JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, 313–320, doi:10.1029/2012JA018351, 2013
Kinetic analysis of the energy transport of bursty bulk flows
in the plasma sheet
Jinbin Cao,1 Yuduan Ma,1 George Parks,2 Henri Reme,3,4 Iannis Dandouras,3,4
and Tielong Zhang5,6
Received 21 November 2012; revised 29 September 2012; accepted 4 November 2012; published 20 January 2013.
[1] The energy transport of bursty bulk flows (BBFs) is very important to the
understanding of substorm energy transport. Previous studies all use the MHD bulk
parameters to calculate the energy flux density of BBFs. In this paper, we use the kinetic
approach, i.e., ion velocity distribution function, to study the energy transport of an
earthward bursty bulk flow observed by Cluster C1 on 30 July 2002. The earthward energy
flux density calculated using kinetic approach QKx is obviously larger than that calculated
using MHD bulk parameters QMHDx. The mean ratio QKx/QMHDx in the flow velocity range
200–800 km/s is 2.7, implying that the previous energy transport of BBF estimated using
MHD approach is much underestimated. The underestimation results from the deviation of
ion velocity distribution from ideal Maxwellian distribution. The energy transport of BBF
is mainly provided by ions above 10 keV although their number density Nf is much smaller
than the total ion number density N. The ratio QKx/QMHDx is basically proportional to the
ratio N/Nf. The flow velocity v(E) increases with increasing energy. The ratio Nf/N is
perfectly proportional to flow velocity Vx. A double ion component model is proposed to
explain the above results. The increase of energy transport capability of BBF is important
to understanding substorm energy transport. It is inferred that for a typical substorm, the
ratio of the energy transport of BBF to the substorm energy consumption may increase
from the previously estimated 5% to 34% or more.
Citation: Cao, J., Y. Ma, G. Parks, H. Reme, I. Dandouras, and T. Zhang (2013), Kinetic analysis of the energy transport
of bursty bulk flows in the plasma sheet, J. Geophys. Res. Space Physics, 118, 313–320, doi:10.1029/2012JA018351.
1. Introduction
[2] The bursty bulk flows (BBFs) in the inner plasma
sheet of the magnetosphere are important phenomena that
are closely related to magnetospheric activities and transport
of energy and magnetic flux [Baumjohann et al., 1990,
Baumjohann, 2002; Angelopoulos et al., 1992; Chen and
Wolf, 1999; Kepko et al., 2001; Sergeev et al., 2000;
Nakamura et al., 2002; Slavin et al., 2002; Cao et al., 2006,
2008, 2010; Ma et al., 2009; Fu et al., 2011]. The longstanding unsolved problem about the BBFs is how large their contribution is to the transport of mass, energy, and magnetic
1
Space Science Institute, School of Astronautics, Beihang University,
Beijing, China.
2
Space Sciences Lab, University of California, Berkeley, California,
USA.
3
Institut de Recherche en Astrophysique et Planétologie, CNRS,
Toulouse, France.
4
University of Toulouse, UPS, IRAP, Toulouse, France.
5
CAS Key Laboratory of Geospace Environment, University of Science
and Technology of China, Hefei, China.
6
Space Research Institute, Austrian Academy of Sciences, Graz,
Austria.
Corresponding author: J. Cao, Space Science Institute, Beihang
University, 100191, Beijing, China. ([email protected])
©2012. American Geophysical Union. All Rights Reserved.
2169-9380/13/2012JA018351
flux during substorm times. Angelopoulos et al. [1996] estimated the earthward transport of mass, energy, and magnetic
flux of BBFs during a substorm by using a typical BBF with a
duration of 10 min and a cross-section area of 33 R2E . They
found that BBFs are responsible for 60%–100% of measured
earthward transport of mass, energy and magnetic flux past
the satellite in the region of maximum occurrence rate, even
though they last only 10%–15% of the observation time
there. Thus, BBFs represent the primary transport
mechanism in those regions. However, a single BBF can
account for the transport rate of a medium substorm only
when the cross-section area in the Y-Z plane reaches 100 R2E .
Even in this case, the total BBF energy transport is about
10% of the transport of substorm. Thus, Angelopoulos et al.
[1996] concluded that BBF is not the main transport mechanism during a substorm. Alternatively, many BBFs last
longer than their median observed duration of 10 min. However, they are not observed due to the localization of BBFs
and the fact there are only a limited number of satellites in
the BBF occurrence regions. Sergeev et al. [2000] also thought
that the BBF transport rate cannot account for the total transport rate of magnetotail. Paterson et al. [1998, 1999] questioned the results of Angelopoulos et al. [1992, 1994] using
different statistical approach, and came to the conclusion
that bursty bulk flows do not contribute significantly to
transport in the near-earth plasma sheet, at least during quiet
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times. Thus, the contributions of BBFs to the energy transport
in substorms and their role in substorm are still under debate
[Lui et al., 2000].
[3] Almost all previous studies of BBF assume that the
ions in the plasma sheet are equilibrium and consists of a
single ion flow component. Therefore, a BBF can be described by three bulk parameters: density, mean velocity,
and temperature. Although this kind of moment parameter
studies of MHD approach have provided a useful picture,
it is becoming increasingly evident that the MHD approach
may not be able to correctly reveal the energy and magnetic
flux transports. Chen et al. [2000] have mentioned that in the
frame of MHD approach, the combination of two counterstreaming fast flows may yield a very small mean velocity.
[4] For ions with a velocity distribution function. f(v,t),
the number density N, the mean velocity V, the pressure tensor P, number flux density G, and the energy flux density Q
can be defined by the following equations:
Z
N¼
f ðv; t Þd 3 v
(1)
f ðv; t Þv d 3 v=N
(2)
Z
V¼
Z
P¼m
f ðv; t Þðv-VÞðv-VÞd 3 v
(3)
f ðv; t Þv d 3 v ¼ N V
(4)
Z
G¼
Z
Q¼
f ðv; t Þ
1 2
mv v d 3 v
2
(5)
[5] In the framework of MHD theory, the energy flux density of particles QMHD is given by [Angelopoulos et al.,
1992]
1
5
QMHD ¼ NmV 2 V þ P V
2
2
(6)
where the two terms on the right side represent the dynamic
energy of fluid motion and the thermal energy, respectively.
Besides the energy flux density of particles, the total energy
flux density also includes Poynting vector representing the
energy transport rate of electromagnetic energy, which can
c
be expressed by 4p
E B [Angelopoulos et al., 1994]. Because
three components of electrical field are not available for the
Cluster mission, we do not give discussions on electromagnetic
energy transport in the present paper.
[6] In fact for an ideal Maxwellian velocity distribution
function, the two expressions in equations (5) and (6) are
equivalent, which allows the energy flux density deduced
by MHD approach in equation (6) to be able to correctly represent energy transport of particles. However, in case of nonMaxwellian velocity distribution, the two expressions in
equations (5) and (6) are not equivalent. The in situ measurements of hot ions in the magnetosphere shows that the velocity distribution functions of ions in the magnetosphere are
more or less non-Maxwellian. Therefore, the energy transport rates of bursty bulk flows in the framework of MHD
in previous studies, which are obtained by using bulk parameters of MHD, cannot correctly represent the energy transport of particles.
[7] Considering the problem of energy transport of substorm is still unsolved, it is appropriate to recheck the energy
transport of bursty bulk flow using ion velocity distribution
function. The purpose of this paper is to study the energy
transport of bursty bulk flows in the plasma sheet using the
original definition of energy flux density in equation (5),
which can better describe the energy transport of bursty bulk
flows than the previous MHD approach. The results show
that for the bursty bulk flow events studied here, the energy
flux density derived by using ion velocity distribution function is much larger than the energy flux density derived by
using bulk parameters of MHD approach. This conclusion
indicates that previous estimation of energy transport of
BBF is underestimated and the fast flows in the plasma sheet
play a more important role in the energy transport of substorms than anticipated in previous studies.
[8] In this study, the data of plasma, magnetic field, and
electric field come from the HIA (hot ion analyzer) [Reme
et al., 2001] and fluxgate magnetometer [Balogh et al., 2001]
on board the Cluster satellites, and all have a time resolution
of 4 s. The HIA instrument measures three-dimensional distribution functions of the ions between 5 eV/q and 32 keV/q
without mass discrimination. The whole energy range is
divided into 62 energy channels. The angle resolution is
11.25 5.625 . The first-order moments give the average
velocity V of the ions, referred to as the bulk velocity. Converting the momentum flux tensor and the energy flux vector
to the frame where the average velocity is zero, one obtains
the pressure tensor [see Reme et al., 2001]. Here all the
moments are calculated by integrating over the energy range
5 eV to 32 keV. The satellite positions are given in geocentric
solar magnetospheric coordinates. In addition, for the sake of
description, throughout this paper, the energy flux density
deduced directly from velocity distribution function as in
equation (5) is referred to as kinetic energy flux density QK
and the energy flux density deduced using MHD bulk parameters (N, V, and P) as in equation (6) is referred to as MHD
energy flux density QMHD.
2. Observations
[9] On 30 July 2002, Cluster satellites crossed the tail
plasma sheet and observed a BBF in a substorm expansion
phase. The peak velocity of this fast flow reached 1800 km/
s, allowing us to study energy transport of from low speed
to high-speed BBFs.
[10] Figure 1 shows the bulk velocity (Vx, Vy, and Vz), magnetic field (Bx, By, and Bz), ion number density N, ion temperature and plasma beta observed by C1 satellite during the interval 1745–1800 UT on 30 July 2002. The decreases of
magnetic field and the increase of ion temperature around
1747:00 UT indicate that C1 was crossing into plasma sheet
from the lobe. C1 entered the plasma sheet proper (b >0.3)
at about 1752:00 UT and then stayed within it except for several short-time outward excursions. In the plasma sheet, C1
observed a bursty bulk flow whose velocity starts to increase
at 1748:00 UT. Within this BBF, there are four flow bursts
with peak velocities of 910, 1800, 1100, and 1550 km/s.
These flow bursts show typical features of bursty bulk flows
in the plasma sheet: high velocity, low density, high temperature and large perpendicular velocity component.
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CAO ET AL.: ENERGY TRANSPORT OF BBFS
Figure 1. The plasma parameters of BBF observed by Cluster satellite during the interval
1745:001810:00 UT on 30 July 2002. From top to bottom: the bulk velocity, magnetic field, ion number
density, temperature, and plasma beta.
[11] Figure 2 shows the earthward kinetic energy flux density QKx and earthward MHD energy flux density QMHDx and
the ratio QKx/QMHDx. Here the parameters QKx and QMHDx
represent the kinetic energy flux density and MHD energy
flux density passing the satellite C1 per cross-sectional area
R2E in the YZ plane. Both the kinetic energy flux density and
MHD energy flux density basically increase with the increase
of flow velocity. This indicates that the flow velocity is a
main factor in determining the energy transport of flows.
The most important result shown in Figure 2 is that the kinetic energy flux density QKx is much larger than the MHD
energy flux density QMHDx. The ratio QKx/QMHDx ranges
from 1 to 6 for ion flow with a velocity larger than 100 km/
s. During the intervals without fast flow, for example from
1801:00UT to 1804:00UT, the ratio QKx/QMHDx fluctuates
highly and sometimes even reaches tens. This is because
the denominator QMHDx is very small when the ion flow
velocity is close to zero.
[12] Figure 3 shows the scatter plot of the ratio of kinetic
energy flux density to MHD energy flux density QKx/QMHDx
as a function of the x component of flow velocity Vx. The
solid line denotes the mean value of QKx/QMHDx for each
100 km/s velocity window. As seen in Figure 3, the ratio
QKx/QMHDx decreases with the increase of bulk flow velocity. The mean ratio QKx/QMHDx is between 1.5 and 2 when
the flow velocity exceeds 1000 km/s. When the flow velocity
is smaller than 1000 km/s, the mean ratio QKx/QMHDx is
Figure 2. The X component of ion flow velocity Vx, kinetic
earthward energy flux density QKx, and MHD earthward
energy flux density QMHDx, and the ratio QKx/ QMHDx.
For more detail see the text.
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CAO ET AL.: ENERGY TRANSPORT OF BBFS
Figure 3. The scatter plot of the ratio of kinetic energy flux
density to MHD energy flux density QKx/QMHDx as a function of x component of ion flow velocity Vx. The solid line
denotes the mean value with a window of 100 km/s.
larger than 2.0. According to Ma et al. [2009], over 80% of
fast flows in the plasma sheet have a velocity smaller than
750 km/s. Thus, to know the ratio for most frequently
observed BBFs, we calculate the mean ratio QKx/QMHDx in
the velocity range 200–700 km/s. It is found that the mean
ratio QKx/QMHDx in the velocity range from 200 to 700 km/s
is about 2.7.
[13] To further study the kinetic characteristics of BBF,
we plot the count per second, velocity vx(E), ion number n
(E), mass flux density gx(E), and energy flux density qx(E)
for each energy channel in Figure 4. Here
N¼
G ¼ NV ¼
Q¼
X
E
X
E
qðEÞ ¼
X
nðE Þ
E
gðEÞ ¼
X1
E
2
X
(7)
nðEÞvðEÞ
(8)
E
nðE Þm v2 ðE ÞvðE Þ
(9)
[14] The velocity, density, and mass flux density calculated
by MHD approach are also plotted in Figure 4 for the purpose
of comparison. The energy spectrum of flow velocity in Figure 4e shows that the flow velocity of ions increases with the
increase of energy. The ions in the energy range 10–35 keV
(red and yellow colors) can have a very high speed (~1100–
2000 km/s), and the ions in the energy range 5–10 keV (green
colors) have a smaller speed (~400–1100 km/s). The ions
below 1 keV (blue color) have only a velocity fluctuating
around zero. The two super high-speed ion flow bursts with
a velocity larger than 1000 km/s (1754:30–1757:00 UT and
1805:30–1807:00 UT) consist of only the ions above
15 keV. Because the maximum flow velocity of ions is about
2000 km/s, it can be inferred that the upper velocity limit of
BBF in the inner plasma sheet should not exceed 2000 km/s.
[15] Figure 4f shows the energy spectrum of ion number
density. We first use it to study the energy features of lobe
and plasma sheet ions during quiet times. Prior to the
entrance to the plasma sheet at 1747:00UT, C1 is in the lobe.
In the lobe, the energy of lobe ions ranges from 600 eV to
4 keV. In the quiet plasma sheet (1800:30UT–1804:00UT),
most ions (red color) are still in the energy range 600 eV to
4 keV. However, the lower edge of ion energy extends down
to 400 eV and the upper edge extends up to 10 keV. It is interesting to find that during the interval 1747:30–1753:00,
there is a clear gap in the energy spectrum of ion number
density (see Figure 4f). This implies that the ions within this
interval should be described by two ion components: one is
the fast flow ions in the higher energy band (5–15 keV) and
the other is background quiet ions with a velocity close to
zero in the lower energy band (500 eV to 4 keV). This result
clearly shows that plasma sheet ions cannot be considered as
one ideal single fluid.
[16] The energy transport in the plasma sheet is mainly
provided by the ions above 10 keV (see the green, yellow,
and red colors in Figure 4h). This result is consistent with
previous conclusions that BBFs represent the primary transport mechanism in the plasma sheet [Angelopoulos et al.,
1996] because only bursty bulk flows can have ions above
10 keV in the present case. The mass transport in the plasma
sheet can be provided by ions with energy down to 1 keV.
During the interval of (1752:30 UT to 1754:00 UT, which
is characterized by high ion number density and moderate
earthward velocity, the energy transport is mainly provided
by the ions in the energy range 10–20 keV while the mass
transport is provided by the ions in a broader energy range
1–15 keV. This implies that although the lower energy ions
may not contribute much to the energy transport, they make
a significant contribution to the mass transport.
3. Interpretation
[17] It is easy to understand why the MHD approach cannot give correct energy transport of fast flows. The origin of
error comes from the ideal single fluid assumption of the
MHD approach. In fact, the MHD bulk parameters density,
average velocity, and temperature can describe correctly ion
flow only when the velocity distribution function of ions is
Maxwellian. This assumption is obviously not consistent
with satellite observations. In fact, the velocity distribution
function of plasma sheet ions is not Maxwellian and plasma
sheet ions may also consist of multiple components as
pointed out by Parks et al. [2001] and as displayed in
Figure 4f. Here we first use a simple example to display the
deficiency of MHD in estimating energy transport. For a onedimensional fluid, which consists of two counter streaming
ion flows: one has a number density N1, bulk velocity V1, and
temperature T1 and the other flow has a number density N2,
bulk velocity V2, and temperature T2. If N2 = 2 N1, V1 = –2 V2
and T1 = T2, the MHD bulk velocity of fluid V (=N1 V1+ N2
V2) will be zero. This implies that there is neither mass transport
nor energy transport according to equation (6) due to a zero
bulk velocity. However, in fact for this fluid, although there
is no mass transport, there is energy transport, which should
be described by the following energy flux density:
Z
Q¼
ð f1 ðv; t Þ þ f2 ðv; t ÞÞ
1 2
mv
2
3
v dv ¼ N1 mV1 3
8
[18] This simple one-dimensional example clearly shows
the deficiency of the single-fluid assumption of MHD.
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CAO ET AL.: ENERGY TRANSPORT OF BBFS
Cluster C1 satellite
(a)
(b)
(c)
log c/s
(d)
vx(E)
(e)
n(E)
(f)
gx(E)
(g)
qx(E)
(h)
UT on 30 July 2002
Figure 4. The energy distribution of (d) the count per second, (e) earthward ion flow velocity Vx(E), (f)
ion number density n(E), (g) earthward mass flux density g(E), and (h) kinetic energy flux density qx(E)
from 1745:00 to 1810:00 UT on 30 July 2002. The bulk ion parameters of (a) earthward bulk velocity
Vx, (b) number density N, and (c) earthward mass flux density Gx are also plotted for comparison.
However, MHD single fluid assumption is now being
adopted by almost all the teams of instruments measuring
the low energy ions to provide single fluid bulk parameters:
velocity, density, and temperature.
[19] We then use a simple one dimensional model to explain
semi-qualitatively the difference between kinetic energy flux
density and MHD energy flux density. In this model, there are
two ion components: one is cold core static components with
a number density Nc and a velocity Vc = 0; and the other is cold
high speed flow with a density Nf and velocity Vf. For the sake
of simplification, their temperatures are assumed to be zero.
[20] The velocity distribution function f is then given by
f ¼ Nc dðvÞ þ Nf d v Vf
where d(v) and d(v – Vf) are two delta functions. In the frame
of MHD theory, the bulk parameters
are then given by
Z
317
the total density of ions N ¼
fdv ¼ Nc þ Nf ;
(10)
Z
the bulk velocity V ¼
Z
the pressure P ¼
fv dv=N ¼ Vf Nf =N :
(11)
h
2 i
mðv V Þ fdv ¼ m Nc V 2 þ Nf Vf V
(12)
2
CAO ET AL.: ENERGY TRANSPORT OF BBFS
Figure 5. The ratio of kinetic energy flux density to MHD
energy flux density QKx/QMHDx as a function of number
density ratio Nf/N derived from one-dimensional model.
Nc. Only during the interval of high speed flow bursts, the
two ion number densities Nf and N are comparable. Comparing the waveforms of Vx in Figure 6d and Nf/N in Figure 6e,
we find that although the ion number density above 10 keV
is very small, it plays a key role in determining the bulk velocity of ion flow. This is because the higher energy ions
have a large speed and small anisotropy in their directional
distribution can lead to a remarkable velocity increase of entire ion flow.
[26] The waveforms of Vx and Nf/N in Figures 6d and 6e
are almost identical. The velocity of ion flow is perfectly
proportional to the ratio of density Nf / N, which is consistent
with equation (11). Also, the waveforms of QK/QMHD and 1/
[3Nf/N – 2(Nf/N)2] in Figures 6f and 6g are very similar,
which is in agreement with equation (15). The high consistency between the simple double-flow model and observations clearly indicates that the single fluid assumption of
[21] If we ignore the electromagnetic energy transport, the
MHD energy transport density QMHD is then given by
1
3
1
N
QMHD ¼ NmV 3 þ PV ¼ NmV 3 3 2
2
2
2
Nf
(13)
[22] However, if one uses directly the velocity distribution
function, the energy flux density QK can be simply given by
1
QK ¼ Nf mVf3
2
(14)
[23] Then we can obtain the following ratio of kinetic energy flux density to MHD energy flux density:
"
QK =QMHD
2 #
Nf
Nf
¼ 1= 3 2
N
N
(15)
[24] Figure 5 shows the ratio QK/QMHD as a function of
number ration Nf /N calculated from Equation (15). When
Nf/N =1, the fluid consists only one high-speed flow and
therefore QK/QMHD =1. With the decrease of Nf/N, the energy flux density ratio increases. When Nf/N =0.2, QK/
QMHD = 1.92. When Nf/N <0.2, the ratio QK/QMHD
increases rapidly with the decrease of Nf/N. The equation
(15) can well explain why the energy flux density ratio during quiet times is sometimes very large. This is because during quiet times, if the bulk velocity is very small, the number
of ions in higher energy Nf is very small and the total number density N is relatively large. Therefore, the ratio Nf/N is
very small, which will lead to a very large ratio QK/QMHD.
[25] To check if the result of this simple model agrees well
with the observations, we plotted Nc, Nf, Nf/N and N/Nf in
Figure 6. The X component of velocity Vx, total ion number
density N and the ratio QK/ QMHD are also plotted for comparison. The ion number density Nc and Nf are defined as the
ion number density below and above the energy 10 keV.
Figures 6a and 6b show that ion number density below
10 keV is at most times very close to the total number density. The ion number density above 10 keV Nf is generally
much smaller than the ion number density below 10 keV
Figure 6. (a) Total ion number density N, (b) ion number
density of lower energy ions Nc, (c) number density of
higher energy ions Nf, (d) the X component of flow velocity
Vx, the ion number density ratio Nf/N, the energy flux density
ratio QKx/QMHDx, the parameter S. Nc and Nf are defined as
the ion number density below and above the energy 10 keV,
respectively. The parameter S is defined by equation (15)
and equals to 1/[3(Nf/N) – 2(Nf/N)2].
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CAO ET AL.: ENERGY TRANSPORT OF BBFS
-2000
2000
2000
-2000
(a)
2000
Log c/s
4.0
(b)
Vy(km/s)
3.6
3.1
-2000
1755:01
(c)
2.7
(d)
2.3
Vy(km/s)
2000
1800:21
1.9
1.4
-2000
1801:25
1800:49
Vx(km/s)
Vx(km/s)
1.0
Figure 7. Four typical velocity distributions in the X-Y plane during the interval of 1745:00–1810:00 UT
on 30 July 2002.
MHD approach is not consistent with observations in the
plasma sheet and will lead to the underestimation of energy
transport of BBFs.
[27] It should be mentioned that the above double ion
component model does not necessarily require plasma sheet
ions to consist of two ion components. For example, when
Nf ~0, there is only one core component, which approximately corresponding to quiet plasma sheet ions. On the
contrary, when Nc = 0, there is only one high-speed ion component, which approximately corresponds to a high-speed
ion flow in the plasma sheet. The velocity distribution functions of plasma sheet ions are complex and do not always
display the features of two components. Figure 7 shows four
typical ion velocity distributions in the X-Y plane during this
interval. For quiet plasma sheet ions (see Figure 7d), the ion
velocity distribution is almost isotropic but superposed with
a week earthward ion flow. For moderate fast flows in the
plasma sheet (see Figures 7b and 7c), the plasma sheet ions
are composed of isotropic static ions and several earthward
fast moving ion flows. The number of higher energy ions
with an earthward moving velocity >1500 km/s increases remarkably. For high speed flows in the plasma sheet (see
Figure 7a), the ion velocity distribution is highly anisotropic.
The core static ions component almost completely disappear
(Nc ~0) and there is only one ion component, i.e., fast earthward moving ions with a speed >1500 km/s.
4. Conclusions
[28] In this paper, we use the data of three-dimensional
velocity distribution function of ions recorded by HIA/
Cluster to study the energy transport of bursty bulk flows
in the plasma sheet observed on 30 July 2002, and compare the results with those obtained using bulk parameters
of MHD approach. The kinetic energy flux density, which is
derived directly from three-dimensional velocity distribution
function of ions, is obviously larger than the energy flux density calculated using the bulk parameters (density, velocity,
and temperature) of the MHD approach. The ratio of kinetic
energy flux density to MHD energy flux density QKx/QMHDx
during the interval of BBF ranges from 1.5 to 6. The mean ratio QKx/QMHDx in the flow velocity range 200–800 km/s is
about 2.7. This implies that the energy transports of BBF
calculated using the MHD approach in many previous studies
are much underestimated. The origin of the underestimation
of energy transport comes from the single-fluid assumption
of MHD, or in other words comes from the deviation of
velocity distribution of plasma sheet ions from ideal Maxwellian distribution function.
[29] The energy spectrum of energy flux density shows
that the energy transport of BBF is mainly provided by ions
above 10 keV although their number density is much smaller
than that of ions below 10 keV. The ratio QKx/ QMHDx is basically proportional to the ratio N/ Nf, where N is the total ion
number density N and Nf is the number density of ions above
10 keV. The energy spectrum of ion flow velocity shows that
the flow velocity v(E) decreases with the decrease of energy.
The ions in the energy range 10–35 keV (red color) basically
have a very high earthward speed (~1800–2000 km/s) while
the ions below 1 keV (blue color) have only a velocity close
to zero. The ratio Nf/ N is perfectly proportional to the ion
flow velocity Vx, indicating that the ratio Nf/ N is a dominant
factor in determining the flow bulk velocity.
[30] To understand the reason of the underestimation of
energy transport flux in the frame of MHD, we establish a
double ion component model in which there are two ion
components: high speed flow and core stationary ions. The
theoretical analysis indicates that the single fluid assumption
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CAO ET AL.: ENERGY TRANSPORT OF BBFS
of MHD can cause the underestimation of energy transport
density. This model also well explains the correlation between Nf/ N and Vx.
[31] In fact, there are some other interpretations for the
difference between Qk and QMHD. It is possible that the use
of larger heat capacity ratio would reduce the difference
between Qk and QMHD. Here we used only the heat capacity
ratio g= 5/3 because this value is extensively used in previous
studies of energy transport of BBFs and we can easily compare previous results with ours. In a real gas, the heat capacity
ratio g is actually a function of temperature and pressure.
Therefore, it is difficult to find a heat capacity ratio that can
be used for all fast flows. Thus, the most credible method
to calculate energy transport of BBFs is to use directly the velocity distribution function.
[32] The heat flux q, which is neglected in previous estimation of energy transport, is another important factor that
causes the difference between QK and QMHD. Because the
heat flux q is a third-order moment and cannot be expressed
in terms of average MHD parameters (density, average velocity, and temperature), almost all previous studies of BBFs
assume it to be zero. However, the heat flux for a non-Maxwell velocity distribution is not zero.
[33] Therefore, the previous method to calculate the energy transport of BBFs is only correct in the case of Maxwellian velocity distribution. This is because for Maxwellian
velocity distribution, the heat capacity ratio g= 5/3 and the
heat flux q = 0. Physically speaking, the underestimation of
energy transport of BBFs in previous studies comes from
the deviation of ion velocity distribution from Maxwellian
velocity distribution.
[34] The increase of energy transport capability of BBFs is
important to the understanding of substorm energy transport.
Previous studies show that the bursty bulk flows in the
plasma sheet can provide neither sufficient energy transport
rate nor sufficient energy transport needed for a typical substorm [Angelopoulos et al., 1996]. Angelopoulos et al.
[1994] compared the energy transport of a BBF with
expected energy transport of a typical substorm. They found
that although BBFs represent the major energy transport
mechanism in the plasma sheet, a BBF with a duration of
10 min can only provide 70% of energy transport rate of a
substorm and only 5% of energy consumption of a substorm.
However, they all use an MHD approach to estimate the
energy transport of bursty bulk flow. Our results indicate
clearly that the energy transport of BBFs obtained from
MHD approach is much underestimated. Figure 3 shows
that the mean ratio QKx/QMHDx in the velocity range from
200 to 700 km/s is about 2.7. Therefore, if one uses kinetic
approach, a BBF may possibly provide 190% (70%2.7)
of energy transport rate and 14% (5%2.7) of energy consumption of a substorm. Considering a typical substorm
may last 2 h (120 min) and the occurrence rate of BBF in
the inner plasma sheet is 19.4% [Cao et al., 2006], the
time of BBF may be 24 min within the 2 h of a typical
substorm. Therefore, BBFs can provide 34% (14%2.4)
of energy consumption of substorm.
[35] In Cao et al. [2006], the occurrence frequency of
BBFs in the central plasma sheet is 9.5% for single satellite
C1 and 19.4% for three satellites (C1, C3, and C4) that are
separated by a distance from hundreds to tens of thousands
of kilometers. It is reasonable to infer that if more satellites
are distributed in a wider range along the Y axis in the inner
plasma sheet, the occurrence rate of BBFs can be further
increased. Thus, it is inferred that for a typical substorm,
the ratio of the energy transport of BBF to the substorm
energy consumption may be larger than 34%.
[36] Acknowledgments. This work was supported by NSFC Grant
40931054 and 973 program 2011CB811404.
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