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GEOPHYSICAL RESEARCH LETTERS, VOL. 37, L03106, doi:10.1029/2009GL041941, 2010
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Magnitude of the Hall fields during magnetic reconnection
A. Le,1 J. Egedal,1 W. Daughton,2 J. F. Drake,3 W. Fox,1 and N. Katz1
Received 25 November 2009; accepted 14 January 2010; published 11 February 2010.
[1] In situ observation of the Earth’s magnetosphere has
identified Hall magnetic fields as a key signature of
collisionless magnetic reconnection. The inflow portion of
the reconnection diffusion region is further characterized
by strong electron pressure anisotropy. These two features
are tightly linked in a quantitative model, which is
verified using fully kinetic simulations. The model
predicts the Hall field strength and the maximum electron
pressure anisotropy as functions of the upstream ratio of
electron fluid and magnetic pressures. Citation: 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.
1. Introduction
[2] Magnetic reconnection allows an often violent reconfiguration of magnetic field lines within a plasma. It
accompanies diverse phenomena including magnetic substorms in the Earth’s magnetosphere, solar flares, coronal
mass ejections, and sawtooth crashes and disruptions in
tokamaks. In plasmas with negligible collisions, found in the
magnetosphere and the solar wind, reconnection involves
the decoupling of electrons and ions in the diffusion region.
Sonnerup [1979] predicted that this leads to the formation of
a quadrupolar Hall magnetic field centered on the reconnection region. The Hall field structure has now been
observed in spacecraft data and numerical simulation, and
it has been measured in laboratory experiments [Øieroset et
al., 2002; Borg et al., 2005; Drake et al., 2008; Daughton
et al., 2006; Hesse et al., 2008; Ren et al., 2005; Brown et
al., 2006]. Despite the fact that the Hall fields are now considered to be a key signature of collisionless reconnection in
space data, there has heretofore been no quantitative theory
for the strength of the Hall fields.
[3] Recent in situ observation of the Earth’s magnetosphere reveals two more characteristic features of the diffusion region: Chen et al. [2008] found electron pressure
anisotropy in the inflow plasma with pk > p? (where directions are with respect to the magnetic field), and Phan et al.
[2007] observed evidence for an electron outflow jet near
the X‐line similar to one observed in their simulations. The
outflow electrons stream faster than the E × B speed, and the
associated current produces the Hall field.
1
Plasma Science and Fusion Center, Massachusetts Institute of
Technology, Cambridge, Massachusetts, USA.
2
Plasma Theory and Applications, Los Alamos National Laboratory,
Los Alamos, New Mexico, USA.
3
Laboratory for Plasma Research, University of Maryland, College
Park, Maryland, USA.
Copyright 2010 by the American Geophysical Union.
0094‐8276/10/2009GL041941$05.00
[4] In reconnection research, emphasis has typically been
placed on the component of the electron momentum balance
equation,
nme
due
¼ neðE þ ue BÞ r P;
dt
ð1Þ
in the direction of the reconnection electric field. To study
the Hall currents that develop with reconnection, however, it
proves more fruitful to first consider what balances the
−neue × B ≈ J? × B force on the electrons. This force is of
course perpendicular to both the magnetic field and the
electron flow. We find that averaged over the electron jets
J? × B ’ r · P. Considering, for example, a pressure
anisotropy with pk − p? ’ 4nTe directly outside the jets of
width ∼2re (typical of values observed in kinetic simulations), ∣r · P/(ne)∣ ’ 4nTe/(ne 2re) = vth,eB. This term is much
larger than the reconnection electric field, Erec ’ 0.1VAB. The
electron jets streaming near the thermal speed across the
magnetic field are thus controlled by the electron pressure
anisotropy. Based on analytical results and data from three
PIC simulation runs (on two different codes), the electron
pressure anisotropy and Hall fields and currents are found to
scale with the upstream plasma conditions.
[5] We begin with the pair of equations of state for the
parallel and perpendicular electron pressure originally
derived by Le et al. [2009] for collisionless reconnection
with a guide magnetic field. The equations of state are based
on an approximate solution of the Vlasov equation for
magnetized electrons that takes into account adiabatic particle trapping by a parallel electric field. For ~n = n/n∞ ≈ 1
~ = B/B∞ < 1 (∞ refers to the upstream ambient plasma
and B
conditions), the equations of state resemble CGL double‐
adiabatic scalings for the electrons: the pressure components
~ 2 and p?/p∞ ≈ ~
~ A
nB.
are approximated by pk/p∞ ≈ p~n3/6B
2 ~3
pressure anisotropy with pk/p? ≈ p~n /6B therefore develops,
where typically pk/p? 1.
[6] In anti‐parallel reconnection, the initial magnetic
geometry contains a neutral sheet where the field vanishes.
During the reconnection process, regions of very weak
magnetic field persist around neutral points. In principle,
this precludes the use of the above equations of state, which
assume the particles are magnetized. We find, however, that
the electron pressure anisotropy predicted by the equations
of state just outside the neutral sheet, where the electrons are
still magnetized, imposes useful relationships between the
upstream electron pressure and the Hall magnetic fields.
2. Fully Kinetic Simulation Results
[7] We use three particle‐in‐cell (PIC) simulation runs to
explore the application of the equations of state to anti‐
parallel reconnection. Figure 1 shows results from one of the
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Figure 3b shows the PIC pressure anisotropy pk/p? divided
by the fluid model estimates to illustrate their agreement in
the inflow region. In the outflow layer (the rectangular
outlines in Figures 3a and 3c), where the equations of state
are inapplicable, the pressure becomes nearly isotropic
because the electrons are effectively pitch‐angle scattered by
passing through the very weak magnetic field. A typical
trapped electron trajectory, as shown in Figure 2, crosses the
region of weak magnetic field repeatedly as it moves
through the outflow region. As noted by Chen et al. [2008],
strong electron pressure anisotropy may therefore help
identify the inflow region in space data.
3. Electron Momentum Balance
Figure 1. PIC simulation results: (a) plasma density,
(b) magnetic field strength, (c) out‐of‐plane Hall magnetic
field Bz, (d) z‐directed electron current, and (e) x‐directed
electron current.
simulations. The simulation is translationally symmetric in
the z‐direction, has a total domain of 2560 × 2560 cells =
400de × 400de, tracks roughly 2 × 109 particles, and implements open boundary conditions for both the particles
and the fields described by Daughton et al. [2006]. 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. We also use results from two other similar simulation
runs with the same mass ratio of mi/me = 400. One was performed on the same code with open boundary conditions, but
used a temperature ratio of Ti/Te = 1, and the last was run on
the code P3D.
[8] 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 quadrupolar out‐of‐plane Hall magnetic field
Bz is shown in Figure 1c. We focus on the inner electron
diffusion layer where strong electron currents jz and jx
(Figures 1d and 1e) flow in a narrow channel. The electron
jets flow at nearly the electron thermal speed (based on the
upstream temperature), and are localized to a layer around
4de wide.
[9] As stated above, the equations of state apply to the
inflow region where the electrons are magnetized. They
agree with the PIC simulation to within ∼10% up to a layer a
few de = c/wpe wide, where the full electron pressure tensor
Pij is used to define p? = 12[Pij(dij − bibj)]. The pressure
anisotropy in the inflow region is substantial: in the present
simulation (which had the strongest anisotropy of the three
simulations), it reaches nearly pk/p? ∼ 7 (see Figure 3a).
[10] The strong pressure anisotropy predicted by the
equations of state contributes to electron momentum balance
in the inner electron region. Here, the electrons carry almost
all of the current (more than 90% throughout the region) and
correspondingly nearly all of the J × B force exerted by the
magnetic field on the plasma. We highlight the pressure
anisotropy by writing steady‐state electron momentum balance, assuming −neue = J = r × B, in the form
0 ¼ ri ðB2 =2 þ p? Þij þ pk p? B2 bi bj þ Fj ;
ð2Þ
where Fj contains the electric field, non‐gyrotropic pressure,
and inertia contributions. In the electron layer, the magnetic
field lines are strongly curved and ribibj is large. To leading
order, its coefficient must be small, or (pk − p? − B2) ≈ 0.
Physically, this means the magnetic tension force across the
layer associated with the bent field lines, indicated schematically in Figure 3d, is largely balanced by the anisotropic
electron pressure.
[11] We consider x‐momentum balance for differentially
narrow fluid elements extending ∼4de across the outflow jet
(for example, the small shaded box in Figure 3c). The
largest contributions come from B2bxby and (pk − p?)bxby
evaluated immediately outside the jet and are plotted in
Figure 3e. A smaller contribution is shown from x gradients
in the stress tensor integrated across the layer. The other
electron terms that are not plotted are a similar size or
smaller. These terms become important where the electron
flows peak and then terminate roughly 40de downstream
from the X‐line. Finally, the integrated magnetic force on
Figure 2. Typical trajectory of a trapped electron overlaid
on contours of constant ∣B∣. In the outflow, the electron repeatedly crosses a region of weak magnetic field where m is
not conserved.
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LE ET AL.: HALL FIELDS DURING RECONNECTION
2pe∞/B2∞. This relation is shown in Figure 4b. An approximate
form good to 10% for be∞ < 1 is
BH
B1
Figure 3. (a) Ratio pk/p? from PIC code. (b) Ratio of PIC
results and fluid model prediction for pk/p?. (The value 1 represents exact agreement.) (c) In‐plane projection of magnetic field lines (black) with in‐plane electron flow vectors
(red). The large magenta rectangle is the electron outflow
layer with width ∼4de. (d) Magnetic field lines and their tension force. (e) Terms in the integrated momentum balance
equation (for an electron fluid element similar to the shaded
box in Figure 3c of width ∼4de). The magnetic force on the
ions (green) is neglected when assuming J ∼ −neue in the
electron outflow layer.
3
1=4
~
n e1
:
12
ð4Þ
The above scaling is confirmed by the three PIC simulations
of reconnecting current sheets with varying electron be∞. BH
is evaluated where the out‐of‐plane electron current reaches
40% of its maximum [roughly (2–4)de from the peak] and is
marked in Figure 4b for three numerical studies. The middle
simulation is from the code P3D using fully periodic
boundary conditions [Shay et al., 2007], and the other two
runs were performed on the open boundary code [Daughton
et al., 2006].
[14] The momentum balance constraint pk − p? = B2
applies everywhere along the length of the electron outflow
jets (as seen in Figure 3d). For a given be∞ and a roughly
uniform density equal to its value in the ambient plasma,
~n = n/n∞ = 1, the equations of state predict a unique
value of BH that satisfies the momentum balance condition.
This implies another result consistent with the simulations:
the magnetic field strength is nearly uniform along the current
sheet. Although its magnitude is roughly constant and equal
to the fixed value B = BH, the magnetic field may rotate along
the outflow. The component rotated out of the plane is the
Hall field Bz. The value of BH determined from our equations
of state outside the current layer is therefore an upper bound
for ∣Bz∣ in the inner diffusion region.
[15] Similarly, the equations of state provide an estimate
for the maximum electron pressure ratio pk/p?. As visible
the ions is plotted. It is small, and we neglect it when we
assume J = −neue. Thus, averaged over the layer,
Z
I
dV ðJe? BÞ dA P;
ð3Þ
and the volume‐averaged perpendicular electron Hall currents are supported by the pressure anisotropy directly outside
the jets, where P ’ (pk − p?)bb + p?I with pk p?.
4. Predicted Scalings
[12] Together with the equations of state pk(n, B) and
p?(n, B), the main result from force balance considerations,
pk − p? ≈ B2 all along the electron layer, determines important Hall physics parameters. Figure 4 shows pk − p? and
B2 as functions of y along a typical cut 15de to the right of
the X‐line using both the simulation data and our equations
of state. By solving pk(n, B) − p?(n, B) = B2 (where the two
dashed lines in Figure 4 intersect), we find the value of the
magnetic field strength immediately outside the electron jet,
which we denote by BH.
[13] Taking the density as approximately uniform with
n ≈ n∞, we obtain BH as a function only of the ratio of
electron to magnetic pressure at the inflow boundary, be∞ =
Figure 4. (a) Pressure anisotropy pk − p? from PIC simulation and predicted by fluid model, and ∣B∣2. Model (curve)
compared to PIC simulation results of quantities that depend
on the upstream electron beta be∞ : (b) magnetic field
strength BH just outside outflow layer normalized to reconnecting field B∞, (c) maximum pressure ratio pk/p?, and
(d) maximum acceleration potential normalized to electron
temperature. The *’s are from two runs on an open‐boundary
code, one with Ti/Te = 5 and the other with Ti/Te = 1. The °’s
are taken from a run on the code P3D using fully periodic
boundary conditions.
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in Figure 4a, the equations of state break down slightly
before B reaches the predicted value of BH. We find empirically from the codes, however, that evaluating the
equations of state pk(n, B) and p?(n, B) at n ∼ n∞ and B =
1.25BH, where the equations of state are valid, gives a
good estimate for the maximum pk/p?. The scaling plotted
in Figure 4c is approximately
pk
p?
max
1
3
4~ne1
1=4
ek
Te1
1
2
max
"
4~n
e1
1=4
currents beyond the E × B drift speed, and these currents in
turn generate the Hall magnetic field.
[18] Acknowledgments. This work was funded at MIT in part by DOE
grant DE‐FG02‐06ER54878 and DOE/NSF grant DE‐FG02‐03ER54712.
References
:
ð5Þ
[16] As explained by Le et al. [2009], the enhanced parallel
pressure predicted by the equations of state results largely
from electron trapping and heating by a parallel electric
field. The effect of the parallel electric field is parameterized
by the acceleration
potential defined by Egedal et al. [2009]
R1
as Fk(x) = x E · dl, where the integral is taken along the
magnetic field from the point x to the ambient plasma
where E · B = 0. In Figure 4d, we plot the value of Fk
predicted by our model at the point of maximum upstream
pressure anisotropy. Note that at low be∞, a large Fk develops,
scaling roughly as
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#2
1
;
2
ð6Þ
and the majority of inflow electrons are trapped. Because
be is typically low in Earth’s magnetotail, electrical trapping in the inflow region is likely a crucial mechanism for
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particular, a modification is made to the underlying distribution function on which the equations of state are based
[Egedal et al., 2010].
5. Summary
[17] Substantial electron pressure anisotropy thus develops
in the inflow region during anti‐parallel reconnection. The
pressure anisotropy is described by equations of state originally derived for guide‐field reconnection. The equations of
state then link the electron pressure immediately outside the
reconnection region to the characteristic strength of the Hall
magnetic field BH through a momentum balance condition,
pk − p? ≈ B2H, and they set the parameters of the model in
terms of the upstream value of be. A self‐consistent model
results in which upstream pressure anisotropy and the curvature of the Hall magnetic field drive perpendicular electron
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J. F. Drake, Laboratory for Plasma Research, University of Maryland,
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