Formation of a localized acceleration potential during
magnetic reconnection with a guide field
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Egedal-Pedersen, J. et al. “Formation of a Localized Acceleration
Potential During Magnetic Reconnection with a Guide Field.”
Physics of Plasmas 16.5 (2009): 050701. © 2009 American
Institute of Physics
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http://dx.doi.org/10.1063/1.3130732
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PHYSICS OF PLASMAS 16, 050701 共2009兲
Formation of a localized acceleration potential during magnetic
reconnection with a guide field
J. Egedal,1 W. Daughton,2 J. F. Drake,3 N. Katz,1 and A. Lê1
1
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
3
University of Maryland, College Park, Maryland 20742, USA
2
共Received 16 March 2009; accepted 17 April 2009; published online 1 May 2009兲
Magnetic reconnection near the surface of the sun and in the Earth’s magnetotail is associated with
the production of highly energetic electrons. Direct acceleration in the reconnection electric field has
been proposed as a possible mechanism for energizing these electrons. Here, however, we use
kinetic simulations of guide-field reconnection to show that in two-dimensional 共2D兲 reconnection
the parallel electric field, E储 in the reconnection region is localized and its structure does not permit
significant energization of the electrons. Rather, a large fraction of the electrons become trapped due
to a sign reversal in E储, imposing strict constraints on their motions and energizations. Given these
new results, simple 2D models, which invoke direct acceleration for energizing electrons during a
single encounter with a reconnection region, need to be revised. © 2009 American Institute of
Physics. 关DOI: 10.1063/1.3130732兴
Magnetic reconnection is a basic plasma physics process
that allows magnetic field lines to change topology in the
presence of a plasma. Although it is localized it controls
large scale explosive plasma phenomena such as coronal
mass ejections, magnetic storms in the Earth’s magnetotail
and internal disruptions in magnetic fusion devices.1,2
During reconnection on the sun, the foot points where
reconnecting magnetic loops intersect the photosphere light
up in x rays. In the Earth’s magnetotail, energetic electrons
up to 300 keV have recently been observed by the Wind
spacecraft.3 This is conclusive evidence that reconnection is
associated with a significant production of energetic electrons. It has been proposed that the electrons are energized
by direct acceleration in the reconnection electric field along
the magnetic X-lines.4,5
However, numerical simulations of reconnection provide
significant information about the physics and structure of the
reconnection layer, which is inconsistent with the direct acceleration models. In particular particle in cell 共PIC兲 simulation is a useful tool for such investigations because it includes the kinetic effects of both the electrons and the ions.
Here we introduce the acceleration potential, ⌽储, as a measure of the work done by parallel electric fields on electrons
streaming along magnetic field lines. This provides a new
and more complete framework for understanding the energization of electrons by direct acceleration in guide field reconnection. We define the reconnection region as the area
where two fluid effects are important, extending about 50de
upstream of the X-line and about 200de downstream of the
X-line. In this region a localized structure in ⌽储 is documented, caused by a sign reversal in E储. The analysis shows
how this localized structure limits direct acceleration and
causes extensive trapping of thermal electrons.
The electron dynamics and energization are
studied based on profiles of an open boundary PIC
simulation6 including 2 ⫻ 109 particles in a domain
1070-664X/2009/16共5兲/050701/4/$25.00
xy = 3072⫻ 3072 cells= 569⫻ 569 c / ␻ pe. The simulation is
translationally symmetric in the z-direction and is characterized by the following parameters: mi / me = 360, Ti / Te = 2,
Bguide = 0.5B0, ␻ pe / ␻ce = 2.0, background density= 0.30n0
共peak Harris density兲, vth,e / c = 0.20, and an initial current
sheet width of about 0.4 di.
We consider the fields observed at the single time slice
of Fig. 1, during a time interval without significant evolution
in the profiles. As seen in Fig. 1 the asymmetric current
profile is not aligned with the superimposed contours of
magnetic flux. This type of current profile is typical of kinetic simulations of reconnection with a guide magnetic
field. The enhancement in the plasma density is also asymmetric, exhibiting a quadrupolar structure similar to that discussed in Refs. 7 and 8. In the inflow region the electrons are
isotropic with T储 = T⬜. However, as the electrons enter the
reconnection region T储 is increased 共by a factor of about 4兲,
while T⬜ is reduced proportionately to the decrease in the
magnetic field, B. In turn, the profile of B is strongly influenced by the strong quadrupolar structure in Bz. The inductive electric field, Ez is nearly uniform throughout the ion
diffusion region and the in-plane potential includes the familiar quadrupolar structure required for having E · B ⬃ 0 outside the reconnection region.
In Ref. 9 the details of the electron distributions observed by the Wind spacecraft are accounted for by hypothesizing the trapping of electrons by an in-plane electrostatic
potential. The simulation presented here 共with Bguide / B0 and
Ti / Te relevant to the Wind event兲 confirms that trapping is
important. However, the trapping is not caused only by an
electrostatic potential, but also by the inductive electric field,
Ez, in combination with the details of the magnetic field
geometry.
To explore the role of E储 we introduce the parallel acceleration potential ⌽储 as a “pseudopotential” that measures the
work done by E储 on a passing electron as it escapes the
16, 050701-1
© 2009 American Institute of Physics
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050701-2
Phys. Plasmas 16, 050701 共2009兲
Egedal et al.
FIG. 1. 共Color online兲 Color contours of jz, n, T储, T⬜, B, Bz, Ez, and ⌽,
respectively. The overlayed contours of ⌿ coincide with magnetic field
lines.
simulation box in a straight shot along a magnetic field line.
Therefore, mathematically ⌽储 is defined as
⌽储共x兲 =
冕
⬁
E · dl,
共1兲
x
where the integration is carried out from the point x along
the magnetic field to the boundary of the simulation box.
Note that in contrast with the electrostatic potential ⌽ in Fig.
1共h兲, ⌽储, in addition to the in-plane electric fields, includes
important contributions from the inductive electric field, Ez.
We find that ⌽储 is nearly independent of whether dl is
taken parallel or antiparallel to B. This is because the integral
of E · dl over its full length in the simulation domain is small,
⬁
E · dl ⬍ Te / e. This result can be understood on the basis of
兰−⬁
the adiabatic fluid model of Ref. 10, which shows that
⌽储 ⯝ ⌽储共n , B兲 is a function of only n and B and increases
monotonically with n and 1 / B. As a static magnetic field line
is followed through the reconnection region, its two ends
FIG. 2. 共Color online兲 共a兲 Contours of constant ⌽储. A zoomed-in view for
the region close to the X-line is given in Fig. 3共b兲. 共b兲 Contours of the
adiabatic variable J0. The contours of constant J0 represent the loci of electron orbit bounce points. J0 is normalized such that J0 ⬍ 1 characterizes the
regions where free streaming of electrons is significantly limited by trapping. 共c兲 Close-up of the orbit in 共b兲. The J0 = 0 contour marks the locations
where the electrons obtain their maximal parallel velocity. 共d兲 Bounce orbit
length 共measured along field lines兲 for the highlighted J-contour in 共c兲 as a
function of the field line coordinate ⌿. Note that ⌿ = 0 and ⌿ = 1 are defined
in 共c兲. 共e兲 Parallel velocity along the orbit in 共b兲 and 共c兲.
outside the region will have similar values of n and B. The
values of ⌽储 outside the region will therefore also be similar.
The localized maximum of ⌽储 occurs because of the localized maxima of both n and 1 / B in the region. This, in turn,
dictates the sign reversal in E储, which develops to boost the
electron density in the regions of high ion density and low
magnetic field strength.
In Fig. 2共a兲 contours of constant e⌽储 are shown. Outside
the reconnection region the contributions to ⌽储 from Ez and
the in-plane electric fields cancel reflecting that E储 ⬃ 0. However, as seen in Fig. 3共a兲, inside the region E储 is finite and
changes sign where n and 1 / B peak. Thus, consistent with
the discussion above the largest values of ⌽储 coincide with
the regions of density enhancement and small B; here e⌽储
reaches an amplitude of about 5Te. Considering a number of
different simulations 共not included兲, we find similar localized
structures for ⌽储.
In the reconnection region where the density is enhanced
nearly all electrons bounce multiple times due to the localized structure and large amplitude 共⬎Te / e兲 of ⌽储. The
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050701-3
Phys. Plasmas 16, 050701 共2009兲
Formation of a localized acceleration potential…
FIG. 3. 共Color online兲 共a兲 Contours of constant values of E储. 共b兲 Zoomed-in
view of ⌽储, for the area indicated in Fig. 2共a兲. 共c兲 Number of electrons with
E ⬎ 0.5mc2 evaluated on a grid with de resolution. 共d兲 Energy distributions
for the location marked by “䊊” in 共c兲. The dashed lines represent f共E兲
= f 0共E兲 and f共E兲 = f 0共E − 5Te兲, respectively. Note that while the simulation is
fully relativistic, it applies that f 0 ⬀ exp关−共v2 / v2th兲兴, where v is the particle
velocity and vth is the thermal velocity.
kinetic properties of these electrons are analyzed by considering the action integral
J=
冖
v储dl =
冖冑
2E储
m
dl.
共2兲
This integral is over one particle bounce motion, where l
represents the length measured along the orbit and E储 is the
parallel kinetic energy. J is an adiabatic invariant and is conserved if the bounce motion is sufficiently rapid. Following
the notation of Ref. 11, J is expanded in a series of increasing powers of ⑀ = me / e giving J = J0 + ⑀J1 + . . .. In Ref. 11 it is
shown that J1 oscillates and vanishes at the orbit bounce
points. Furthermore, J0 is obtained simply by integration
along field lines 共not the actual electron orbits兲. Let x0 be a
bounce point 共E储 = 0兲 of an electron. The parallel kinetic
energy at a point x on the same field line can then be calculated as
E储共x,x0兲 = e关⌽储共x兲 − ⌽储共x0兲兴 + ␮关B共x0兲 − B共x兲兴.
共3兲
It is here assumed that ␮ is conserved, which will be justified
below. The action integral can then be expressed as
J0共x0兲 =
冖冑
2E储共x⬘,x0兲
m
dl⬘ .
共4兲
For a given bounce point, x0, the points of integration, x⬘, are
those on the field line where E储共x⬘ , x0兲 ⬎ 0.
In Fig. 2共b兲 contours of constant J0共x0兲 are shown for x0
represented by the points in the xy-plane. The contours coincide with loci of the electron bounce points and therefore
provide a direct visualization of how the trapped trajectories
are channelled by ⌽储 through the ion diffusion region. The
asymmetry in ⌽储 causes the electrons entering the upper
共lower兲 inflow region to exit through the left 共right兲 outflow
region generally without making direct contact with the magnetic X-line. Contours for one particular value of J0 are highlighted by the white lines. An electron orbit corresponding to
this value of J0 共and ␮ = Te / Bx-line兲 is shown in yellow. Inside
the reconnection region the bounce points coincide with the
white contours of J0 and we conclude that J0 is conserved in
this region. In turn, given J0 is obtained based on E储 in Eq.
共3兲, this emphasizes the importance of ⌽储 to the motion of
the trapped electrons.
In Fig. 2共b兲 J0 is normalized such that in the areas along
the separator where J0 ⬎ 1 the level of trapping is insignificant and free streaming electrons can carry parallel
currents.8,12 Meanwhile, in the extended areas where J0 ⬍ 1
the dynamics of the trapped electrons dominate. To evaluate
the heating of the trapped electrons we consider the adiabatic
invariants, ␮ and J0. From conservation of ␮ = E⬜ / B it is
clear that the perpendicular particle energy E⬜ must be reduced in the inner reconnection region where the lowest values of B are observed. A comparison of Figs. 1共d兲 and 1共e兲
clearly shows that T⬜ is proportional to B 共consistent with
␮-conservation for most electrons兲. The evolution of the parallel kinetic energy is controlled by J0. As is evident in Fig.
2共d兲, when the trapped electrons enter the reconnection region their bounce lengths 共measured along field lines兲 are
reduced. As a consequence 关see Fig. 2共e兲兴 the parallel velocity is increased in order for J0 to be conserved. Consistently,
in Fig. 1共c兲 we observed that T储 increases by about a factor of
4 inside the current channel.
The acceleration potential is also important for the energy balance of the passing electrons. In the limit where the
electrons strictly follow the field lines, ⌽储共x兲 is a measure of
the gain in energy of the passing electrons. Because
Bx = By = 0 at the X-line, it follows that 兩dl兩 / 兩dlxy兩
= B / 冑B2x + B2y → ⬁ at this point. Thus, ⌽储 as defined in Eq. 共1兲
has a logarithmic singularity which can be observed in the
zoomed-in view of ⌽储 given in Fig. 3共b兲. In a narrow region
around the singularity less than de / 4 wide, we have e⌽储
⬎ 6Te along two of the four separators, and electrons can
therefore acquire large energies by direct acceleration.13
However, because the singularity is only logarithmic and
very narrow the number of electrons accelerated here is
small.
⬁
E · dl
Excluding the singular region we find that 兰−⬁
⬍ Te / e, which suggests that electrons will gain very little
energy in a single pass through the reconnection region.
However, because the reconnecting field lines are “moving,”
the passing electrons do not strictly follow the static field
lines. This allows a significant group of passing electrons to
be accelerated as they enter the reconnection region, while
avoiding the decelerating electric field as they leave the region. The electron trajectories in Fig. 3共a兲 are representative
for this heating mechanism. The locations in the simulation
domain where the heating mechanism is effective are shown
Fig. 3共c兲, which contains a color map of the number of electrons recorded in 1de ⫻ 1de cells with energies Ek ⬎ 0.5mc2.
The combination of the sign reversal of E储 and the drift of the
field lines causes energization of the electrons along all four
separators 共rather than just the two separators of the logarithmic singularity兲.
The passing electrons that enter the simulation domain at
optimal locations will acquire an energy gain on the order of
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050701-4
Phys. Plasmas 16, 050701 共2009兲
Egedal et al.
e⌽储max ⬃ 5Te, where ⌽储max is the maximal value of ⌽储 in the
bulk of the ion diffusion region. Thus, for any point in the
simulation domain, the kinetic energy of a passing electron is
characterized by
E共x兲 ⱕ E0 + e⌽储max ,
共5兲
where E0 is the kinetic energy the electron had when it entered the simulation at the domain boundary. This can be
tested directly against the distribution of electrons in the
simulation. According to Liouville’s theorem, df / dt = 0, the
phase space density, f along a particle orbit is conserved
such that f关E共x兲兴 = f 0共E0兲, where f 0 is the distribution function applied at the simulation boundary. Thus, since
E0 ⱖ E共x兲 − e⌽储max and df 0 / dE ⬍ 0 we find that
f共E兲 ⱕ f 0共E − e⌽储max兲.
共6兲
In Fig. 3共d兲 the full line represents f共E兲 obtained directly
from the particle data within the small region 共3de ⫻ 3de兲
marked by the circle in Fig. 3共c兲. The two dashed theoretical
curves are f 0共E兲 and f 0共E − 5Te兲, respectively. Consistent with
Eq. 共6兲, we verified that throughout the computational domain these two curves bracket the simulated f共E兲, as expected from Eq. 共6兲.
Although a significant relativistic population is present
in Fig. 3共d兲 this cannot account for observations of fast electrons in nature. Rather, the relativistic electrons in the simulation are just a result of the large value of vth / c = 0.2 共typical
for PIC-simulations兲 combined with the energy boost of
e⌽储 ⬃ 5Te. For example, a boost of 5Te for the electrons observed in the Wind event, would only increase the temperature to 2 keV and is not consistent with the observation of
electron acceleration up to 300 keV. Other possible mechanisms, such as Fermi acceleration14 or perhaps threedimensional effects in reconnection, must therefore be explored to account for electron heating. Interestingly,
observations consistent with Fermi acceleration have recently been reported for an event where the Cluster spacecraft encountered a series of magnetic islands in the Earth’s
magnetosphere.15
The formation of a localized acceleration potential and
the described physics of the trapped electrons are not dependent on the choice of boundary conditions. This is evident
from Fig. 4, which illustrates that ⌽储 in a P3D code simulation with periodic boundaries16 is similar to ⌽储 in Fig. 3共a兲.
In summary, the documented sign reversal in E储 leads to
the formation of a localized acceleration potential, ⌽储, which
controls the energization of both trapped and passing elec-
FIG. 4. 共Color online兲 Acceleration potential ⌽储 from a simulation with
periodic boundary conditions 共Ref. 16兲. This simulation is characterized by
Ti / Te = 1, Bguide = 0.5B0, mi / me = 225, background density= 0.5n0, and about
2 ⫻ 108 particles in a domain of 2048⫻ 1024 cells= 392⫻ 196c / ␻ pe.
trons. In contrast with models where the in-plane electric
fields are neglected, we find that single pass energization is
limited and is bounded by e⌽储max ⬃ 5Te 共per electron兲. However, given the sign reversal in E储 this level of energization is
possible along all four separators.
This work was funded in part by DOE Junior Faculty
Grant No. DE-FG02-06ER54878.
R. Giovanelli, Nature 共London兲 158, 81 共1946兲.
T. Nagai, I. Shinohara, M. Fujimoto, M. Hoshino, Y. Saito, S. Machida,
and T. Mukai, J. Geophys. Res. 106, 25929, DOI:10.1029/2001JA900038
共2001兲.
3
M. Oieroset, R. Lin, and T. Phan, Phys. Rev. Lett. 89, 195001 共2002兲.
4
Y. Litvinenko, Astrophys. J. 462, 997 共1996兲.
5
P. Browning and G. Vekstein, J. Geophys. Res. 106, 18677, DOI:10.1029/
2001JA900014 共2001兲.
6
W. Daughton, J. Scudder, and H. Karimabadi, Phys. Plasmas 13, 072101
共2006兲.
7
R. G. Kleva, J. F. Drake, and F. L. Waelbroeck, Phys. Plasmas 2, 23
共1995兲.
8
P. L. Pritchett and F. Coroniti, J. Geophys. Res. 109, A01220,
DOI:10.1029/2003JA009999 共2004兲.
9
J. Egedal, M. Oieroset, W. Fox, and R. Lin, Phys. Rev. Lett. 94, 025006
共2005兲.
10
A. Le, J. Egedal, W. Daughton, W. Fox, and N. Katz, Phys. Rev. Lett. 102,
085001 共2009兲.
11
T. G. Northrop, C. S. Liu, and M. D. Kruskal, Phys. Fluids 9, 1503
共1966兲.
12
C. Cattell, J. Dombeck, J. Wygant, J. F. Drake, M. Swisdak, M. L. Goldstein, W. Keith, A. Fazakerley, M. Andre, E. Lucek, and A. Balogh, J.
Geophys. Res. 110, A01211, DOI:10.1029/2004JA010519 共2005兲.
13
P. L. Pritchett, Geophys. Res. Lett. 33, L13104, DOI:10.1029/
2005GL025267 共2006兲.
14
J. F. Drake, M. Swisdak, H. Che, and M. A. Shay, Nature 共London兲 443,
553 共2006兲.
15
L. J. Chen, A. Bhattacharjee, P. A. Puhl-Quinn, H. Yang, N. Bessho, S.
Imada, S. Muehlbachler, P. W. Daly, B. Lefebvre, Y. Khotyaintsev, A.
Vaivads, A. Fazakerley, and E. Georgescu, Nat. Phys. 4, 19 共2008兲.
16
A. Zeiler, D. Biskamp, J. F. Drake, B. N. Rogers, M. A. Shay, and M.
Scholer, J. Geophys. Res. 107, 1230, DOI:10.1029/2001JA000287
共2002兲.
1
2
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