Electron phase-space holes in a two-dimensional plasma
Q. M. Lu1,2, J. B. Tao1, B. Lembege3 and S. Wang1,2
1
CAS Key Laboratory of Basic Plasma Physics, School of Earth and Space Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China
2
Key Laboratory of Space Weather, Center for Space Science and Applied Research,
Chinese Academy of Sciences, Beijing, China.
3
Centre d’étude des Environnements Terrestre et Planétaires, CNRS Université de
Versailles-Saint Quentin, Vélizy, France
The parallel cut of the parallel component of the electric field ( E|| ) in electron
phase-space holes has bipolar structures, and recent satellite observations found that
the parallel cut of their perpendicular electric field ( E ) is unipolar. In this letter, we
performed two-dimensional (2D) particle simulations to investigate the nonlinear
evolution of electron beam instability. The results show that the electron holes are
unstable to electrostatic whistler waves, and inside the holes a series of islands with
alternate positive and negative E can be formed in the direction perpendicular to
the background magnetic field. It can explain the observed unipolar structures of E
along the parallel cut of the background magnetic field. The influence of the
amplitude of the background magnetic field is also considered.
1. Introduction
Electrostatic solitary waves (ESWs), which move along the background
magnetic field, have been observed in different space environments, such as the
magnetotail [Matsumoto et al., 1994], the auroral region [Ergun et al., 1998a; Cattell
et al., 2002; Franz et al., 1998], the Earth’s foreshock region [Bale et al., 1998], the
magnetosheath [Pickett et al., 2004] and the solar wind [Mangeney et al., 1999]. They
are positive potential pulses and their parallel cut of the parallel electric field along
the background magnetic field ( E|| ) is bipolar. These structures are modeled as
electron phase-space holes which are stationary Bernstein-Greene-Kruskal (BGK)
solutions of the Vlasov and Poisson equations [Bernstein et al., 1957].
One-dimensional (1D) particle simulations have shown that such electron holes can
be formed through nonlinear evolution of electron beam instability [Omura et al.,
1994; Lu et al., 2005]. In their nonlinear evolution, electron holes coalesce with
adjacent holes and merge into larger, more intense and isolated holes, which are
thought as solitary waves with bipolar electric field structure. The characteristics of
electron phase-space holes in higher dimensions are a subject of ongoing study. The
2D and 3D particle simulations of electron beam instability have confirmed the
formation of the electrostatic solitary waves with bipolar E|| along the parallel cut of
the background magnetic field [Goldman et al., 1999; Oppenheim, et al., 1999; 2001].
Munschietti et al. [2000] found that when the amplitude of the background magnetic
field is sufficiently small, the electron holes, which remain stable in 1D, are unstable
to the transverse instability and quickly dissipate in multi-dimensional particle
simulations. However, in a strongly magnetized plasma, 2D and 3D particle
simulations showed that in the long time evolution of the electron beam instability the
electron holes decay after hundreds of plasma periods and emit electrostatic whistler
waves [Goldman et al., 1999; Oppenheim et al., 2001].
Recently, observations by the Polar and FAST satellites found that electrostatic
solitary waves have multi-dimensional structures. The parallel cut of the parallel
component of the electric field ( E ) is bipolar and that of the perpendicular
component ( E ) is unipolar [Ergun et al., 1998a, 1998b; Franz et al., 1998, 2005].
Based on data from the Polar Plasma Wave Instrument Franz et al. [2000] indicated
that the perpendicular size of electron solitary waves is finite, and the ratio of the
parallel dimension ( L|| ) to the perpendicular dimension ( L ) can be described as:
2
L|| L  E E||  (1   pe
e2 )1/ 2 (where  pe and e are the electron plasma
frequency and cyclotron frequency, respectively). Chen and Parks [2002] constructed
a set of analytical BGK solitary wave solutions by solving the coupled 3D Possion
and 1D Vlasov equations. Their results indicated that for a single humped electric
potentials, the parallel cut of the perpendicular component of the electric field ( E ) is
unipolar and that of the parallel component ( E|| ) bipolar. In this Letter, we perform
2D particle-in-cell (PIC) simulations to investigate the nonlinear evolution of electron
beam instability, and the emphasis is placed on the perpendicular structure of the
electrostatic solitary waves. The implications on the satellite observations are also
discussed.
2. Simulation Model
A 2D electrostatic PIC code with periodic boundary conditions is employed in
our simulations [Decyk, 1995; Lu and Cai, 2001]. The simulation system is taken in
the x  y plane with a uniform embedded magnetic field B0 in the x direction.
The amplitude of the background magnetic field B0 is chosen so that the ratio of the
electron cyclotron frequency, e  eB0 / me , to the electron plasma frequency,
 pe  (n0e2 me 0 )1/ 2 , is 1 if there is no explicit statement. We initiate the simulations
with counter-streaming electron beams, and their initial velocity distributions are
assumed to satisfy Maxwellian function. These two electron components have equal
density and temperature. Initially, the first component has a mean velocity of vx  0
and the second with vx  5vte , where vte is the initial thermal velocity of each
component. A Maxwellian distribution of ions with a mean velocity of vx  0 is also
introduced, and their temperature is same as that of electrons. The ion to electron
mass ratio is defined as 1836 ( mi me  1836 ). In the simulations, the densities and
velocities of the electrons and ions are expressed in units of the total unperturbed
density n0  ni (where ni is the number density of ions) and thermal velocity vte ,
respectively. The space and time are normalized to the electron Debye length
D  ( 0 k BTe n0e 2 ) (where Te is the initial temperature of electrons) and to the
inverse of the electron plasma frequency  pe . The electric field is normalized to
me pe vte / e . Grid cells 128  256 with grid size D  D are used in the simulations,
1
and the time step is 0.02 pe
. The number of particles employed for each component
is 3,276,800.
3. Simulation Results
Fig. 1 shows the time evolution of the electric field energy Ex2 and Ey2 . The
energy corresponds to the electrostatic waves excited by the electron beam instability.
The evolution can be divided into the linear growth stage and nonlinear evolution
stage. In the linear growth stage, the electric field energy Ex2 begins to increase
rapidly from about  pet  10 , and it saturates at about  pet  30 . At the beginning of
the nonlinear evolution stage, the electric field energy Ex2 begins to decrease rapidly,
while the electric field energy Ey2 begins to increase at first and then saturates at
about  pet  60 . From about  pet  300 to about  pet  3000 , both the electric
field energy Ex2 and Ey2 is kept as almost constant. At about  pet  3000 , the
electric field energy Ey2 begins to increase and saturates at about  pet  3600 .
During the linear growth stage of the two-dimensional electron beam instability,
the excited waves are nearly monochromatic and have a substantial degree of
coherence perpendicular to the background magnetic field. The evolution of the
excited waves follows the similar evolutionary path described in Goldman et al.
[1999], and it forms an ensemble of closely tubes elongated along y direction. After
the amplitude of the waves is sufficiently large, part of electrons are trapped by the
waves and nonlinear kinetic effects develop. In this nonlinear evolution stage of the
instability, these tubes interact with adjacent tubes and begin to coalesce, and the
process is similar to 1D simulations [Omura et al., 1994; Lu et al., 2005]. However, in
2D simulations the tubes are not perfectly straight, the merging of the tubes begins at
the point of closest approach between two tubes. Usually such merging is not
complete, and many tube segments are formed. The phenomena can be found in Fig.
1(a), which describes the electric field Ex and E y at  pet  80 . There is no regular
structure for the electric field E y .
As the merging of tubes proceeds, only one tube remains at about  pet  280
and this ESW is an electron phase-space hole with the bipolar electric field Ex along
the background magnetic field. The phase-space hole is unstable to electrostatic
whistler waves, which are a generalized Langmuir waves in a highly magnetized
plasma. The whistler waves propagate oblique to the background magnetic field and
have the fluid dispersion relation in the electron center-of-mass frame    pe cos  ,
where  is the angle between wave vector k and the background magnetic field
B0 and is near 900 . Over the next few thousands of plasma periods, the electrostatic
whistler waves are excited, and gradually we can find that there are regular structures
inside the ESW for the electric field E y . Fig. 1(b) shows the electric field Ex and
E y at  pet  1580 . In the figure, we can find that one ESW is located around
x  64D with width about 30D . Inside the ESW, there exists a series of islands
with comparable scales in x and y directions, and these islands have alternate
positive and negative E y . Fig. 2 shows the parallel cut of the electric field Ex and
E y along (a) y  107D and (b) y  130D . The Ex has a bipolar electric field
structure while E y has a unipolar structure, and their ratio E y E x is about 0.5. The
unipolar E y structure is positive along y  107D and negative along y  130D .
The results can explain the satellite observations: the bipolar E and unipolar E
along the parallel cut of the background magnetic field. [Ergun et al., 1998a, 1998b;
Franz et al., 1998, 2005]
The unipolar E y structure continues to exist until about  pet  2400 , and then
the electric field E y forms a striated structure, which can be found in Fig. 1(c). Fig.
1(c) shows the electric field Ex and E y at  pet  3240 . Although Ex still has
obvious bipolar structure, the tube is broken into several segments. The striated
structure for the electric field E y can be obviously found, and the structure is
oblique to the background magnetic field. We cannot find obvious unipolar structure
for the electric field E y at that time.
We also calculate the case e  pe  5 . In this case the unipolar structure of the
electric field E y along the parallel cut of the background magnetic field can also be
formed. However, it lasts shorter time, which begins at about  pet  420 and
disappears at about  pet  900 . Fig. 4 shows the electric field Ex and E y at
 pet  500 . Along the parallel cut of the background magnetic field, Ex has the
bipolar structure with width about 45D . Inside the ESW, E y consists of a series of
islands with alternate positive and negative E y , and their x scale is larger than the
y scale. At last the whole simulation domain is also full of striated E y structure
(not shown), which is similar to the case e  pe  1 .
4. Discussion and Conclusions
In this letter, using two-dimensional particle simulations, we investigated the
nonlinear evolution of electron beam instability. Similar to the simulation results of
Oppenheim et al. [1999], we found that electron phase-space tubes are formed during
the nonlinear stage of the instability, and then these tubes merge with the adjacent
tubes until at last only one ESW exists. The ESW aligns primarily perpendicular to
the background magnetic field, and its parallel cut of the parallel electric field ( E|| )
has bipolar structure. The ESW is unstable to electrostatic whistler waves, which
propagate oblique to the background magnetic field. Compared with the simulation
results of Oppenheim et al. [1999], we found that before the electrostatic whistler
waves are fully developed and the perpendicular electric field forms a striated
structure in the whole simulation domain, inside the ESW the perpendicular electric
field ( E ) forms a series of islands with alternate positive and negative electric field.
Of course, in our simulation we choose e /  pe  1 , which is smaller than what
Oppenheim et al. [1999] used, and such structure is easier to be form than larger
e /  pe . If we choose the x and y scales as the parallel dimension ( L|| ) and the
perpendicular dimension ( L ) of the ESW, we can observe that their L|| L increase
with the increase of e  pe in our simulations, which is qualitatively consistent
with the observations of Franz et al. [2000].
In our simulation results, the parallel cut of the perpendicular electric field ( E )
has unipolar structure, which is consistent with the recent observations [Ergun et al.,
1998a, 1998b; Franz et al., 1998, 2005]. Chen and Parks [2002] indicated that such a
unipolar structure is the results of a single humped electric potential, which is the
stationary solution of the coupled 3D Possion and 1D Vlasov equations. However, our
simulations indicates that such unipolar structure of the perpendicular electric field is
the results of the nonlinear evolution of the electrostatic whistler waves, which are
unstable to the electron phase-space hole, and it is dynamic and usually lasts hundreds
to thousands of the plasma periods.
Acknowledgments
This research was supported by the Program for New Century Excellent Talents in
University (NCET-05-0554) and the National Science Foundation of China (NSFC)
under grants 40336052.
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Figure Captions
Fig. 1 The time evolution of the electric field energy Ex2 and Ey2 . Here e  pe is
chosen as 1.
Fig.2 The electric field Ex and E y at  pet  80 , 2400, 3240. Here e  pe is
chosen as 1.
Fig. 3 The parallel cut of the electric field Ex and E y along (a) y  107D and (b)
y  130D . Here e  pe is chosen as 1.
Fig. 4 The electric field Ex and E y at  pet  500 . Here e  pe is chosen as 5.