
alternative
would
Newton-like
would
bebe
to to
useuse
thethe
so-sohave
βpotential
= −0.5
and µAn
=alternative
0.4. During
head-on
collisions,
Newton-like
method.
would
be
to
the
socalled
method
(Schamel,
1971).
In
Fig.
18use
wewe
discalledpotential
potentialmethod
method(Schamel,
(Schamel,
1971).In
In
Fig.
18
discalled
1971).
Fig.
18
we
disthe
two
holes
merge
and
form
a
new
electron
hole,
which
called
potential
method
(Schamel,
1971).
In
Fig.
18
we
display
the
electron
hole
Mach
number
Me asas
a function
of thethe
playthe
theelectron
electronhole
holeMach
Mach
number
a function
e a function
play
number
MM
of of
the
e as
survives
the
rest
of potential
the
simulation
As a of
complay the through
electron
hole
Mach
number
M
as time.
a function
maximum
amplitude
ψ
of
the
for
values
ofthe
edifferent
maximumamplitude
amplitude
potential
different
values
maximum
ψψ
ofof
thethe
potential
forfor
different
values
of of
parison,
shown
in
Fig.
22,
we
changed
the
relativistic
facmaximum
amplitude
ψ
of
the
potential
for
different
values
of
the
parameter
µ.
We
see
that
for
larger
values
of
µ,
the
amplitheparameter
parameter
We
see
that
larger
values
amplithe
µ.µ.We
see
that
forfor
larger
values
ofof
µ,µ,
thethe
amplithe
parameter
µ.
We
see
that
for
larger
values
of
µ,
the
amplitude
of
the
electron
holes
drastically
increases.
Note
that
the
tor
to
µ
=
0.01,
while
keeping
the
same
trapping
paramtudeofofthe
theelectron
electronholes
holes
drastically
increases.
Note
that
tude
drastically
increases.
Note
that
thethe
electron
hole
potential
hasinitial
been
normalized
by T
/e, so
tude
the
electron
drastically
increases.
Note
that
the
eter
βof=
−0.5
and holes
the
speeds
of the
holes.
Inthe
this
electron
hole
potential
hasbeen
beennormalized
normalized
electron
hole
potential
has
byby
TeeT/e,
soso
thethe
e /e,
curves
in Fig.
18
show
pure
relativistic
effects
due
toand
the
relelectron
hole
potential
has
been
normalized
by
T
/e,
so
the
case,
the
electron
holes
are
considerably
smaller,
do
not
curvesininFig.
Fig.1818show
show
pure
relativistic
effects
due
curves
pure
relativistic
effects
due
toe to
thethe
rel-relcurvesasinthey
Fig.did
18 in
show
pure relativistic
effects
due
to numerthe relmerge
the strongly
relativistic
case.
The
284
16
B.
dynamics of
of electron
electron and
and ion
ion holes
holes
B. Eliasson
Eliasson and
and P.
P. K.
K. Shukla:
Shukla: The
The dynamics
Fig.
Fig.22.
22.Interactions
Interactionsbetween
betweenweakly
weaklyrelativistic
relativisticelectron
electronholes.
holes. IniInitially,
tially,the
theparameters
parametersare
areµµ =
= 0.01
0.01 and
and ββ =
= −0.5
−0.5 for
for both
both holes,
holes,
and
andthe
theleft
leftand
andright
rightholes
holes have
have the
the Mach
Mach number
number M
Mee =
= 0.7
0.7 and
and
MMe e==−0.2,
−0.2,respectively.
respectively.Times:
Times:t t== 00(upper
(upper left
left panel),
panel), tt =
= 50
50
(upper
(upperright
rightpanel),
panel),tt =
= 100
100(middle
(middle left
left panel),
panel), tt =
= 150
150 (middle
(middle
rightpanel),
panel),t t == 200
200(lower
(lowerleft
left panel)
panel) and
and tt =
= 250
250 (lower
(lower right
right
right
panel).
panel).
ical method used in our simulations was a pseudo-spectral
method to approximate the x-derivatives, a fourth-order compact
Padémass
scheme
to approximate
the pInderivative,
and the
ativistic
increase
of the electrons.
the non-relativistic
fourth-order
Runge-Kutta
for theresults
time stepping.
The
limit (µ = 0),
we recovermethod
the previous
of Bujarbarua
integral
over p (1981);
space insee
Eq. their
(29) was
withprofiles
a simpleofsum
and Schamel
Fig.done
1. The
the
representation
of the integral.
electrostatic potential
and the electron density are shown in
Fig. 19 for a few sets of parameters. To convert the potential
to Volts,
the values
of φ should bewaves
multiplied
by the factor
5.1
Trapping
of electromagnetic
in relativistic
elec5.1 ×tron
105holes
µ2 ; accordingly, the amplitude of the potential for
the case a) (ψ ≈ 33, µ = 0.4) is ≈ 2.4 × 106 Volts, while
6
for here
the case
b) (ψ
≈ relativistic
9, µ = 0.2)nonlinear
it is ≈ 0.18
× 10
Volts.
We
present
fully
theory
and
comThe
profiles
of
the
electron
holes
in
phase
space
is
shown
in
puter simulations for intense electromagnetic envelope soliFig.
20
for
the
same
sets
of
parameters
as
in
Fig.
19.
We
tons in a relativistically hot electron plasma, by adopting the
have solved the Vlasov-Poisson
system
(29)–(30)
and have
Maxwell-Poisson-relativistic
Vlasov
system
which accounts
investigated
interactions
between
two
large-amplitude
elecfor relativistic electron mass increase in the electromagnetic
tron holes.
results are
shown in
Fig. 21. The force,
left hole
fields
and theThe
relativistic
radiation
ponderomotive
in
has initially
the Mach
number
Me =
0.7 and
right relahole
addition
to trapped
electrons
which
support
thethe
driven
has a negative
Me = −0.2,
while both
holes
tivistic
electron Mach
holes.number
The importance
of trapped
electrons
have
β been
= −0.5
and µ =
0.4. During2001)
head-on
collisions,
has
also
recognized
(Montgomery,
in the
context
two holesCompton
merge and
form a of
new
electron
hole,
which
ofthestimulated
scattering
laser
light off
electron
survives through
the rest of the simulation
time. As our
a comquasi-modes
in a non-relativistic
situation. However,
obparison,
shown
Fig. 22,
we changed
the relativistic
jective
here
is to in
consider
nonlinear
interactions
betweenfacintor toelectromagnetic
µ = 0.01, while
keeping
the same trapping
tense
waves
and relativistic
electronparamholes.
eterthis
β =
−0.5 and
the initial
speeds of the holes.
this
For
purpose,
we derive
the electromagnetic
wave In
equacase,bythe
electronthe
holes
are considerably
smaller,
and do
not
tion
including
nonlinear
current density
arising
from
merge
as theyofdid
the stronglyelectron
relativistic
case.velocity
The numerthe
coupling
thein relativistic
quiver
and
icaldensity
methodvariation
used in of
ourthe
simulations
a pseudo-spectral
the
relativisticwas
electron
holes. The
method
approximate
the relativistic
x-derivatives,
a fourth-orderforce.
comlatter
aretomodified
by the
ponderomotive
pactfind
Padé
scheme
approximate
the p derivative,
and the
We
that
such to
nonlinear
interactions
between electrofourth-order
Runge-Kutta
methodelectron
for the time
magnetic
waves
and relativistic
holesstepping.
produce The
inintegral
over p space inenvelope
Eq. (30)solitons
was done
with a simple
sum
tense
electromagnetic
composed
of localrepresentation
the integral.
ized
light wave of
envelope
and large amplitude localized posi-
tive
potential
distributions. waves
The latter
may contribute
5.1 electric
Trapping
of electromagnetic
in relativistic
electo accelerating
electrons
to
extremely
high
energies. We
tron holes
also present numerical studies of the dynamics of interacting
electromagnetic
solitonsnonlinear
by meanstheory
of a fully
relaWe here present envelope
fully relativistic
and comtivistic
Vlasov simulations.
nonlinear
features
of
puter simulations
for intenseInteresting
electromagnetic
envelope
solilaser
light
intensificationhotand
corresponding
density
cavitatons in
a relativistically
electron
plasma, by
adopting
the
tion
have been observed.
Maxwell-Poisson-relativistic
Vlasov system which accounts
present the
relevant
equations
the action of
forWe
relativistic
electron
mass
increasedescribing
in the electromagnetic
intense
laser
on the electrons
a relativisticallyforce,
hot colfields and
thelight
relativistic
radiationinponderomotive
in
lisionless
well as the
electromagnetic
wave equaaddition toplasma,
trappedaselectrons
which
support the driven
relation
accounting
the relativistic
mass of
increase
the electivistic
electron for
holes.
The importance
trappedofelectrons
trons
andbeen
the electron
density
modification
dueintothe
thecontext
radiahas also
recognized
(Montgomery,
2001)
tion
relativisticCompton
ponderomotive
force
1986)
of stimulated
scattering
of (Shukla,
laser light
off electron
quasi-modes in a non-relativistic situation. However, our ob2 ∂γ
jective
here
inF
= −m
e c is to, consider nonlinear interactions between(41)
∂z
tense electromagnetic
waves and relativistic electron holes.
For this purpose, we derive the electromagnetic wave equawhere
tion by including the nonlinear
!1/2 current density arising from
2
2 |A|2
the couplingpof
the erelativistic
electron quiver velocity and
γ = 1+ 2 2 + 2 4
(42)
the densitymvariation
mofe cthe relativistic electron holes. The
ec
latter are modified by the relativistic ponderomotive force.
is
relativistic
gamma
factor.interactions
Here, p is between
the x component
Wethefind
that such
nonlinear
electroof
the electron
momentum,
and weelectron
have used
thatproduce
the perpenmagnetic
waves
and relativistic
holes
indicular
(to the x direction)
momentum
= eA/c, of
where
A
tense electromagnetic
envelope
solitonsp⊥composed
localis
thelight
vector
potential
of the
polarized
electromagized
wave
envelope
andcircularly
large amplitude
localized
posinetic
waves.potential
The dynamics
of the The
coupled
tive electric
distributions.
latterelectromagnetic
may contribute
waves
and relativistic
electron
holes with
immobile
is
to accelerating
electrons
to extremely
high
energies.ionsWe
governed
by numerical studies of the dynamics of interacting
also present
electromagnetic
envelope
Z ∞ solitons by means of a fully rela∂ 2 A Vlasov
1 ∂ 2 Asimulations.
fe Interesting nonlinear features of
tivistic
dp A = 0,
(43)
− 2 2 +
2
∂t
µ
∂x
−∞ γand corresponding density cavitalaser light intensification
tion have been observed. 2
∂feWe present
p ∂fe the∂(φ
− γ /µ ) ∂fe
the action(44)
of
+
+ relevant equations=describing
0,
∂t
γ
∂x
∂xelectrons
∂p in a relativistically hot colintense laser light on the
lisionless plasma, as well as the electromagnetic wave equaand
tion accounting for the relativistic mass increase of the elecZ ∞
trons
electron density modification due to the radia∂ 2 φ and the
=
fe ponderomotive
dp − 1,
(45)
tion
force (Shukla, 1986)
∂x 2 relativistic
−∞
∂γ
where A is normalized by me c/e,
φ by Te /e, p by me VT e
(42)
F = −me c2 ,
and x by rD . We have denoted γ =∂z(1 + µ2 p2 + |A|2 ).
Far away from the relativistic electron hole, where φ =
where
µ are assumed
¶ a Jüttner-Synge
|A| = 0, the electrons
to obey
2 1/2
p2
e2 |A|
γ = 1(de
+ Groot
+ al., 21980)
(43)
distribution function
et
m2 c2
me c4 q e
1
the relativistic
gamma
p is,the x component
e0 (p)
1 + µ2 Here,
p2 − 1)
(46)
fis
= a0 exp −
( factor.
µ2
of the electron momentum,
and we have used that the perpendicularthe
(tonormalization
the x direction)constant
momentum
where
is p⊥ = eA/c, where A
is the vector
potential
of
the
circularly
polarized
nR
h
p
io−1 electromag∞ The dynamics
1
netic
waves.
of
the
coupled
2
2
a0 = −∞ exp − µ2
1 + µ p − 1 electromagnetic
waves and relativistic
electron holes with immobile ions is
exp(−µ−2 )
governed by
= µ2K
.
−2
1 (µ )
Z ∞
2
2
fe
1 ∂ A
∂ A
For the Jüttner-Synge
function,
in
− 2 distribution
+
dp A the
= 0,last term
(44)
2
2
γ value
the left-hand∂tside ofµEq.∂x(43) takes
−∞the
Z ∞
−2 ) − γ/µ2 ) ∂f
p ∂fK
fe0∂f
(p)
e
e (µ∂(φ
e
+ = 0+
= 0,
(45)
p
≡ 2p ,
(47)
−2
∂t
γ
∂x
∂x
∂p
2
2
K1 (µ )
−∞
1+µ p
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
where K0 and K1 are the modified Bessel functions of second
kind. The frequency p represents the normalized (by ωpe )
relativistic plasma frequency at equilibrium.
We now investigate the properties of driven relativistic
electron holes which move with a constant speed. Accordingly, we use the ansatz f (p, ξ ) for the relativistic electron
distribution function, and assume that φ and |A|2 depend on
ξ only, where ξ = x − Me t. Then, Eqs. (44) and (45) take
the form
p ∂fe
∂(φ − γ /µ2 ) ∂fe
+
= 0,
(48)
−Me +
γ ∂ξ
∂ξ
∂p
and
d 2φ
=
dξ 2
Z
∞
fe dp − 1,
(49)
−∞
respectively. The general solution of Eq. (48) is fe = f0 (E),
where f0 is some function of one variable and
1
1
E = −Me p + 2 γ −
−φ
(50)
γ0
µ
is p
the energy integral. Here, we have denoted γ0 =
1/ 1 − µ2 Me2 . We note that trapped electrons have negative
energy while untrapped (free) electrons have positive energy.
In in the slowly varying envelope approximation, viz. A =
(1/2)A(x, t)(b
y + ib
z) exp(−iω0 t + ik0 x)+ complex conjugate, Eq. (43) can be written as
∂A
∂A
1 ∂ 2A
2iω0
+ vg
+ 2 2
∂t
∂x
µ ∂x
Z ∞
fe
−
dp − 2p A = 0,
(51)
−∞ γ
where ω0 = (2p + k02 /µ2 )1/2 is the electromagnetic wave
frequency,
vg = k0 /µ2 ω0 is the group velocity, and γ =
p
2
2
1 + µ p + |A|2 . Introducing A = W (ξ ) exp(−i2t +
iKx) into Eq. (51), where W is a real-valued function, we
obtain K = µ2 ω0 (Me − vg ) and
d 2W
− λW − µ2 2 − 2p W = 0,
2
dξ
(52)
where λ = −2ω0 µ2 2 + µ4 ω02 (Me2 − vg2 ) represents a nonlinear frequency shift, and the gamma
p
R ∞ factor becomes γ =
1 + µ2 p2 + W 2 . Here, 2 = −∞ (fe /γ ) dp represents
the square of the local electron plasma frequency that accounts for the relativistic electron mass increase in the electron hole potential and the electromagnetic wave fields.
A condition for the untrapped electron is that far away
from the relativistic electron hole, the electron distribution
function should smoothly connect to the Jüttner-Synge distribution function. In order to impose this condition for the
free electrons, we use the solution
fe = f0 (E) = fe0 [e
p(E)],
E > 0,
(53)
285
e(E) is a function of the energy such that p
e(E) → p
where p
when W → 0 and φ → 0. Such a function can be found
with the help of the energy integral by setting
q
1
1
e2 (E) −
e(E) + 2
1 + µ2 p
= E.
(54)
− Me p
γ0
µ
e(E), we have
Solving for p
e± (E) =
p
γ02 Me
1
±
µ E+
γ0
µ
2
v
u
γ02 u
t µ2 E
1
+
γ0
2
−
1
, (55)
γ02
e+ (E) and p
e− (E) correspond to a modified momenwhere p
tum for free electrons on each side of the trapped electron
e(E) = p
e+ (E) and
population in momentum space. Using p
e(E) = p
e− (E) in Eq. (53), we obtain the distribution funcp
tion for free electrons. In the limit of vanishing energy,
e± (0) = γ0 Me , and the value of the distriE = 0+ , we have p
bution function is
h
i
fe |E =0+ =f0 (0)=fe0 (γ0 Me )=a0 exp −(γ0 −1)/µ2 , (56)
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,” viz.
γ0 − 1
fe = a0 exp −
− βE , E < 0.
(57)
µ2
The separatrix between the free and trapped electron distributions is found where E = 0 in the energy integral. Solving
for p in the energy integral with E = 0, we have
v
u
2
γ u
1
1 2 1+W 2
p± =γ02 Me µ2 φ+
± 0 t µ2 φ+
−
, (58)
γ0
µ
γ0
γ02
where p− and p+ constitute the limits between the trapped
and free electron distributions in momentum space. Using
these limits and combining Eqs. (53), (55) and (57), we can
now write

q

1
2
2

e+ (E)−1) , p>p+ ,

 a0 exp − µ2 ( +µ p


h
i
fe = a0 exp − γ0µ−1
p− ≤p≤p+ , (59)
2 −βE ,

q



2 (E)−1) , p<p .

e−
 a0 exp − µ12 ( 1+µ2 p
−
Integrating the distribution function (59) over the momentum space, we obtain the total electron number density as a
function of φ and W , which are calculated self-consistently
by means of the Poisson and Schrödinger equations, respectively.
Figure 23 exhibits the influence of intense electromagnetic
waves on relativistic electron holes, described by the coupled
system of Eqs. (49) and (52). In the Schrödinger equation
(52), λ represents the eigenvalue, and the square of the local
electron plasma frequency, 2 , enters as a “potential.” We
e
0.5
0
−80
1.5
−40
−20
0
20
40
60
−60
−40
−20
0
20
40
60
1
2
Ω
286
18
−60
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
0.5
0
−80
2
W
electromagnetic waves, corresponding to the eigenvalues λ1 =
0.088 (solid line), λ2 = 0.053 (dashed line) and λ2 = 0.013 (dashdotted line). The parameters used are the same as for the dotted lines
in Fig.
B. Eliasson
and 23.
P. K. Shukla: The dynamics of electron and ion holes
80
80
ξ
1
0
−80
20
W
−60
−40
−20
0
20
40
60
80
Fig. 23. Large-amplitude trapped electromagnetic wave envelope
panel), the potential (second panel), the electron number
10
φ (upper
density (third panel), and the square of the local electron plasma
0
frequency
(lower−40
panel) −20
for large 0amplitude
waves
−80
−60
20 electromagnetic
40
60
80
1.5
with a maximum amplitude of Wmax = 1.5 (solid lines) and
1
NeWmax = 1.0 (dashed lines), and as a comparison a relativistic elec0.5
tron hole with small-amplitude electromagnetic waves which have
0
W−80
1 (dotted
shift for80the
max ¿
−60
−40 lines).
−20 The 0nonlinear
20 frequency
40
60
1.5
W
max = 1.5 case is λ = 0.099, and for the Wmax = 1.0 case
2 1
Ω it is λ = 0.095, to be compared with the small-amplitude case
0.5
which
has λ = 0.088. Parameters are: Me = 0.7, µ = 0.4, and
0 −0.5.
β −80
=
−60
−40
−20
0
20
40
60
80
−80
−60
−40
−20
0
20
40
60
80
ξ
Fig. 24. Small-amplitude trapped electromagnetic waves in a relativistic electron hole. Upper panel: Three eigenstates of trapped
electromagnetic waves, corresponding to the eigenvalues λ1 =
0.088 (solid line), λ2 = 0.053 (dashed line) and λ2 = 0.013 (dashdotted line). The parameters used are the same as for the dotted lines
in Fig. 23.
ξ
where p− and p+ constitute the limits between the trapped
and free electron distributions in momentum space. Using
Fig. 23. Large-amplitude trapped electromagnetic wave envelope
these
limits
andpotential
combining
Eqs.
(54), (56)
(58),
we can
Fig.
23.panel),
Large-amplitude
trapped
electromagnetic
wave
envelope
(upper
the
(second
panel),
the and
electron
number
now
write
(upper
panel),
the
potential
(second
panel),
the
electron
number
density (third panel), and the square of the local electron plasma
density (third
panel),
the
square
of the electromagnetic
localielectron plasma
(lower
h and
frequency
panel)
forq
large
amplitude
waves
1for large amplitude
2 (E)electromagnetic
2p

frequency
(lower
panel)
a
exp
−
1
+
µ
(
e
−
1) (solid
, p >lines)
p+waves
, and

2
0
+
with
a maximum
amplitude
of
W
=
1.5
µ

max

i
h
with a=maximum
amplitude
of as
Wamax
= 1.5 (solid
lines) elecand
γ0 −1 and
W
1.0a(dashed
a relativistic
max
− βE
,comparison
p− ≤ p elec≤ p+ ,
exp −lines),
2
e == 1.0
0 (dashed
Wfmax
lines),
and
as
a
comparison
a
relativistic
µ
q electromagnetic waves
i
h
tron hole
with small-amplitude
which have

tron holewith small-amplitude
electromagnetic
waves which have
1
1+
µ2 pe2− (E) frequency
− 1)
, p shift
< p−for
. the
Wmax 1a(dotted
lines).
The
nonlinear
2(
0 exp −
µ
Wmax ¿ 1 (dotted lines). The nonlinear frequency shift for the
Wmax
= 1.5 case is λ = 0.099, and for the Wmax = 1.0 case
it
(60)
Wmax = 1.5 case is λ = 0.099, and for the Wmax = 1.0 case
is λIntegrating
= 0.095, tothe
be compared
with
the
small-amplitude
case
which
function
over the momenit is λ = 0.095, todistribution
be compared
with the(60)
small-amplitude
case
hastum
λand
=space,
0.088.
Parameters
are:
Me =electron
0.7, µ =number
0.4, anddensity
β =holes
−0.5.
B. Eliasson
P. K.
Shukla:
The
dynamics
we
obtain
the
total
as a
which
has
λ =
0.088.
Parameters
are:ofMelectron
µ =ion
0.4, and
e = 0.7,and
β function
= −0.5. of φ and W , which are calculated self-consistently
by means of the Poisson and Schrödinger equations, respectively.
W 0
where
p− and
p+ constitute
the limits
between
the trapped
Figure
23 exhibits
the influence
of intense
electromagnetic
and
free
electron
distributions
in
momentum
space.
Using
waves
on
relativistic
electron
holes,
described
by
0
80
−80
−60
−40
−20
0
20
40
60 the cou80
these
andofcombining
(54),(53).
(56) In
andthe
(58),
we can
pledlimits
system
equations Eqs.
(50)ξ and
Schrödinger
now
write (53), λ represents the eigenvalue, and the square
equation
 local electron
q
h
of the
plasma frequency, Ωi2 , enters as a “po1
2 (E) − 1) , p > p ,

a
exp
−
1for
+ larger
µ2 peelectromagnetic
(

0
80
0 We seeµ2that trapped
W,
Fig.tential.”
24.
waves+fields
in a rel Small-amplitude
i + electromagnetic
h
γ0 −1
Fig.
24.
Small-amplitude
trapped
electromagnetic
waves
in
a
relthe
relativistic
electron
hole
potential
φ
becomes
larger
and
ativistic
electron
hole.
Upper
panel:
Three
eigenstates
of
trapped
fe = a0 exp − µ2 − βE ,
p− ≤ p ≤ p+ ,
ativistic
electronwaves,
Upper
panel:
Three
eigenstates
of trapped
q
ieigenvalues
hhole.
the 
relativistic
electron
hole
wider,
larger

electromagnetic
corresponding
toadmitting
the
λeigen1 =

1
2

2
electromagnetic
waves,
corresponding
to
the
eigenvalues
=
(
1
+
µ
p
e
(E)
−
1)
,
p
<
p
a
exp
−
1 pon2 be
− . λ(dash0 line),
values
λ.
Thisλcan
explained
that
0.088
(solid
0.053
(dashed
line)
andthe
λ3relativistic
= 0.013
− in
2 µ=
0
80
0.088
(solid
line),
λ
=
0.053
(dashed
line)
and
λ
=
0.013
(dash2
2
dotted
line). The
parameters
used electromagnetic
are the same as for
the pushes
dotted
deromotive
force
of localized
waves
(60)
dotted
The parameters used are the same as for the dotted lines
lines
in line).
Fig. 23.
Integrating
the distribution function (60) over the momenin Fig. 23.
tum space, we obtain the total electron number density as a
function of φ and W , which are calculated self-consistently
0
80
by means
the Poisson
and Schrödinger
respecsee
that foroflarger
electromagnetic
fields equations,
W , the relativistic
tively. hole potential φ becomes larger and the relativistic
electron
Figurehole
23 exhibits
the influence
of intense
electromagnetic
electron
wider, admitting
larger
eigenvalues
λ. This can
waves
on relativistic
electron
holes,ponderomotive
described by the
cou-of
be
explained
in that the
relativistic
force
pled system
of equations (50)
andpushes
(53). In
Schrödinger
envelope
localized
electromagnetic
waves
thethe
electrons
away
n number
equation
(53), λofrepresents
the eigenvalue,
and the
squareto
from
the center
the relativistic
electron hole,
leading
n plasma
of increase
the local of
electron
plasma frequency,
Ω2distribution
, enters as a and
“po-a
an
the electrostatic
potential
tic waves
tential.” of
Wethe
see
that for larger
electromagnetic
fields
,
widening
relativistic
electron
hole. We see that
theWdenes) and
the relativistic
electrondensity
hole potential
φ becomes
larger and
pletion
of
the
electron
in
the
relativistic
electron
hole
stic elecrelativistic
hole
wider,
admitting
eigen-
isthe
only
minimal,electron
while the
local
electron
plasmalarger
frequency
hich have
values
λ.
This
can
be
explained
in
that
the
relativistic
is strongly reduced owing to the increased mass of theponelecft for the
deromotive
of localized
electromagnetic
waveshole
pushes
1.0 case
trons
that areforce
accelerated
by the
relativistic electron
po-
ude case
0.4, and
0
tential. The linear trapping of electromagnetic waves in rela-
Fig. 25. Phase space plots of the electron distribution function fe
Fig. panels)
25. Phase
plots of the
electron
fe
(left
andspace
the amplitude
of the
vectordistribution
potential Afunction
(right pan(left
panels)
and
the
amplitude
of
the
vector
potential
A
(right
panels)
for
t
=
0,
t
=
50,
t
=
125
and
t
=
162.
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
els) for t = 0, t = 50, t = 125 and t = 162.
the electrons away from the center of the relativistic electron hole, leading to an increase of the electrostatic potential
distribution and a widening of the relativistic electron hole.
We see that the depletion of the electron density in the relativistic electron hole is only minimal, while the local electron plasma frequency Ω is strongly reduced owing to the increased mass of the electrons that are accelerated by the relativistic electron hole potential. The linear trapping of electromagnetic waves in relativistic electron holes is displayed in
Fig. 24, where we have assumed a zero electromagnetic field
Fig.
plots of thefor
electron
function
(W25.=Phase
0) in space
the expression
γ useddistribution
in Eq. (53)
and infthe
e
(left
panels)
and the The
amplitude
of the problem
vector potential
pan-set
energy
integral.
eigenvalue
admitsAa(right
discrete
els)
t = 0, t eigenfunctions
= 50, t = 125 and
t =positive
162. eigenvalues, and in
offor
localized
with
this case we found two even and one odd eigenfunction corresponding to three different eigenvalues. Equation (53) has
solvedaway
as a from
linearthe
eigenvalue
for W , elecwhere
thebeen
electrons
center ofproblem
the relativistic
thehole,
amplitude
the electromagnetic
wavespotential
was kept
tron
leadingWto
an of
increase
of the electrostatic
max
distribution and a widening of the relativistic electron hole.
We
depletionof of
electron
density
in the
Fig.see
26.that
Thethe
amplitude
thethe
vector
potential
(upper
left relpanel),
ativistic
electron
hole
is
only
minimal,
while
the
local
Fig. 26. (upper
The amplitude
of the
vector
potential
leftelecpanel),
potential
right panel),
squared
local
plasma(upper
frequency
(lower
tron
plasma
frequency
Ω
is strongly
reduced
owing
to the
inpotential
right panel),
squared
frequency
(lower
left
panel)(upper
and
electron
density
(lowerlocal
rightplasma
panel)
for
two
colliding
creased
mass
ofelectron
the holes.
electrons
are right
accelerated
bytwo
the colliding
relaleft panel)
and
densitythat
(lower
panel) for
relativistic
electron
relativistic
electron
tivistic
electron
holeholes.
potential. The linear trapping of electromagnetic waves in relativistic electron holes is displayed in
Fig.
24, where
weholes
have assumed
a zero
field
tivistic
electron
is displayed
in electromagnetic
Fig. 24, where we
have
fixed,
toin
obtain
new valuesforonγW
andin(W
λ.Eq.
Then,
the
procedure
(W
= 0)
the expression
used
and
inexpresthe
assumed
a zero
electromagnetic
field
=(53)
0) in
the
of solving
for The
φ in
and
W (52)
was and
repeated
convergence.
The
energy
integral.
eigenvalue
problem
admits
a discrete
sion
for
γ used
Eq.
in theuntil
energy
integral.setThe
second
derivatives
were
approximated
with
a
second-order
ofeigenvalue
localized eigenfunctions
with
positiveset
eigenvalues,
andeigenin
problem admits
a discrete
of localized
centered
witheven
the and
function
values set to zerocorat the
this
case wescheme
found
two
one odd
functions
with
positive
eigenvalues,
and eigenfunction
in this case we found
boundaries.
responding
to three
eigenvalues.
Equation (53)tohas
two even and
one different
odd eigenfunction
corresponding
three
In
order
to
study
the
dynamics
of
interacting
been
solved
as
a
linear
eigenvalue
problem
forsolved
Welectromag, where
different eigenvalues. Equation (52) has been
as a linthe
amplitude
Wproblem
of the
electromagnetic
waves electromagwasWkept of
netic
envelope
solitons
composed
ofthe
localized
max
ear
eigenvalue
for
W , where
amplitude
max
netic
waves
and
relativistic
electron
holes,
we
have
the
the electromagnetic waves was kept fixed, to obtainsolved
new valtime-dependent, relativistic Vlasov equation (45) together
with the Schrödinger equation (52) numerically. The results
are displayed in Figs. 25 and 26. As an initial condition to the
simulation, we used solutions to the quasi-stationary equa-
waves
t > 15
the com
and rel
netic w
linear i
hole ha
6 Co
In this
related
lisionle
holes m
to the a
tential.
fore an
tron dis
acceler
Langm
muir w
hole, a
the La
equatio
Electro
esting d
enough
due to
ion den
hole, w
stant sp
ion den
cally an
ion and
namics
with a v
Knowle
B. Eliasson and P. K. Shukla: The dynamics of electron and ion holes
ues on W and λ. Then, the procedure of solving for φ and W
was repeated until convergence. The second derivatives were
approximated with a second-order centered scheme with the
function values set to zero at the boundaries.
In order to study the dynamics of interacting electromagnetic envelope solitons composed of localized electromagnetic waves and relativistic electron holes, we have solved the
time-dependent, relativistic Vlasov equation (44) together
with the Schrödinger equation (51) numerically. The results
are displayed in Figs. 25 and 26. As an initial condition to the
simulation, we used solutions to the quasi-stationary equations described above, where the left electron hole initially
has the speed Me = 0.7 and is loaded with electromagnetic
waves with Wmax = 1.5, while the right electron hole has
the speed Me = −0.3, and is loaded with electromagnetic
waves with Wmax = 2.5. Further, we used k0 = vg = 0 in
the initial condition for A and in the solution of Eq. (51). In
Fig. 25 we show the phase space distribution of the electrons
and the electromagnetic field amplitude at different times.
We see that the relativistic electron holes loaded with trapped
electromagnetic waves collide, merge and then split into two
electron holes, while there are a few strongly peaked electromagnetic wave envelopes at x ≈ 70 remaining after the splitting of the relativistic electron hole, and where a population
of electrons has been accelerated to large energies. The time
development of the electromagnetic wave amplitudes, relativistic electron hole potential, the squared local plasma frequency and the electron number density is shown in Fig. 26.
We observe collision and splitting of the relativistic electron
holes and the creation of localized electromagnetic solitary
waves at z ≈ 70; clearly visible in the left two panels at
t > 150. The electromagnetic solitary waves are created by
the combined action of the relativistic electron mass increase
and relativistic ponderomotive force of localized electromagnetic waves, which have been further intensified due to nonlinear interactions where the collapsing relativistic electron
hole has deposited its energy into the plasma.
6 Conclusions
In this paper, we have presented a review of recent findings
related to the dynamics of electron and ion holes in a collisionless plasma. We find that interacting (colliding) ion
holes may create non-Maxwellian electron distributions due
to the acceleration of electrons in the negative ion hole potential. This can create streams of accelerated electrons before and during the ion hole collisions, and a flat-topped electron distribution between the ion holes, after collisions. The
accelerated streams of electrons can excite high-frequency
Langmuir waves due to a streaming instability, and Langmuir waves can be trapped in the density well of the ion
hole, as described by a nonlinear Schrödinger equation for
the Langmuir field, coupled nonlinearly with the Poisson
equation for the slow timescale ion hole potential response.
Electron holes in a plasma with mobile ions show an interesting dynamics when the speed of the electron hole is small
287
enough. A standing electron hole accelerates locally the ions
due to the positive electron hole potential. The self-created
ion density cavity, on the other hand, accelerates the electron
hole, which leaves the ion cavity and propagates with a constant speed. The propagating electron hole can be trapped at
ion density maxima, where it again accelerates the ions locally and a new ion density cavity is formed, etc. Thus, both
ion and electron holes show an interesting and complex dynamics during their interaction with the background plasma,
with a variety of couplings between fast and slow timescales.
Knowledge of dynamically evolving electron and ion holes
and their interaction with the background plasma is required
for understanding the properties of localized electric pulses
in the Earth’s auroral zone and in the magnetosphere.
We have also presented a theory for relativistic electron
holes and their interactions with high-amplitude electromagnetic fields, taking into account the relativistic electron mass
increase in a hot plasma. Due to the relativistic effect, the
size of the electron hole as well as the amplitude of the
electron hole potential increase dramatically with increasing
relativistic electron temperature. Large-amplitude electromagnetic fields also modify the electron hole owing to the
relativistic ponderomotive force and the electron mass increase due to the relativistic quivering velocity in the highfrequency electromagnetic fields. Simulation studies of interactions between relativistic electron holes show that they
interact in a complex fashion, and may merge when they collide, resulting in accelerated particles and a release of the
electromagnetic radiation. We stress that the present results
should help to understand the salient features of localized
intense electromagnetic and electrostatic pulses in relativistically hot plasmas, such as those in inertial confinement fusion as well as in compact astrophysical objects in which
gamma-ray bursts are produced by acceleration of electrons
to extremely high energies.
Acknowledgements. This work was partially supported by the
Deutsche Forschungsgemeinschaft (Bonn) through the Sonderforschungsbereich 591 “Universelles Verhalten Gleichgewichtsferner Plasmen, International Space Science Institute at Bern, as
well as by the European Commission (Brussels) through Contract
No. HPRN-CT-2001-00314 for carrying out the task of the research
training network entitled “Turbulent Boundary Layers in Geospace
Plasmas”. The Swedish National Supercomputer Centre (NSC)
provided the computer resources to carry out this work.
Edited by: J. F. McKenzie
Reviewed by: two referees
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