Download 3D electron-acoustic solitary waves introduced by phase space

GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L07803, doi:10.1029/2004GL019533, 2004
3D electron-acoustic solitary waves introduced by phase space electron
vortices in magnetized space plasmas
P. K. Shukla, A. A. Mamun,1 and B. Eliasson
Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, Bochum, Germany
Received 20 January 2004; revised 15 March 2004; accepted 18 March 2004; published 10 April 2004.
[1] We derive a multi-dimensional nonlinear equation for
electron-acoustic waves in magnetized plasmas composed
of stationary ions, magnetized cold electrons with a
significant fraction of electron beams, and magnetized hot
electrons which have a non-isothermal vortex-like
distribution. We find three-dimensional (3D) electronacoustic solitary waves which have a stronger nonlinearity
(in comparison with that associated with the traditional
harmonic generation) due to the trapping of hot electrons in
the localized electron-acoustic wave potential. The
relevance of our investigation to nonlinear structures (e.g.,
the bipolar parallel electric fields) in the Earth’s
I NDEX T ERMS : 2772
magnetosphere is discussed.
Magnetospheric Physics: Plasma waves and instabilities;
2720 Magnetospheric Physics: Energetic particles, trapped; 2704
Magnetospheric Physics: Auroral phenomena (2407).
Citation: Shukla, P. K., A. A. Mamun, and B. Eliasson (2004),
3D electron-acoustic solitary waves introduced by phase space
electron vortices in magnetized space plasmas, Geophys. Res.
Lett., 31, L07803, doi:10.1029/2004GL019533.
[2] It is well known that in two-electron-temperature
plasmas, which are found to occur both in space [Dubouloz
et al., 1991, 1993; Pottelette et al., 1999; Berthomier et al.,
2000, 2003; Singh and Lakhina, 2001; Lakhina et al., 2003]
and laboratory [Henry and Treguier, 1972] environments,
there exist a large variety of wave modes whose spatiotemporal scales are quite broad. One of these most important wave modes is the electron-acoustic waves (EAWs) in
which the restoring force comes from the pressure of the hot
electrons, while the inertia comes from the mass of the cold
electron component [Watanabe and Taniuti, 1977]. Thus,
the phase speed of the EAWs lies between the thermal
speeds of the hot and cold electron components, and their
frequencies are larger than the ion plasma frequency. The
EAWs can also be excited by electron and laser beams
[Gary and Tokar, 1985; Montgomery et al., 2001].
[3] The EAWs have received a great deal of renewed
interest not only because they are excited in space and
laboratory plasmas, but also because of their potential role
in interpreting electrostatic component of the broadband
electrostatic noise (BEN) observed in the cusp region of the
terrestrial magnetosphere [Tokar and Gary, 1984; Singh and
Lakhina, 2001], in the geomagnetic tail [Schriver and
Ashour-Abdalla, 1989], in the dayside auroral acceleration
1
Visiting from Department of Physics, Jahangirnagar University, Savar,
Dhaka, Bangladesh.
Copyright 2004 by the American Geophysical Union.
0094-8276/04/2004GL019533$05.00
region [Dubouloz et al., 1991, 1993; Pottelette et al., 1999],
etc. The EAWs have been used to explain various wave
emissions in different regions of the Earth’s magnetosphere
[Dubouloz et al., 1991, 1993]. It was first applied to
interpret the hiss emissions observed in the polar cusp
region in association with low energy (100 eV) upward
moving electron beams [Lin et al., 1984]. Dubouloz et al.
[1991] rigorously studied the BEN observed in the dayside
auroral zone and showed that because of the very high
electric field amplitudes (100 mV/m) involved, the nonlinear effects must play a significant role in the generation of
the BEN in the dayside auroral zone.
[4] The nonlinear propagation of finite amplitude EAWs
in an unmagnetized two electron plasma has been considered by several authors [Dubouloz et al., 1991, 1993;
Berthomier et al., 2000; Mamun and Shukla, 2002]. Recently, attempts have been made to generalize the nonlinear
theory of localized EAWs to include the effects of an
external magnetic field [Mace and Hellberg, 2001] and
magnetic field-aligned electron beams [Berthomier et al.,
2003], by assuming that the hot electrons follow a Boltzmann distribution in the wave potential f. However, in most
space plasmas, in addition to cold and hot electrons,
electron beams can act as a main energy source [De-Yu,
2003] for the excitation of EAWs with sufficiently large
amplitudes for which the hot electrons can be trapped in the
EAW potential. Then, the hot electrons follow a vortex-like
distribution [Schamel, 1972, 2000; Jovanović et al., 2002]
due to the formation of phase space holes caused by the
trapping of the hot electrons in the EAW potential. The
electron trapping is common not only in space plasmas, but
also in laboratory experiments, viz. a diffraction limited
laser-plasma interaction experiment [Montgomery et al.,
2001] where the low-velocity wave, that corresponds to
stimulated scattering from an EAW and implies strong
electron trapping, is observed. Therefore, motivated by the
observations of the electron trapping and electron beams in
both space and laboratory plasmas, in this Letter, we study
the nonlinear properties of finite amplitude low-frequency
(in comparison with the electron gyrofrequency) EAWs in a
magnetized plasma whose constituents are stationary ions,
magnetized cold electrons with a significant fraction of
beam electrons, and magnetized hot electrons which have
non-isothermal distribution due to their being trapped in
large amplitude EAW potential. Specifically, we show that
the oblique propagation of the EAWs is governed by a
modified multi-dimensional Korteweg-de Vries (KdV)
equation [e.g., Verheest et al., 2002], in which we have a
stronger trapped particle effect induced nonlinearity (proportional to f3/2), contrary to the f2 nonlinearity arising
from the harmonic generation in plasmas.
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SHUKLA ET AL.: 3D ELECTRON-ACOUSTIC SOLITARY WAVES
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[5] We consider a multi-electron space plasma in an
^, where B0 is the strength
external magnetic field B0 = B0z
^ is the unit vector along the z
of the geomagnetic field and z
axis. The dynamics of the EAWs in our plasma is governed
by Poisson’s equation
[8] We first consider the linear approximation in our
basic equations which, for wce w k vb0 and u0 =
uD = 0, yield a linear dispersion relation for the EAWs
w ¼ kk
r2 f ¼ 4peðnh þ nc þ nb n0 Þ;
ð1Þ
where nh and nc are the number densities of the hot and cold
electron components, nb is the number density of the beam
electrons, and n0 is the number density of stationary ions.
We note that the ions play the role of a neutralizing
background only, since the EAW frequency is much larger
than the ion plasma frequency. At equilibrium we have n0 =
nh0 + nc0 + nb0, where the subscript 0 stands for the
corresponding unperturbed quantity.
[6] We use the expression for nh that has been derived by
Jovanović et al. [2002] by employing the gyrokinetic
Vlasov equation for the hot electrons, which suffer the
E B0 and the polarization drifts, in addition to undergoing
magnetic field-aligned trapping in the wave potential, where
E = rf. Thus, we express nh as [Jovanović et al., 2002]
"
nh ’ nh0
3=2 #
2
VTe
ef
ef
4
ef
2
þa
b
;
1 2 r?
wce
Tek
Tek
3 Tek
ð2Þ
where VTe = (Tek/me)1/2 is the electron thermal speed, Tk is
^, me is the electron mass,
the electron temperature along z
wce = eB0/mec is the electron gyrofrequency, e is the
magnitude of the electron charge, and c is the speed of
light in vacuum. The coefficients a and b are
Z
1
a¼
2 Þ1=2
ð2pVTe
1
1
wdw
w2
exp 2 ;
w U0
2VTe
1
U02
U02
b ¼ pffiffiffi 1 b 2 exp 2 ;
p
VTe
2VTe
where U0 = u0 uD, u0 is the speed of the frame of reference,
uD is the equilibrium drift of the free hot electrons, and jbj is
the ratio between free and trapped hot electron temperatures.
We note that b < 0 represents a vortex-like excavated trapped
electron distribution corresponding to an under-population
of trapped electrons, whereas b = 1 (b = 0) represents a
Maxwellian (flat-topped) distribution. We are interested in
the case with b < 0. It is obvious from equations (3) and
(4) that for
pffiffiffi u0 = 0 and uD = 0, we have a = 1 and b =
(1 b)/ p.
[7] The dynamics of cold and beam electrons is governed
by
@ns
þ r ðns vs Þ ¼ 0;
@t
@
e
rf wce vs ^z;
þ vs r vs ¼
@t
me
ð5Þ
ð6Þ
where s = c (s = b) stands for the cold (beam) electrons and
vs is the fluid velocity of the electron species s.
Ca2 þ Cb2
1 þ k?2 r2 þ kk2 l2Dh
!1=2
;
ð7Þ
where w is the wave frequency, kk and k? are the parallel
and perpendicular components of the wave vector, respectively, and vb0 is the equilibrium streaming velocity of the
electron beam. Furthermore, we have denoted Ca = lDhwpc,
2
, re = VTe/wce, ra = Ca/wce,
Cb = lDhwpb, r2 = r2e + ra2 + rb2 + lDh
wce, rb = Cb/wce, lDh = (Tek/4pnh0e2)1/2, wpc = (4pnc0e2/me)1/2,
wph = (4p nh0e2/me)1/2, and wpb = (4pnb0e2/me)1/2. Equation (7)
shows that the EAWs have differential group dispersion in
two electron magnetoplasmas.
[9] We now derive the modified KdV equation in order to
investigate the nonlinear properties of the EAWs. We follow
the reductive perturbation technique [Washimi and Taniuti,
1966] and construct a weakly nonlinear theory for the
EAWs by using the stretched coordinates: x0 = 1/4x, y0 =
1/4y, z0 = 1/4(z v0t), and t0 = 3/4t, where is a small
parameter measuring the weakness of the dispersion and v0
is the wave phase velocity. We expand the perturbed
quantities about their equilibrium values in powers of as
3/2 (2)
(1)
+ 3/2f(2) + , vsz =
ns = ns0 + n(1)
s + ns + , f = f
(1)
3/2 (2)
5/4 (1)
vs0+ vsz + vsz + , vsx = vsx + 3/2vsx(2)+ , and
vsy = 5/2vsy(1) + 3/2vsy(2) + . We note that vc0 = 0, but vb0 6¼
0. Now, substituting the stretched coordinates and expansion series into our basic equations, we develop equations in
various powers of , and finally obtain
2
@j
@
@2
pffiffiffi @j 1 @ @ 2
j ¼ 0;
þ
D
þ
þ Ab j
þ A
@t
@x 2 @x @x2
@z2 @h2
ð3Þ
ð4Þ
L07803
ð8Þ
where j = ef(1)/Tek, t = t0 wpc, z = x0/lDh, h = y0/lDh, x =
z0/lDh, A = [Ca3/(U0 + v0)3+ CaCb2/(U0 + v0 vb0)3]1, and
2
2
2
2
+ wpb
+ wph
)/wce
. Equation (8) is our modified
D = 1 + (wpc
KdV equation in three dimensions.
[10] To infer the properties of 3D EAW solitary structures, we have numerically solved equation (8) for the
electrostatic potential. The numerical results are displayed
in Figures 1 – 3. We have used typical parameters
corresponding to the Earth’s auroral zone [Dubouloz et
al., 1993; Singh and Lakhina, 2001]: nh0/nc0 = 4 and
nb0/nc0 = 2. The values of other parameters are taken as
B0 ’ 1.1 105 T, wce/wpc = 25, b = 0.7, U0/Ca = 0.1, v0/
Ca = 1, and vb0/Ca = 0.1, giving A = 0.364, b = 0.959 and
D = 1.01. To compare our results with that observed by
satellite observations [Ergun et al., 1998a, 1998b] in the
Earth’s auroral zone, we choose a set of available parameters (Tek ’ 700 eV, nh0 ’ 6 cm3, lDh = 68 m) along with
an observed electric field amplitude E0 ’ 50– 100 mV/m
[Ergun et al., 1998a, 1998b].
[11] The upper left panel of Figure 1, where the initial
condition jt=0 = (15u/8Ab)2sech[(x2 + r2/D)1/2(u/8A)1/2)]
with u = 0.07 is used, corresponds to an almost spherically
symmetric potential hump with an amplitude of jmax 0.14. During the temporal evolution, the pulse decreases its
amplitude to jmax 0.12, whereafter an almost steady-state
pulse propagates with a constant speed, as seen in the upper
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SHUKLA ET AL.: 3D ELECTRON-ACOUSTIC SOLITARY WAVES
Figure 1. The electrostatic potential associated with an
electron hole at t = 0 (upper left panel), t = 335 (upper right
panel), t = 670 (lower left panel), and t = 1000 (lower right
panel). The lower (left and right) panels show the
development of a steady state structure.
right and lower panels of Figure 1. The lower right panel of
Figure 1 shows that the diameter of the solitary structures is
10lDh 700 m; such a structure, propagating with a
speed 5 106 m/s would have a typical duration of
0.1 ms, in agreement with satellite observations [Ergun et
al., 1998b]. In order to express the electric potential in volts,
the dimensionless potential given in Figure 1 should be
multiplied by a factor 700, giving a peak potential of 85 V,
which is consistent with 100 eV streaming electrons
associated with the solitary waves observed by the FAST
satellite [Ergun et al., 1998b].
[12] Figure 2 shows the electron density perturbation
(ne/neh = r2j) which is associated with the potential hump
plotted in Figure 1. The electron density is characterized
by a cavity localized at the center of the solitary wave,
L07803
Figure 3. The electric field parallel (upper panel) and
perpendicular (lower panel) to the magnetic field direction
at a point 210 m from the axial center of the solitary wave
for Tek = 700 eV and lDh = 68 m. The time is normalized by
1
.
wpc
while the maxima of the electron density appear on the
flanks of the electron density cavity.
[13] We have given the electric field in dimensional units
as a function of time in Figure 3, where the time is
1
. The electric field is measured at the
normalized by wpc
axial coordinate x = 17 and r = 3.1, simulating satellite
measurements at a radial distance of 3.1 68 m = 210 m
from the axial center of the solitary wave. The parallel
electric field in the upper panel shows a typical bipolar
structure, with a peak amplitude of 120 mV/m, while the
lower panel shows the perpendicular electric field with a
typical monopolar structure, with a peak amplitude of
around 200 mV/m. Our results are thus in good agreement
with the FAST satellite observations in the Earth’s auroral
zone where the same electric field amplitudes have been
observed [Ergun et al., 1998a, 1998b].
[14] To summarize, we have carried out a theoretical
investigation for understanding the properties of 3D EAW
solitary waves in a magnetized plasma whose constituents
are stationary ions, magnetized cold electrons with a significant fraction of beam electrons, and magnetized hot
electrons obeying a vortex-like distribution. We have derived and numerically solved the modified 3D KdV equation, and showed how the 3D EAW solitary waves evolve
with time. It is found that the EAW solitary waves have a
positive potential, which corresponds to a dip (hump) in the
cold (hot) electron number density. Our present theoretical
results are good agreements with the FAST satellite observations [Ergun et al., 1998a, 1998b] of the electrostatic
solitary structures in the Earth’s auroral zone.
[15] Acknowledgments. This research was partially supported by the
Deutsche Forschungsgemeinschaft (Bonn) through the Sonderforschungsbereich 591, as well as by the European Commission (Brussels) through
grant No. HPRN-CT-2001-00314. The constructive suggestions of the
reviewers are gratefully acknowledged.
Figure 2. The electron density perturbation ne/neh associated with an electron hole at t = 0 (upper left panel), t =
335 (upper right panel), t = 670 (lower left panel), and t =
1000 (lower right panel).
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B. Eliasson, A. A. Mamun, and P. K. Shukla, Institut für Theoretische
Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany. ([email protected]; [email protected];
[email protected])
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