Cluster observations of bidirectional beams caused by
electron trapping during antiparallel reconnection
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Egedal, J. et al. “Cluster Observations of Bidirectional Beams
Caused by Electron Trapping During Antiparallel Reconnection.”
Journal of Geophysical Research 115.A3 (2010). Copyright ©
1999–2013 John Wiley & Sons, Inc.
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http://dx.doi.org/10.1029/2009ja014650
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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, A03214, doi:10.1029/2009JA014650, 2010
for
Full
Article
Cluster observations of bidirectional beams
caused by electron trapping during antiparallel reconnection
J. Egedal,1 A. Lê,1 N. Katz,1 L.‐J. Chen,2 B. Lefebvre,2 W. Daughton,3 and A. Fazakerley4
Received 15 July 2009; revised 19 September 2009; accepted 6 October 2009; published 23 March 2010.
[1] So‐called bidirectional electron beams have been observed by a number of spacecraft
missions mainly in the inflow of reconnection regions. Here we show that these beam‐like
features in the electron distribution function are explained by electron trapping. The
trapping is mainly controlled by a positive acceleration potential, Fk, which is related to
the structure of the parallel electric fields in the vicinity of the reconnection region. Guided
by the results of a kinetic simulation, we extend a recent analytical model for the electron
distribution function applicable to the inflow region in antiparallel reconnection. The
model is successfully compared to data observed by the four Cluster spacecraft inside an
active reconnection region. The anisotropy recorded in the electron distributions is
consistent with mainly electric trapping of electrons by Fk. In the analysis we determine
the profiles of Fk along the paths of the Cluster spacecraft during their encounter with a
reconnection region. Typical values of eFk are in excess of 1 keV (much higher than the
electron temperature in the ambient lobe plasma) and Fk traps all thermal electrons. This is
important for the internal structure of the Hall current system associated with the ion
diffusion region because extended trapping significantly alters the electrical and kinematic
properties of the electron fluid. Finally, at the boundary between the inflow and exhaust
regions the values of eFk in the inflow region smoothly approach the shoulder energies (up to
15 keV) of the so‐called flat‐top distribution observed in the reconnection exhaust. This
suggests that Fk may be important also to the formation of the flat‐top distributions.
Citation: Egedal, J., A. Lê, N. Katz, L.‐J. Chen, B. Lefebvre, W. Daughton, and A. Fazakerley (2010), Cluster observations of
bidirectional beams caused by electron trapping during antiparallel reconnection, J. Geophys. Res., 115, A03214,
doi:10.1029/2009JA014650.
1. Introduction
[2] Magnetic reconnection [Dungey, 1953] plays a fundamental role in magnetized plasmas as it permits the rapid
release of magnetic stress and energy through changes in the
magnetic field line topology. It controls the spatial and
temporal evolution of explosive events such as solar flares,
coronal mass ejection, and magnetic storms in the Earth’s
magnetotail driving the auroral phenomena [Taylor, 1986;
Vasyliunas, 1975; Phan et al., 2000; Masuda et al., 1994].
Because of its importance to understanding violent macroscopic phenomena in nearly all magnetized plasmas, reconnection has been the subject of an increasing number of
numerical investigations. Such studies have shown that the
Hall effect plays an important role in determining the rate of
1
Plasma Science and Fusion Center and Physics Department,
Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.
2
Space Science Center, University of New Hampshire, Durham, New
Hampshire, USA.
3
Los Alamos National Laboratory, Los Alamos, New Mexico, USA.
4
Mullard Space Science Laboratory, University College London,
Dorking, UK.
Copyright 2010 by the American Geophysical Union.
0148‐0227/10/2009JA014650
reconnection [Birn et al., 2001, 2005]. However, a detailed
understanding of the electron dynamics is still needed to
explain the spontaneous onset of reconnection and to understand how magnetic field energy is converted into kinetic
energy in the electron and ion fluids.
[ 3 ] The Hall effects were first discussed by Sonnerup
[1979] and are related to the current pattern that results
when the inertia of the ions causes the electron and ion fluids
to decouple within the ion diffusion region. The characteristic
quadrupole Hall magnetic fields predicted have since been
observed by numerous spacecraft in the Earth’s magnetotail
[Nagai et al., 2001; Øieroset et al., 2001; Borg et al., 2005]
and also in laboratory experiments [Ren et al., 2005; Brown
et al., 2006].
[4] Measurements taken by the Geotail and the Cluster
spacecraft [Nagai et al., 2001; Chen et al., 2008a] have
revealed that the bulk electrons are heated during reconnection. Within the reconnection outflow the electrons often
have a characteristic isotropic flat‐top distribution, where
the phase space density, f, of the electrons is nearly constant
from thermal energies up to several keV. The associated
heating could be caused by the thermalization of electron
beams flowing toward the reconnection X line as a part of
the Hall current system [Nagai et al., 2001; Hoshino et al.,
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2001; Manapat et al., 2006; Asano et al., 2008]. Numerous
investigators [Nagai et al., 2001; Øieroset et al., 2002;
Vogiatzis et al., 2006; Nakamura et al., 2006; Asnes et al.,
2008; Asano et al., 2008] also report on related bidirectional
beams mainly within the inflow regions in magnetotail
reconnection.
[5] Egedal et al. [2005] accounted for the bidirectional
beam structure observed by the Wind spacecraft in the deep
magnetotail by a numerical study of electrons trapped in
parallel electric fields. Recently, this mechanism has been
confirmed in kinetic simulations of reconnection with a
guide magnetic field [Le et al., 2009]. The anisotropy/beam
structure develops due to constraints on the electron distribution function imposed by constant of motion variables
and Liouville’s equation (df/dt = 0 along electron orbits).
For the case of antiparallel reconnection additional investigations by Le et al. [2010] show that the strength of the out‐of‐
plane Hall magnetic fields is directly linked to the electron
anisotropy. This emphasizes the important role of the electron
pressure anisotropy in regulating the Hall current system.
[6] In this paper we investigate the details of the bidirectional beam distributions observed by Cluster during reconnection in the magnetotail [Chen et al., 2008a, 2009].
The pressure anisotropy for this antiparallel reconnection
geometry is also caused by trapping of electrons mainly in
parallel electric fields. The starting point for the analysis is a
theoretical form for the electron distribution function, f,
derived by Egedal et al. [2008]. This model distribution
includes the effects of both magnetic trapping and electric
trapping by the acceleration potential Fk related to parallel
electric fields in the reconnection region [Egedal et al.,
2009]. While the model is in excellent agreement with the
electron distributions observed in kinetic simulations [Le et
al., 2009, 2010], for the event encountered by the Cluster
mission the values of Fk are so large that the theoretical
form previously derived for f needs to be generalized to
include additional physics influencing the electron dynamics.
A new extended model for f is derived here. This new model
is in good agreement with the Cluster observations and allows
for a detailed characterization of Fk along the paths of the four
Cluster spacecraft. In turn, the magnitude of Fk appears to be
related to the heating of the isotropic flat top distribution
within the reconnection exhaust.
[7] The paper is organized as follows. Section 2 addresses
the role of the acceleration potential in a kinetic simulation of
antiparallel reconnection. In section 3 we revisit the previously derived theoretical form for f, followed in section 4 by
a discussion of why a positive acceleration potential is generic in collisionless reconnection. In section 5 we derive an
extended model for f which is applied in section 6 to account
for the anisotropic electron distributions measured by Cluster.
The role of the pressure anisotropy in on the structure of the
reconnection region is discussed in section 7 and the paper is
concluded in section 8.
2. Acceleration Potential in a Kinetic Simulation
[8] In this section we will examine the characteristics of
the reconnection geometry in a neutral sheet predicted by a
kinetic simulation. In particular we will focus on the parallel
electric fields observed in the simulation and their influence
on the electron distribution. The overview of the recon-
nection geometry provided by the simulation will bring into
perspective the results of the subsequent analysis of electron
distribution functions measured by the Cluster spacecraft.
[9] Reconnection around an X‐type neutral point is
studied using a fully kinetic particle‐in‐cell (PIC) simulation
[Daughton et al., 2006]. Figure 1 shows results from the run.
The PIC code is translationally symmetric in the z direction
and has a total domain of 2560 × 2560 cells = 400 de × 400de
(note that our z axis corresponds to the GSM y axis generally
used in magnetotail data analysis). The initial state is a Harris
neutral sheet with gradients in the y direction and is characterized by the following parameters: mi/me = 400, Ti/Te = 5,
wpe/wce = 2, and background density = 0.3 n0 (peak Harris
density). Magnetic reconnection with a single X line evolves
from a small perturbation, and we consider a time with
approximately steady‐state reconnection.
[10] In the simulation the density n is fairly uniform in the
vicinity of the X line, while the value of B becomes very low
(Figures 1a and 1b). The characteristic quadrupolar out‐of‐
plane Hall magnetic field Bz is shown in Figure 1c. The
temperature anisotropy that develops in the inflow region is
of special interest for the present analysis. As seen in
Figures 1d and 1e, the parallel temperature is greatly enhanced approaching the neutral sheet, whereas the perpendicular temperature decreases. Meanwhile, in the outflow
region the parallel and perpendicular temperatures are identical. We will later use the abrupt change in the temperature
anisotropy to identify the inflow and outflow regions when
analyzing experimental data of spacecraft missions [Chen et
al., 2008a].
[11] Le et al. [2010] demonstrated that the temperature
anisotropy of the inflow region in antiparallel reconnection
is controlled by the formation of a positive acceleration
potential. This potential is defined as
Z
k ðxÞ ¼
1
x
E dl;
ð1Þ
and it should not be confused with the regular (in‐plane)
electrostatic potential F. The integral in equation (1) is taken
along the magnetic field lines, following the field lines all
the way out to the ambient ideal plasma where E · B = 0.
Thus Fk is only a “pseudopotential” that measures the work
done by the total electric field as an electron escapes the
reconnection region in a straight shot along a magnetic field
line. In a three‐dimensional reconnection geometry Fk includes contributions from both F and the inductive electric
fields, which are primarily in the z direction. Outside the
reconnection region these contributions cancel, while in the
reconnection region small, but finite, values of the parallel
electric field Ek develop. Because the integral in equation (1)
is over relatively large length scales, the magnitude of Fk
becomes significant.
[12] For the kinetic simulation we can evaluate Fk directly
because the magnetic and electric fields are known throughout the reconnection domain. Figures 2a and 2b show contours of constant in‐plane potential, F, and the reconnection
electric field, Ez, respectively. We note that F is negative in
the outflow region and has a structure that is consistent with
that inferred by Wygant et al. [2005] from Cluster observations in the magnetotail. As mentioned above, Ez provides an
important contribution to the parallel electric field. This is
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Figure 1. Results of particle‐in‐cell (PIC) simulation: Color contours of n, ∣B∣, Bz, Tk, and T?. The overlaid contours of Y coincide with magnetic field lines.
evident in Figure 2c, which shows contours of constant
Ek = E · B/B. The quadrupole structure of Ek results from the
contributions of the nearly uniform Ez and the characteristic
quadrupole structure of the Hall magnetic field Bz. We note
that the magnitude of Ek is nearly a factor of 10 less than
that of Ez. Nevertheless, because of the long path of integration in equation (1), the magnitude of Fk shown in
Figure 2d (reaching 4Te/e) is comparable to that of F.
Another important point is that a positive Fk trapping
electrons and a negative F in the outflow region are not
mutually exclusive. This may appear counterintuitive, but as
demonstrated here, it follows as a result of the contributions
to Fk from the inductive electric fields.
[13] The local minimum in the magnetic field centered on
the reconnection region (see Figure 1b) provides an effec-
tive potential well that can trap electrons which have pitch
angles Q = ∠(B, v) close to 90°. The strength of the magnetic mirror force scales with the perpendicular energy, and
for thermal electrons one may quantify the “depth” of the
magnetic well to be on the order of Te. Thus for trapping of
thermal particles the acceleration potential eFk ∼ 4Te is
much more important, and it applies to all the thermal
electrons independent of their pitch angle. The parallel
motion of the electrons in the inflow region is therefore
mostly dictated by Fk, which results in the close correlation
between Tk and Fk apparent when comparing Figure 1d and
Figure 2d.
[14] The trajectory in Figure 3 is typical for electrons
encountering the reconnection region. The trajectory is
overlaid on contours of constant values of B = ∣B∣. In the
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Figure 2. (a) Color contours of constant in‐plane potential F, (b) the reconnection electric field Ez,
(c) the parallel electric field, and (d) the acceleration potential Fk.
inflow region the electrons are well magnetized such that the
magnetic moment m = mv2?/(2B) is an adiabatic invariant.
This effectively controls the pitch angles of the incoming
electrons and causes v2? to decrease proportionally to B as
the electron diffusion region is approached. The decreasing
values of T? in the inflow region of Figure 1e are therefore
directly related to the magnetic field strength in Figure 1b.
[15] Meanwhile, as seen in Figure 3, the electrons pass
through a layer of negligible B in the outflow region. Here
the electrons are totally demagnetized, and their pitch angles
scatter randomly each time they pass through the layer. This
is the mechanism that makes the parallel and perpendicular
temperatures equal in the outflow.
[16] The present simulation tracks the positions and velocities of ∼2 · 109 electrons, which is sufficient for reconstructing
reliable electron distribution functions throughout the simulation domain. For the two points marked by circles in
Figure 2d, we provide (in Figure 4) contours of constant f in
the (vk, v?) plane. For the point in the outflow the contours
of f are approximately half circles corresponding to the
complete randomization of the pitch angles as the electrons
scatter in the layer of small B.
[17] The contours of f in Figure 4b are for the point
marked in the inflow region where eFk ∼ 2Te. The distribution has a characteristic central part where the contours
are nearly independent of vk and can therefore be characterized as a bidirectional beam distribution [Asano et al.,
2008]. The transition from this region of straight contour
lines to the rounded lines approximately coincides with the
Figure 3. Typical trajectory of a trapped electron overlayed on contours of constant ∣B∣.
4 of 17
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Figure 6. Illustration of the general reconnection geometry
considered for deriving the form of the theoretical electron
distribution.
Figure 4. Contours of constant f in the (vk, v?) plane.
(a) The distribution is from the point marked in Figure 2d
in the flow region of the PIC simulation. (b) The distribution is for the point marked in the inflow region in Figure 2d.
two dashed lines that represent the trapped‐passing boundary obtained by solving for vk in the equation
E ðxÞ ek ðxÞ B1 ¼ 0:
ð2Þ
This equation expresses the physical condition that marginally trapped electrons will deplete their parallel energy
(E k∞ ∼ 0) as they move along field lines into the more
uniform plasma at the simulation boundaries. Thus eFk is
the energy at which the bidirectional beams terminate.
[18] In the following sections we will study the anisotropy
in the electron distributions by considering graphs similar to
those of Figure 5, which show the distributions of Figure 4
as functions of energy evaluated at Q = 0° and Q = 90°. For
the isotropic distribution of the outflow region we see (as
expected) in Figure 5a that log10(f) is nearly identical for the
two pitch angles. Meanwhile, in Figure 5b for Q = 0° we
find that log10(f) is constant in the interval 0 < E < eFk and
only falls off for E > eFk. For Q = 90° the values of log10(f)
have the form of a regular Maxwellian distribution.
3. Adiabatic Kinetic Theory
[19] Above we indicated a strong link between the parallel
electron temperature and the acceleration potential, Fk. This
relationship was recently quantified by an adiabatic theory
for the electron distribution [Egedal et al., 2008]. The model
assumes magnetized electrons and is valid in the limit where
the electron thermal speed is much larger than the Alfvén
speed, vth,e vA. The model is derived for a general reconnection configuration outlined in Figure 6 where the
X line geometry is embedded in a current sheet. When
following a static field line away from the reconnection
region it is assumed that it reaches an ideal region where
E · B = 0 and where the electrons are characterized by
an isotropic distribution f∞(E ∞); we will denote this as the
ambient plasma (for the Cluster observation this corresponds
to the lobe plasma).
[20] From Liouville’s theorem (df/dt = 0 along the trajectory) it follows that the phase space density f(x, v) for a
point (x, v) inside the reconnection region is identical to
f∞(x∞, v∞), where (x∞, v∞) is the phase space point in the
ambient plasma. Because the distribution in the ambient
plasma is isotropic, to determine f(x, v) = f∞(x∞, v∞) =
f∞(E ∞) we need only characterize the kinetic energy E ∞ that
the electron had before it entered the reconnection region.
For this we need to distinguish between trapped and passing
electrons. An electron will only become trapped if its initial
∣vk∣ in the ambient plasma is small. Thus for most trapped
electrons E ∞ ∼ mB∞, where B∞ is the strength of the magnetic field in the ambient plasma. Meanwhile, for passing
electrons, which enter and leave the reconnection region in a
single shot along a magnetic field line, the kinetic energy in
the ambient plasma is approximately given by E ∞ = E − eFk.
From these relatively simple considerations regarding the
energy of the individual electrons we obtain the following
form of the electron distribution function:
Figure 5. Phase space density as a function of E for the
distributions in Figure 4, evaluated for Q = 0° and Q = 90°.
5 of 17
8
< f1 ðB1 Þ
f ðx; vÞ ¼
: f E e 1
k
;
trapped
;
passing:
ð3Þ
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EGEDAL ET AL.: ELECTRON TRAPPING IN RECONNECTION
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Figure 8. Phase space density as a function of E for the
distributions in Figure 7, evaluated for Q = 0° and Q = 90°.
pendent of E into a rapid decline. This is also consistent with
the numerical distribution in Figure 5b.
Figure 7. Contours of constant f evaluated in the (vk, v?)
plane for theoretical distributions predicted by equation (3).
(a) The distribution is obtained with B∞/B = 2 and Fk = 0. The
shaded area correspond to trapped electrons. (b) The distribution is obtained with B∞/B = 2 and a finite Fk > 0 which
enhances the number of trapped electrons. The elongated
form is consistent with the so‐called bidirectional beam
distributions.
Again, the trapped‐passing boundary is found by solving
E − eFk − mB∞ = 0. One remarkable feature of this distribution is that it depends only on the local value of Fk and
the local value of B (through m = mv2?/(2B)) and is therefore
independent of the detailed geometry of the reconnection
region.
[21] Figure 7a illustrates the theoretical distribution in the
case where Fk = 0 and B∞/B = 2 by contours of constant f in
the (vk, v?) plane. For this case the trapped‐passing
boundary is simply given by the cone v2k/v2 = 0.5 (or Q =
45° and Q = 135°). The trapped electrons are found for 45° <
Q < 135° where f is independent of vk. Elsewhere f is identical
to the ambient distribution f∞. We note that for the trapped
region f < f∞, and it follows that pure magnetic trapping
(Fk = 0) leads to a decrease in the electron density, n = ∫ f d3v,
and the perpendicular pressure, p? = 0.5me ∫ f v2? d3v.
[22] A distribution obtained with a positive Fk and B∞/B = 2
is illustrated in Figure 7b. Here the trapped‐passing boundary
starts at v2k = 2eFk/m for v? = 0 and asymptotes to Q = 45°
(and Q = 135°) for v? → ∞. Thus the main impact of Fk
is to elongate the trapped section of the distribution in both
the negative and positive directions along the vk axis, leading
to an increased electron density and parallel pressure, pk =
me ∫ f v2k d3v. Good qualitative agreement is found when
comparing the distribution of the kinetic simulation in
Figure 4b with that in Figure 7b. Quantitatively, agreement
between the analytical distributions and those of kinetic
simulation has been verified by Le et al. [2009].
[23] In Figure 8 the distributions of Figure 7 are evaluated
as a function of E for Q = 0° and Q = 90°. For the following
analysis it is important to note that for Q = 0° in Figure 7b
E = eFk determines where f transitions from being inde-
4. Why a Localized Acceleration Potential?
[24] Above we demonstrated the formation of a positive
structure in Fk in the considered simulation of antiparallel
reconnection. Similar structures of Fk have been observed
also in simulations with a guide magnetic field [Egedal et
al., 2009]. The positive structure in Fk does not develop
arbitrarily. In contrast, we find that Fk is the main parameter
that controls the density of the electrons.
[25] To elucidate this point we integrate f in equation (3)
and obtain the density n = ∫ f d3v = n(B, Fk), which clearly is
a function of only the local values of B and Fk. Figure 9a
shows n(B) evaluated for Fk = 0. We notice that n depends
strongly on B and that n declines with small values of B.
Thus in the hypothetical case where Fk is negligible in the
reconnection region the electron density would decline
rapidly as the magnetic X line (with B = 0) is approached.
Meanwhile, in Figure 1a we see that the density is uniform
in the reconnection region which is only possible with the
formation of a positive structure in Fk.
[26] Numerically, n(B, Fk) is readily evaluated for any
values of B and Fk and the curve in (B, Fk) space defined by
n(B, Fk) = n∞ can be identified. The result is shown in
Figure 9b where the value of Fk that produces n = n∞ is
plotted as a function of B. We notice the sharp increase in Fk
as B decreases. The dashed curve in Figure 9b represents the
Figure 9. (a) Density as a function of ∣B∣ for Fk = 0.
(b) Acceleration potential Fk as a function of ∣B∣ for
n = n∞.
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EGEDAL ET AL.: ELECTRON TRAPPING IN RECONNECTION
Figure 10. Solid line represents the values of Fk in the kinetic simulation as a function of y for a vertical
cut close to the X line (x/de = 210). The dashed lines are the value predicted by the adiabatic kinetic theory. This model breaks down in the layer where the electrons become unmagnetized.
~ − 0.5)2, where ñ = n/n∞
approximate from eFk/Te = (p/4)(ñ/B
~ = B/B∞. This scaling emphasizes how the localized
and B
structure of Fk is linked to the local minimum in B centered
on the reconnection region.
[27] The function n(B, Fk) is monotonic in both B and Fk
and can therefore be inverted numerically to give Fk(n, B).
Taking n and B along a cut (x/de = 210) through the simulated reconnection region, we can compare the theoretical
predicted Fk(n, B) with that of Figure 2d found by direct
integration of the electric fields. The result is shown in
Figure 10 where the full curve represents the cut through
Figure 2d and the dashed curve is obtained using Fk(n, B).
The two curves are in good agreement up to a layer about
15 de wide. Within this layer the electrons are unmagnetized
and the theory breaks down. The agreement outside this
layer further validates the theoretical form of f in equation
(3) and shows that Fk is the key parameter that controls
the electron density. The detailed structure of Fk develops
such that the density of the electrons is equal to that of the
ions as required by the condition of quasi‐neutrality.
[28] From the mathematical form of f in equation (3) it is
also possible to derive general equations of state, pk(n, B)
and p?(n, B). For numerical simulations including a guide
magnetic field the equations of state are consistent with the
numerical results in the entire simulation domain [Le et al.,
2009]. While for antiparallel reconnection the equations of
state apply only to the inflow region, in the work of Le et al.
[2010] it has been shown they can be used to relate the
structure of the inner electron diffusion region to the upstream
electron pressure. In fact, it is the upstream pressure normalized to the magnetic field pressure, be∞ = 2m0p∞/B2∞ which
controls the strength of the Hall magnetic fields (Bz), the
temperature anisotropy, and the maximum value of Fk. Le et
al. [2010] also derived a theoretical scaling for the maximum value of the acceleration potential
ek
Te1
1
2
max
"
4~n
e1
1=4
#2
1
;
2
ð4Þ
(where ñ = n/n∞) which was tested against kinetic simulations for a range of values of b e∞ 2 [0.05;0.1].
5. Correction to the Adiabatic Theory
[29] We have found that the theoretical form of f in
equation (3) agrees well with kinetic simulations of reconnection. Nevertheless, we show below that the theory has to
be generalized in order to account for electron distributions
measured in situ by the four Cluster spacecraft.
[30] One of the assumptions in deriving equation (3) is
that in order for an electron to be trapped its parallel velocity
in the ambient plasma, ∣vk∣, must be small. When estimating
the particle energy in the ambient plasma E ∞ = mB∞ + E k∞
this allowed us to apply the approximation E k∞ = 0 relative
to Te. Egedal et al. [2008] obtained the estimate E k∞ ∼
eFkmax/10. Therefore in the kinetic simulation with eFkmax ∼
4Te we have E k∞ < Te/2 and the approximation E k∞ ’ 0 is
applicable.
[31] However, for the reconnection event encountered by
Cluster the upstream electron pressure is very small (b e∞ ’
5 · 10−3) and from the scaling in equation (4) we therefore
expect eFkmax ’ 12 Te. Thus for this event electrons may
become trapped with E k∞ ’ eFkmax/10 > Te and the approximation E k∞ ’ 0 may not be justified.
[32] To include the effect of a finite E k∞, we generalize
equation (3) as
f ðx; vÞ ¼
8
<
f1 B1 þ E k1
;
trapped
:
f1 E ðxÞ ek ðxÞ þ E k1
;
passing;
ð5Þ
where the trapped passing boundary is still found by solving
E − eFk − mB∞ = 0. Here we have introduced the effective
parallel energy hE k∞i. This term is included also in the part
of the expression for the passing electrons, corresponding
to the physical condition that f must be continuous across
the trapped/passing boundary. When estimating hE k∞i, the
averaging is carried out over the possible range of E k∞ that
is compatible with electron trapping. Below we show that
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EGEDAL ET AL.: ELECTRON TRAPPING IN RECONNECTION
Figure 11. (a) Sketch of the magnetic and eletrostatic geometry applied in a simple numerical study of electron trajectories to determining hE k∞i. (b) Loci of electron bounce
points for a number of different values of J.
this average is not just a simple average over the range of
E k∞ yielding trapping, rather it must be calculated as a
weighted average, where the weight includes the phase‐
space densities f∞ (mB∞ + E k∞) in the abient plasma.
[33] To obtain an approximation for hE k∞i, we study a
simple geometry with a homogeneous magnetic field of
20 nT in the x direction and a 4 mV electric field in the
z direction. In this geometry, electrons E × B drift with constant velocity in the negative y direction. We introduce an
acceleration potential
k ¼ ¼ 0 exp !
x2 y2
;
dx2 dy2
the integral is carried out over one full period of the bounce
motion. This quantity is an adiabatic invariant provided that
the bounce motion is sufficiently rapid. Egedal et al. [2009]
showed how the loci of bounce points can be obtained for an
electron orbit characterized by a given value of J. Figure 11b
is obtained by applying these methods to the present configuration and illustrates the loci of bounce points for a range
of different values of J. The meaning of these loci may
become clearer by considering Figure 12, where five guiding
center orbits are shown all characterized by the same value
of J. We introduce the bounce phase J as a variable by
which we can discriminate between the different trajectories
with a particular value of J. For the present analysis the
details of how the loci of bounce points are determined and
exactly how J is defined are not important. Rather, the
discussion of J and J is included to emphasize that E k∞ =
E k∞(J, J) is a function of J and J only. In turn, there exists
a unique relationship, E kmax(J), between J and E kmax, where
E kmax is the maximum value of E k along a given trajectory.
This maximum occurs at x = 0 and y ’ 0. It follows that by
using the functions E kmax(J) and J = J(x, v) we only need to
obtain g(E kmax, J) in order to characterize g(x, v) for any
phase‐space point (x, v).
[36] Further simplification is possible because any measurement of f by a spacecraft corresponds to the average
value of f over some finite volume in velocity space. We
assume that the phase information, J, of the bounce motion
is averaged out over this volume such that we only need
to determine the bounce‐phase averaged value, g(E kmax) =
hg(E kmax, J)iJ. For each value of E kmax considered we
determine the value of J and numerically track an ensemble
of 400 trajectories for which the phases, J, in their bounce
motion are uniformly distributed (as shown in Figure 12).
We thereby obtain E k∞(E kmax,J) and can then calculate the
ensemble average
g E kmax ¼
ð6Þ
which controls the parallel electron motion. Here the numerical values, F0 = 5 kV, dx = 107 m, and dy = 106 m, are
chosen to be representative of the event observed by Cluster.
Note that since this configuration is characterized by a homogeneous magnetic field and Bz = 0, the regular in‐plane
potential and the acceleration potential are identical, Fk = F.
The geometry is outlined in Figure 11a.
[34] While this configuration is much simpler than that of
an actual reconnection inflow region, it still allows us to
study how electrons become trapped in a localized acceleration potential. Because the magnetic field is uniform and
the magnetic moment is conserved, the perpendicular energy
of the electrons remains constant during their motion. This
greatly simplifies the analysis as we only have to consider
the parallel motion. For a Maxwellian f∞ the distribution of
the trapped electrons in equation (5) can be rewritten as
f(x, v) = f∞(mB∞) g(hE k∞i), where g(hE k∞i) = exp(−hE k∞i/Te).
The approximation for hE k∞i as a function of (x, v) is
obtained below by first determining g as a function of (x,v)
through a numerical study of the guiding center orbits of
the trapped electrons.
[35] The dynamics of the trapped electrons can be analyzed by considering the action integral J = ∮ vkdl, where
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E k1
exp :
Te
J
ð7Þ
The resulting g(E kmax) shown in Figure 13 is obtained for
F0 = 5kV and Ez/(mV/m) = 1, 2, and 4.
Figure 12. Examples of orbits with one particular value of
J and the corresponding loci of the bounce points. Each orbit
is characterized by a particular value of J. Because the drift
speed in the y direction is constant, the angle at which the
electrons enter the trapping region is directly related to E k∞.
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the entire electron trajectory is simply given by E max = E k,m.
For all other cases the exact relationship between E max and
E k,m can only be established through a comprehensive analysis of the second adiabatic invariant, J. However, as a
general approximation we find that E max and E k,m are nearly
proportional
E kmax ’ E k þ e k ;
ð12Þ
where typical values of a are in the range of 1 to 10. Defining
E′c = E c/a as the local critical energy of the particular field line
our final form of hE k∞i is
!
E k þ e k 1:7
:
E k1 ’ 0:7 Te log 1 þ
0
Ec
Figure 13. Numerically determined values of g(E kmax) for
three different values of Ez. The lines approximating the
numerical data points were obtained with the analytical
expression in equation (8).
[37] In Figure 13 the fitted lines correspond to the
approximation
g E kmax ’
!0:7
1
1:7
1 þ E kmax =E c
;
ð8Þ
where the value of the critical energy, E c is given by
1 12 3
k 0 2
Ec
Ez
Te 4
¼ 30
:
1 eV
1 keV
1 mV=m
1 eV
ð9Þ
[38] For the present configuration the approximation in
equation (7) has been verified for a large range of Te, F0 and
Ez. For values applicable to the reconnection event encountered by Cluster (F0 = 5 kV, Ez = 4 mV/m and Te = 50 eV) we
find that E c ’ 600 eV.
[39] From the approximate form of g in equation (8) and
recalling g(hE k∞i) = exp (−hE k∞i/Te) we obtain an equation
for the effective parallel energy in the ambient lobe plasma,
hE k∞i, characteristic for those electrons that later become
trapped in the inflow region
E k1
exp ¼
Te
1
!0:7
1:7
1 þ E kmax =E c
:
ð10Þ
This equation is readily solved yielding
E k1 ’ 0:7 Te log 1 þ ðE=E c Þ1:7 :
ð11Þ
[40] The final step in obtaining an analytical form for
hE k∞i that can be used in the analysis of experimental
spacecraft data is to express E max in terms of the local value
E k. Let Fk,m be the maximum value of Fk along a given field
line. Then, the maximum parallel energy E k,m along that
field line is given by E k,m = E k + e(Fk,m − Fk) = E k + eDFk,
where DFk = Fk,m − Fk is the difference between the local
value of Fk and Fk,m. For the special case where Fk,m is the
global maximum of Fk the maximum parallel energy along
ð13Þ
[41] To summarize, equation (5) and equation (6) now
provide an extended model for f which has three free
parameters, Fk, DFk, and E′c. It is unlikely that this form for
hE k∞i in equation (6) will be exactly the same for an actual
reconnection event. Nevertheless, the described electron
dynamics should occur in nature and the mathematical form
for f is likely to account for the major anisotropic features of
the electron distributions observed in space.
[42] Again, our new model has three free parameters, Fk,
DFk, and E′c, that cannot be directly measured by any
spacecraft. Each of these parameters controls characteristic
features of f. The role of Fk was discussed in sections 2 and
3 as the parameter that determines the maximum energy (for
Q = 0°, 180°) at which electrons are trapped. In cases (such
as the kinetic simulation considered above) where the maximum value of eFk is less than E′c the approximation hE k∞i = 0
is valid and the simpler approximation of equation (3) applies.
[43] Another simple case is when eDFk is larger than both
hE k∞i and eFk. For this case hE k∞i is approximately independent of E k and the result of the finite hE k∞i is to simply
multiply the form of the solution in equation (3) by a
factor, g(hE k∞i), less than unity. Meanwhile, in the case
where DF ’ 0 the distribution f will decrease gradually
with E k for E k < eFk.
6. Application of the Model to Cluster Data
6.1. Reconnection Geometry Encountered by Cluster
[44] Many details of Cluster’s encounter with multiple
reconnection layers on 1 October 2001 have been documented by a number of different investigators [Runov et al.,
2003; Kistler et al., 2005; Wygant et al., 2005; Chen et al.,
2008a, 2008b, 2009]. Here we are concerned with the form
of the measured electron distributions and we will only
consider the electron data from the PEACE instrument’s
high‐energy sensor (HEEA operated with an energy range
30 eV to 26 keV), and low‐energy sensor (LEEA operated
with an energy range 5 eV to 2.5 keV) [Johnstone et al., 1997].
Combining data from HEEA and LEEA accumulated during
a single energy/angle sweep when the magnetic field is in
the sensors’ field of view, one can get a complete pitch angle
distribution at a given gyrophase accumulated in 1/8 s. The
electron data were corrected for the effects of variable
spacecraft potential electric field data from the double‐probe
electric field instrument [Gustafsson et al., 1997] at a
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Figure 14. Cartoon displays the reconnection geometry reconstructed by Chen et al. [2008a] based on
the magnetic field and multipoint constraints of observed distributions. The relative locations of the
spacecraft at equally distributed time intervals between 0948:00 and 0948:43 are specified by the letters A
though K.
5 samples per second resolution. Besides, electron pitch
angles were recalculated on the ground using data from the
fluxgate magnetometer (FGM) [Balogh et al., 2001] at a
22 vectors per second resolution. See Chen et al. [2008a] for
more detailed description of how the data was handled.
[45] The temperature anisotropy of the electron distribution was used by Chen et al. [2008a] for determining the
paths of the four Cluster spacecraft through the reconnection
geometry. Figure 14 is reproduced from Chen et al. [2008a]
illustrating the inferred geometry of the Cluster spacecraft as
they recorded the reconnection event between 0948:00 UT
and 0948:43 UT. Consistent with the relative positions of
the four spacecraft, the recorded temperature anisotropy and
the magnetic field data, it was determined that Cluster 1
passed above the reconnection layer remaining mainly in the
inflow region, while Cluster 3 and Cluster 4 sampled the
inflow region below the reconnection layer. Meanwhile,
Cluster 2 was found to pass nearly straight through the
electron diffusion region and only briefly to make contact
with both of the inflow regions.
6.2. Determination of f∞ and B∞
[46] To verify that the mathematical from of equation (5) is
consistent with the cluster observations, the first task is to
determine the distribution f∞ and the value of B∞. For this we
consider the nearly isotropic electron distributions recorded
by Cluster 1 between 0946:48 UT and 0947:10 UT. During
this time interval Cluster 1 was in the inflow region at a
location characterized by a relatively high magnetic field in
the range 24 nT < B < 35 nT. The electron data recorded by
HEEA for Q 2 7.5°, 82.5°, 97.5°, 172.5° is shown as a
function of E in Figure 15a. Note that other pitch angles are
available, but for now we are only interested in the directions nearly parallel and perpendicular to B. These data show
no evidence of a significant Fk and the weak anisotropy is due
to magnetic trapping for the electrons with Q ’ 90°.
Figure 15. (a) Measured values of the electron distribution, f, by HEEA on Cluster 1 as a function of E
for a time interval where Cluster 1 was in the inflow region. Here f is given in units of s3/km6. (b) The
same data as in Figure 15a but now the points with Q = 82.5° and 97.5° are graphed as a function of
E ∞ = EB∞/B, with B∞ = 46 nT. The line through the data points represents our fit for f∞(E) given in
equation (14).
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[47] Because Fk ’ hE k∞i ’ 0, according to equation (5)
we simply have f(E) = f∞(E) for the passing electrons.
Thus the data points in Figure 15a obtained with Q ’ 0° and
Q ’ 180° represent a direct measurement of f∞.
[48] Furthermore, the data in Figure 15a allow us to
determine B∞. According to equation (5) the trapped electrons obey f(E) = f∞(mB∞). The data obtained with Q ’ 90°
correspond to magnetically trapped electrons, so when
graphing this data as a function of E ∞ = mB∞ = E B∞/B it
should (for the correct value of B∞) coincide with the
measurements of f(E) for Q ’ 0° and Q ’ 180°. This is
confirmed in Figure 15b, which contains the same data as
shown in Figure 15a but now the data with Q 2 82.5°, 97.5°
are graphed as a function of E ∞ = E B∞/B where B∞ = 46 nT
and B is the magnetic field strength recorded simultaneously
with the electron data points. For the identified value of
B∞ = 46 nT the data for the different pitch angles fall
on a well‐defined curve. The line through the data points
is our approximation for f∞(E) which consists of two
Maxwellian terms and a power law term:
f1 ðE Þ ¼ A1 expðE=Te1 Þ þ A2 expðE=Te2 Þ
3
E 1:8
E
;
þ A3
1eV
E þ 30eV
ð14Þ
where A1 = 300 s3/km6, A2 = 22 s3/km6, A3 = 3 · 104 s3/km6,
Te1 = 50 eV, Te2 = 230 eV. The power‐law term proportional
to E −1.8 dominates at high energies (E > 3 keV) and the
factor (E/(E + 30 eV))3 is included to “knock down” the
term at low energies.
6.3. Validity of the Kinetic Model for the Electron
Distribution Function
[49] Having determined f∞ and B∞, we can now apply
the analytical models for f directly to the data recorded by
the four Cluster spacecraft. The fitting of our model to the
observations is done manually by adjusting the free parameters (Fk, DFk, and E′c) until the best visual agreement is
obtained when graphing the data points in the log10( f ) –
log10(E) representation. First, we examine the validity of our
basic model in equation (3) which only has one free
parameter, Fk.
[50] In Figures 16a–16d, measured distributions by each
of the spacecraft are shown. Each of the subplots contains
three lines. The thin dashed line is f∞ given by equation (15)
which is included for reference. Meanwhile, the two thick
lines are f as given by the basic model in equation (3). Of
these, the dashed lines correspond to Q = 90° and since at
this pitch angle all electrons are trapped, these lines are
obtained using the measured value of B for calculating
f∞(mB∞). The full lines represent equation (3) evaluated for
Q = 0° and a fitted value of Fk.
[51] For the data from Cluster 1 shown in Figure 16a the
points with Q = 7.5° and Q = 172.5° nearly coincide with
f∞. The small difference is evident in the modest value of
Fk = 50 V. The temperature anisotropy for this data is
caused mostly by magnetic trapping which yields the
reduced values of f for Q ’ 90°.
[52] The form for f in equation (3) also accurately accounts
for the data in Figure 16b recorded by Cluster 3. Here there
is a noticeable difference between f∞ (the thin dashed line)
and the data points recorded with Q = 7.5°. This enhancement
A03214
of the values of f in the parallel direction is consistent with
equation (3) when evaluated with Fk = 180 V. Thus here the
anisotropy is due to both magnetic and electric trapping of the
electrons.
[53] For the data of Cluster 2 and Cluster 4 in Figures 16c
and 16d it is possible to obtain reliable estimates of
Fk (>600 V) from the data points with Q = 7.5°,172.5°
recorded at energies above 1 keV. However, the reduced
values of f observed for E < 1 keV are not consistent with
equation (3). On the basis of the analysis above we estimate
that E′c ’ 200 eV (correspond to a ’ 3) and given the inferred
values of Fk are much larger than 200 V, it is reasonable that
the effects of finite hE k∞i now must be taken into account.
[54] The experimental data points of Figures 16c and 16d
are repeated in Figures 16e and 16f, but here the fits are
obtained based on the extended model of f given in
equation (5) and equation (6). As seen, this new model
accounts for the observations at values of E′c and DFk
(given in the caption of Figure 16) which are consistent with
our expectations from the analysis above. For both cases the
inferred value of DFk is larger than Fk and in this limit
equation (5) is only sensitive to the value of Fk and the ratio
DFk/E′c. Therefore in practice, the successful fitting of the
data in Figures 16e and 16f is obtained with only two free
parameters.
[55] The level of accuracy of the fits in Figures 16e and 16f
is typical for the measurements in the inflow region. As a
more complete example in Figure 17 we have included all the
data points measured by Cluster 1 at 0948:43.40 UT. It is seen
that equation (5) fully accounts for the anisotropic features
in f for the entire range of pitch angle data recorded at this
time by the HEEA analyzer.
6.4. Profile of Fk Within the Reconnection Region
[56] Next, we explore the evolution of the electron distribution as recorded during a crossing of the boundary
between the outflow region where f is isotropic and the
inflow region where f is anisotropic. For this we consider
the data recorded by Cluster 3 just around 0948:00 shown
in Figure 18. The eight times selected include those in
Figure 14 marked by the letters A, B, and C.
[57] The data in Figure 18a was recorded well inside the
outflow region. Our model does not include pitch angle
diffusion that causes the distribution to become isotropic
and the recorded data cannot therefore be modeled by the
analytical expression for f. We notice the characteristic shelf
in the experimental values for 1 keV < E < 14 keV followed
by an abrupt decline in f; Asano et al. [2008] denoted this a
flat‐top distribution. As seen in Figure 18, with DFk = 0 and
E′c = 300 eV we identify the value of Fk for which the
model best matched the rapid decline observed in f at high
energies (note that the obtained values of Fk are insensitive
to the applied values of DFk and E′c). This provides an
accurate measure of the energy for which the observed high‐
energy tail terminates (referred to as the shoulder of the flat‐
top distribution). We have verified that the inferred values of
eFk obtained by this procedure match the shoulder energies
inferred by visual inspection of the data.
[58] Figures 18b–18h show the data points recorded by
HEEA at each half spin of Cluster 3 corresponding to the ∼2 s
time steps between the subplots. The data in Figure 18e
has the clear anisotropic signature of the inflow region
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Figure 16. Comparison of our analytical models for f with measured distributions in the inflow region.
The time is given in seconds since 0948:00 UT. (a–d) Experimental data are compared to the basic model
for f in equation (3). This basic model is inaccurate in Figures 16c and 16d where Fk > 600 V. (e, f) The
data of Figures 16c and 16d are repeated here and are compared with the extended model in equation (5).
For the data in Figure 16e agreement is documented with Fk = 640 V, DFk = 1.1 kV, and E′c =
193 eV. The model distribution in Figure 16f corresponds to Fk = 1.1 kV, DFk = 2.5 kV, and E′c =
223 eV.
and represents the first measurement of the electron distribution after the inflow/outflow boundary was crossed.
The values recorded for f at Q ’ 180° are lower than those
predicted by the model and the value recorded for Q ’ 90°
are higher than those predicted. This is consistent with a
small level of pitch angle scattering which is unlikely to
influence the accuracy of the estimated value of Fk = 5.4 kV
because pitch angle scattering will not influence the energy
threshold for the rapid decline in f. Thus compared to the
bulk temperature of the electrons in the ambient plasma,
Te’50 eV, we find that eFk reaches values larger than
100 Te.
[59] The data in Figures 18f–18h is recorded while the
spacecraft moves deeper into the inflow region away from
the inflow/outflow boundary. During this process declining
values of Fk are inferred. Again, we notice the good agreement between our model and the observations.
[60] An analysis similar to that outlined in Figure 18 is
carried out for electron data measured by the four Cluster
spacecraft. The analysis is based on data from both HEEA
and LEEA. While we obtain estimates for all the three
free parameters of the model, the main results of the
analysis are the profiles of Fk corresponding to each of the
spacecraft.
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electron energization is much smaller than the values of
eFk reported here. This can be explained by eFk being a
localized acceleration potential such that highly energized
electrons are only observed near the center of the reconnection region where eFk is large. According to the adiabatic theory derived above, a passing electron will not
experience a net gain in energy during a transit along a
stationary magnetic field line through the reconnection
region. However, in reality the magnetic field lines are not
stationary and this allows for finite energy gains of passing
electron: An electron that enters the reconnection region at
an optimal location may be accelerated as it approach the
reconnection region while avoiding most of the decelerating parallel electric fields during its exit out of the region.
This will lead to energization levels at some fraction of the
maximum values of eFk [Egedal et al., 2009].
7. Role of Pressure Anisotropy Within the
Reconnection Region
Figure 17. Detailed comparison between the model distribution in equation (5) and the electron distribution measured
by Cluster 1 at 0948:43.40. The model distribution is
evaluated with Fk = 1.3 kV, DFk = 910 V, and E′c = 40 eV.
[61] In Figure 19 we consider a time interval centered on
the encounter with the reconnection region outlined in
Figure 14. The values of Fk obtained in the outflow region
are marked by red triangles. Thus these values represent the
energy at which the high‐energy tails of the isotropic distributions terminate. Only the HEEA analyzer had the sufficient energy range for determining these points. Meanwhile,
the blue plus symbols represent the values of Fk inferred from
HEEA data in the inflow regions (bidirectional beam distributions) and the green stars are inferred from LEEA data
also recorded in the inflow regions.
[62] For Cluster 1, 3, and 4 we notice how the values of
Fk inferred in the inflow region smoothly join the values of
Fk observed in the outflow region. Thus while our model
does not apply to the isotropic distributions of the outflow
region, the data points in Figure 19 substantiate the physical
meaning of the inferred values of Fk for the outflow regions.
It suggests that the parallel acceleration provided by Fk is
related to the characteristic shelves observed in distributions
of the outflow region (such as in Figure 18a). This aspect
will be analyzed further in future publications.
[63] The inferred time series of Fk are consistent with a
spatial structure of Fk as observed in the kinetic simulation (see Figure 2d). However, the inferred values of eFk ’
15 keV ’ 300Te are much larger than the values of eFk ’
4Te observed in the simulation. On the basis of the scaling
for eFk in equation (4) we can expect that kinetic simulations with smaller values of b e will provide values of eFk
in better agreement with those inferred from the Cluster
observations.
[64] Similar mapping techniques based Liouville’s equation have previously been applied to electron distributions in
the so‐called polar rain and to the electrons observed in the
exhaust of reconnection regions [Alexeev et al., 2006;
Sergeev et al., 2008]. In both of these cases the level of
[65] The good agreement between the adiabatic theory
and the measured electron distribution functions shows that
trapping of electrons in parallel electric field dominates the
properties of the electron fluid within the inflow region. As
mentioned above, for the case where the basic model is
applicable (eFk < 10Te) a complete fluid theory can be
obtained by taking moments over the analytical form for
f in equation (3). Le et al. [2009] showed that in the inflow
region where electron trapping is extensive the parallel and
perpendicular electron pressure obey CGL‐like [Chew et al.,
1956] equations of state: pk / n3/B2 and p? / nB.
[66] The strong pressure anisotropy predicted by these
equations of state affects electron momentum balance in the
inner electron diffusion region. Here, the electrons carry
essentially all of the current and correspondingly all of the
j × B force exerted by the magnetic field on the plasma.
We highlight the role of the pressure anisotropy by writing
steady‐state electron momentum balance in the form
0 ¼ ri B2 =2 þ p? ij þ pk p? B2 bi bj þ Fj ;
where Fj contains the electric field, nongyrotropic pressure,
and inertia contributions. Owing to the substantial current
in the electron diffusion regions, the magnetic field lines
are strongly curved and ribibj is large. The magnetic
tension force across the layer associated with the bent field
lines, is largely balanced by the anisotropic electron pressure,
such that just outside the electron diffusion region (pk − p? −
B2)’0. By combining this condition with the equations of
state Le et al. [2010] derived a scaling law for the strength of
the Hall magnetic fields
BH
’
B1
3
1=4
n e1
:
12n31
While this scaling law is only valid in the limit where eFk <
10Te, it shows that the electron pressure anisotropy is fundamental in regulating the Hall current system and controls
the internal structure of the electron diffusion region.
[67] In future studies we will attempt to generalized the
equations of state to include the regime where eFk > 10Te.
On the basis of the extended model in equation (5) we here
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Figure 18. Sequence of electron distributions measured by Cluster 3 as a boundary between the outflow
region and inflow region was crossed. (a–d) The data were measured in the outflow region. Here the fitted
value of eFk correspond to the energy at which the high‐energy tail of the nearly isotropic outflow distributions terminate. (e–h) These data were obtained in the inflow region. The inferred values of Fk decline as the spacecraft moves away from the boundary. The data in Figure 18e suggest that some level of
pitch angle diffusion was present (not included in our model).
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Figure 19. Inferred values of Fk as a function of time for each of the four Cluster spacecraft. The data
points marked in red were inferred from HEEA data recorded in the outflow regions, while the data points
in blue correspond to HEEA data of the inflow regions. The values of Fk marked in green were inferred
from LEEA data in the inflow regions. Values of Fk based on the electron distributions in Figure 18 are
indicated by the yellow shading.
expect the pressure anisotropy to be even stronger than that
predicted by the CGL‐like equations of state.
8. Conclusion
[68] In this paper we have investigated the role of parallel
electric fields within the reconnection region in antiparallel
reconnection. The parallel electric fields are conveniently
described by the acceleration potential, Fk, as given in
equation (1), which is a measure of the work of the parallel
electric field on an electron which escapes (or enters) the
reconnection region along a magnetic field line. In a kinetic
simulation it is found that positive values of Fk develop in
the reconnection region while negative values of the regular
potential, F, are observed in the outflow region. The negative values of F do not greatly influence the dynamics of
the magnetized electrons but are responsible for driving the
ions through the reconnection region. Meanwhile, the
electrons respond strongly to Fk due to their small inertia
and rapid motion along the field lines.
[69] The positive value of Fk causes extensive trapping of
the thermal electrons within the reconnection region which
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in turn strongly influence the electron distribution function,
f. Previously, it was shown that our basic model, f = f(B, Fk),
for the electron distribution can account for the details of the
temperature anisotropy in the inflow region of the kinetic
simulation [Le et al., 2009]. However, this model is only
valid for values of eFk less than 10Te.
[70] An extended model for f is derived here which is
valid also for large values of Fk. For reasonable values of
the three free parameters of this new model, it accurately
accounts for the highly anisotropic distributions observed in
the reconnection inflow region. We obtain the time evolution of Fk for each of the four spacecraft during Cluster’s
encounter with a reconnection region on 1 October 2001
around 0948:00 UT. Typical values of Fk within the inflow
region are between 0.5 kV and 1 kV. However, as the
boundaries between the inflow and outflow regions are
approached, we identify values of Fk above 5 kV. The data
also suggest that Fk is connected to the characteristic shape
of the isotropic flat‐top distributions in the outflow region.
Here values of eFk on the order of 15 keV ’ 300Te are
inferred.
[71] The large positive values of Fk cause nearly all
electrons to be trapped and the fluid properties of the reconnection region are fully controlled by these trapped
electrons. Therefore a full understanding of antiparallel reconnection cannot be reached without accurate modeling of
the trapped electrons, which have substantially reduced
abilities to conduct currents along the magnetic field lines. It
has previously been shown that the temperature anisotropy
associated with Fk controls the detailed structure of the
electron diffusion region for kinetic simulations with eFk <
10Te [Le et al., 2010]. However, to explore the strong
trapping effects (eFk ’ 300Te) evident in the Cluster
observations additional kinetic simulation with smaller values
of b ∞ are required.
[72] Acknowledgments. This work was supported in part by DOE
Junior Faculty Award ER54878 and NSF/DOE Award PHY‐0613734.
[73] Wolfgang Baumjohann thanks Victor Sergeev and another reviewer
for their assistance in evaluating this paper.
References
Alexeev, I. V., V. Sergeev, C. J. Owen, A. Fazakerley, E. Lucek, and
H. Reme (2006), Remote sensing of a magnetotail reconnection X‐line
using polar rain electrons, Geophys. Res. Lett., 33, L19105,
doi:10.1029/2006GL027243.
Asano, Y., et al. (2008), Electron flat‐top distributions around the magnetic
reconnection region, J. Geophys. Res., 113, A01207, doi:10.1029/
2007JA012461.
Asnes, A., M. G. G. T. Taylor, A. L. Borg, B. Lavraud, R. W. H. Friedel,
C. P. Escoubet, H. Laakso, P. Daly, and A. N. Fazakerley (2008), Multispacecraft observation of electron beam in reconnection region, J. Geophys.
Res., 113, A07S30, doi:10.1029/2007JA012770.
Balogh, A., et al. (2001), The Cluster Magnetic Field Investigation: Overview of in‐flight performance and initial results, Ann. Geophys., 19,
1207–1217.
Birn, J., et al. (2001), Geospace environmental modeling (GEM) magnetic
reconnection challenge, J. Geophys. Res., 106(A3), 3715–3719.
Birn, J., et al. (2005), Forced magnetic reconnection, Geophys. Res. Lett.,
32, L06105, doi:10.1029/2004GL022058.
Borg, A. L., M. Øieroset, T. D. Phan, F. S. Mozer, A. Pedersen, C. Mouikis,
J. P. McFadden, C. Twitty, A. Balogh, and H. Rème (2005), Cluster encounter of a magnetic reconnection diffusion region in the near‐Earth
magnetotail on September 19, 2003, Geophys. Res. Lett., 32, L19105,
doi:10.1029/2005GL023794.
A03214
Brown, M. R., C. D. Cothran, and J. Fung (2006), Two fluid effects on three‐
dimensional reconnection in the Swarthmore Spheromak Experiment with
comparisons to space data, Phys. Plasmas, 13(5), doi:10.1063/1.2180729.
Chen, L. J., et al. (2008a), Evidence of an extended electron current sheet
and its neighboring magnetic island during magnetotail reconnection,
J. Geophys. Res., 113, A12213, doi:10.1029/2008JA013385.
Chen, L. J., et al. (2008b), Observation of energetic electrons within magnetic islands, Nat. Phys., 4(1), 19–23, doi:10.1038/nphys777.
Chen, L. J., et al. (2009), Multispacecraft observations of the electron current
sheet, neighboring magnetic islands, and electron acceleration during magnetotail reconnection, Phys. Plasmas, 16(5), 056501, doi:10.1063/
1.3112744.
Chew, G. F., M. L. Goldberger, and F. E. Low (1956), The boltzmann
equation and the one‐fluid hydromagnetic equations in the absence of
particle collisions, Proc. R. Soc. London, Ser. A, 236, 112–118.
Daughton, W., J. Scudder, and K. Homa (2006), Fully kinetic simulations
of undriven magnetic reconnection with open boundary conditions, Phys.
Plasmas, 13(7), 072101, doi:10.1063/1.2218817.
Dungey, J. (1953), Conditions for the occurence of electrical discharges in
astrophysical systems, Philos. Mag., 44, 725–738.
Egedal, J., M. Øieroset, W. Fox, and R. P. Lin (2005), In situ discovery of
an electrostatic potential, trapping electrons and mediating fast reconnection in the Earth’s magnetotail, Phys. Rev. Lett., 94(2), 025006,
doi:10.1103/PhysRevLett.94.025006.
Egedal, J., W. Fox, N. Katz, M. Porkolab, M. Øieroset, R. P. Lin,
W. Daughton, and J. F. Drake (2008), Evidence and theory for trapped
electrons in guide field magnetotail reconnection, J. Geophys. Res., 113,
A12207, doi:10.1029/2008JA013520.
Egedal, J., W. Daughton, J. F. Drake, N. Katz, and A. Le (2009), Formation
of a localized acceleration potential during magnetic reconnection with a
guide field, Phys. Plasmas, 16(5), doi:10.1063/1.3130732.
Gustafsson, G., et al. (1997), The electric field and wave experiment for the
Cluster mission, Space Sci. Rev., 79(1–2), 137–156.
Hoshino, M., T. Mukai, T. Terasawa, and I. Shinohara (2001), Suprathermal
electron acceleration in magnetic reconnection, J. Geophys. Res., 106
(A11), 25,979–25,997.
Johnstone, A. D., et al. (1997), Peace: A plasma electron and current experiment, Space Sci. Rev., 79(1–2), 351–398.
Kistler, L. M., et al. (2005), Contribution of nonadiabatic ions to the cross‐
tail current in an O+ dominated thin current sheet, J. Geophys. Res., 110,
A06213, doi:10.1029/2004JA010653.
Le, A., J. Egedal, W. Daughton, W. Fox, and N. Katz (2009), Equations of
state for collisionless guide‐field reconnection, Phys. Rev. Lett., 102(8),
doi:10.1103/PhysRevLett.102.085001.
Le, A., J. Egedal, W. Daughton, J. F. Drake, W. Fox, and N. Katz (2010),
Magnitude of the Hall fields during magnetic reconnection, Geophys.
Res. Lett., 37, L03106, doi:10.1029/2009GL041941.
Manapat, M., M. Øieroset, T. Phan, R. Lin, and M. Fujimoto (2006), Field‐
aligned electrons at the lobe/plasma sheet boundary in the mid‐to‐distant
magnetotail and their association with reconnection, Geophys. Res. Lett.,
33, L05101, doi:10.1029/2005GL024971.
Masuda, S., T. Kosugi, H. Hara, and Y. Ogawaray (1994), A loop top hard
X‐ray source, i, A compact solar‐flare as evidence for magnetic reconnection, Nature, 371(6497), 495–497.
Nagai, T., I. Shinohara, M. Fujimoto, M. Hoshino, Y. Saito, S. Machida,
and T. Mukai (2001), Geotail observations of the hall current system:
Evidence of magnetic reconnection in the magnetotail, J. Geophys.
Res., 106(A11), 25,929–25,949.
Nakamura, R., W. Baumjohann, Y. Asano, A. Runov, A. Balogh, C. J.
Owen, A. N. Fazakerley, M. Fujimoto, B. Klecker, and H. Reme
(2006), Dynamics of thin current sheets associated with magnetotail reconnection, J. Geophys. Res., 111, A11206, doi:10.1029/2006JA011706.
Øieroset, M., T. Phan, M. Fujimoto, R. P. Lin, and R. P. Lepping (2001), In
situ detection of collisionless reconnection in the Earth’s magnetotail,
Nature, 412(6845), 414–417.
Øieroset, M., R. Lin, and T. Phan (2002), Evidence for electron acceleration up to similar to 300 kev in the magnetic reconnection diffusion region of Earth’s magnetotail, Phys. Rev. Lett., 89(19), 195001,
doi:10.1103/PhysRevLett.89.195001.
Phan, T. D., et al. (2000), Extended magnetic reconnection at the earth’s
magnetopause from detection of bi‐directional jets, Nature, 404(6780),
848–850.
Ren, Y., M. Yamada, S. Gerhardt, H. T. Ji, R. Kulsrud, and A. Kuritsyn
(2005), Experimental verification of the hall effect during magnetic reconnection in a laboratory plasma, Phys. Rev. Lett., 95(5), 055003,
doi:10.1103/PhysRevLett.95.055003.
Runov, A., et al. (2003), Current sheet structure near magnetic X‐line
observed by Cluster, Geophys. Res. Lett., 30(11), 1579, doi:10.1029/
2002GL016730.
16 of 17
A03214
EGEDAL ET AL.: ELECTRON TRAPPING IN RECONNECTION
Sergeev, V., et al. (2008), Study of near‐Earth reconnection events with
Cluster and Double Star, J. Geophys. Res., 113, A07S36, doi:10.1029/
2007JA012902.
Sonnerup, B. U. Ö. (1979), Magnetic field reconnection, in Solar System
Plasma Physics, vol. 3, edited by L. T. Lanzerotti, C. F. Kennel, and
E. N. Parker, pp. 45–108, North Holland, New York.
Taylor, J. B. (1986), Relaxation and magnetic reconnection in plasmas,
Rev. Mod. Phys., 58(3), 741–763.
Vasyliunas, V. M. (1975), Theoretical models of magnetic‐field line
merging, 1, Rev. Geophys., 13(1), 303–336.
Vogiatzis, I. I., T. A. Fritz, Q. G. Zong, and E. T. Sarris (2006), Two distinct energetic electron populations of different origin in the Earth’s magnetotail: A Cluster case study, Ann. Geophys., 24(7), 1931–1948.
Wygant, J. R., et al. (2005), Cluster observations of an intense normal component of the electric field at a thin reconnecting current sheet in the tail
A03214
and its role in the shock‐like acceleration of the ion fluid into the separatrix
region, J. Geophys. Res., 110, A09206, doi:10.1029/2004JA010708.
L.‐J. Chen and B. Lefebvre, Space Science Center, University of New
Hampshire, Durham, NH 03824, USA.
W. Daughton, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA.
J. Egedal, N. Katz, and A. Lê, Plasma Science and Fusion Center,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
(jegedal@psfc.mit.edu)
A. Fazakerley, Mullard Space Science Laboratory, University College
London, Holmbury St. Mary, Dorking RH5 6NT, UK.
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