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In Situ Discovery of an Electrostatic Potential, Trapping Electrons

and Mediating Fast Reconnection in the Earth’s Magnetotail

J. Egedal,1M. Øieroset,2W. Fox,1and R.P. Lin2,3

1Massachusetts Institute of Technology, Plasma Science Fusion Center, Cambridge, Massachusetts 02139, USA

2Space Sciences Laboratory, University of California, Berkeley, California 94720, USA

3Physics Department, University of California, Berkeley, California 94720, USA

(Received 11 February 2004; revised manuscript received 10 August 2004; published 19 January 2005)

Anisotropic electron phase space distributions, f, measured by the Wind spacecraft in a rare crossing of

a diffusion region in Earth’s far magnetotail (60 Earth radii), are analyzed. We use the measured f to probe

the electrostatic and magnetic geometry of the diffusion region. For the first time, the presence of a strong

electrostatic potential (1 kV) within the ion diffusion region is revealed. This potential has far reaching

implications for the reconnection process; it accounts for the observed acceleration of the unmagnetized

ions out of the reconnection region and it causes all thermal electrons be trapped electrostatically. The

trapped electron motion implies that the thermal part of the electron distributions are symmetric around

vk? 0: f?vk;v?? ’ f??vk;v??. It follows that the field aligned currents in the diffusion region are

limited and fast magnetic reconnection is mediated.

DOI: 10.1103/PhysRevLett.94.025006 PACS numbers: 94.30.–d, 52.20.–j, 52.35.Vd

Magnetic reconnection, the process in a plasma during

which magnetic field lines rearrange and change topology

[1,2], plays a fundamental role in a variety of plasma

phenomena. It controls the evolution of solar flares [3], it

allows the solar Wind to enter Earth’s magnetosphere [4],

and it is an integral part of magnetic substorms and the

aurora phenomena [5,6]. Mounting evidence suggests that

the fast rate at which reconnection proceeds is strongly

influenced by the ion-scale Hall effect [7]. However, for

fast reconnection to be maintained, a hitherto unknown

electron scale process must limit the electron current in the

reconnection region, allowing electrons to diffuse through

the magnetic field at the same or at even a higher rate than

the ions [8].

Recently, the first in situ measurements of the electron

distribution function have been reported for a reconnection

event encountered by the Wind spacecraft in Earth’s mag-

netotail [9,10]. This event, observed in the deep magneto-

tail (60 Earth radii), is characterized by a kinetic regime in

which the electron temperature is high, Te? 400 eV, and

the magnetic field is weak, B ? 6–12 nT, implying an

electron thermal speed, vth, much larger than the Alfve ´n

speed, vA(for the observed density n ? 0:1 cm?3), and

thus larger than the outflow speed, vout, of plasma from the

reconnection region.

Because of the computational requirements posed by

this kinetic regime (vth? 40vout), so far it has not been

accessible to self-consistent particle simulations. Mean-

while, in Ref. [11] the regime was investigated through

the use of Liouville’s theorem applied to the electron

trajectories in prescribed fields. This technique of follow-

ing particle trajectories for understanding particle distribu-

tion functions measured in space was pioneered by Speiser

[12] and has since been applied in numerous investigations

[13–16]. More recently, prompted by experimental obser-

vation in the Versatile Toroidal Facility (VTF) magnetic

reconnection experiment, the kinetic regime was also

studied in Ref. [17]. Applying an approach similar to those

in Refs. [11,17], the Wind electron distributions are used to

‘‘probe’’the electrostatic and the magnetic geometry of the

reconnection region; the presence of a strong electrostatic

potential (?1 kV) is revealed. This potential causes all

thermal electrons to follow trapped trajectories inside the

reconnection region, significantly limiting the number of

‘‘free’’ current carriers. As has been proven experimentally

at the VTF [18,19], the trapped orbit dynamics effectively

mediates fast reconnection.

Figure 1 illustrates the ion flow, magnetic field compo-

nents, and plasma density measured by Wind during its

fortunate encounter with the reconnection region. The

gradual ion flow reversal and the measured magnetic field

components, including Hall-current-generated out-of-

plane (By) magnetic fields, indicate that Wind passed

through the ion diffusion region of a quasisteady collision-

less reconnection region [9,10]. Based on these measure-

ments and assuming a magnetic X-line structure as

characteristic for reconnection, the approximate path of

Wind through the diffusion region can be reconstructed

as shown in the schematic of Fig. 2(a).

Examplesofelectron

Wind passed through the ion diffusion region are dis-

played in Figs. 3(a)–3(c). The measurements provide

the phase-space density, f, of the electrons as a func-

tion of pitch angle, ? ? ??v;B?, and kinetic energy,

Ek. The anisotropy is most pronounced in the distribu-

tion shown in Fig. 3(a), which was obtained near the center

of the diffusion region. The anisotropy gradually decreases

as Wind exits the X-line region [Figs. 3(b) and 3(c)].

Because the Coulomb collision frequency of the elec-

trons, ? ? 10?9s?1, is very small compared to any other

frequency in the magnetotail, f is governed by the colli-

sionless Vlasov equation df=dt ? 0. Here d=dt represents

distributionsmeasuredas

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2005 The American Physical Society

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the total time derivative along particle orbits. The Vlasov

equation states that the distribution is constant along par-

ticle orbits through phase-space ?x;v?. Hence, we can

equate f for a point ?x0;v0? in the reconnection region to

the isotropic distribution, f1, in the ambient plasma sheet

outside the ion diffusion region by following particle orbits

back in time until they reach points ?x1;v1? in the sheet. It

then follows that

f?x0;v0? ? f1?v1?;v1? jv1j:

(1)

Wind trajectory

08:00:22 08:03:3808:06:53

B∞~10.5nT,

By~6nT

+

+

+

⋅ ⋅ ⋅

+

+

+

−

−

−

−

−

−

−

−

−

−

~ 100 km

° ⋅

x

z

y

Earth

(a)

E

Passing Electron

(b)(c)

E

Trapped Electron

FIG. 2 (color).

the trajectory of Wind. The green arrows indicate the flow of

magnetic flux towards and away from the X line. At locations

away from the X line the in-plane magnetic field approaches

values of about 10 nT. Besides the in-plane magnetic fields, an

out-of-plane guide magnetic field, By? 6 nT, was also ob-

served. The red arrows illustrate the electrostatic electric field

trapping the thermal electrons within the ion diffusion region

(the yellow area). (b) An example of a passing electron guiding

center trajectory reaching the location of the Wind spacecraft.

(c) An example of a trapped electron trajectory. The electron

bounces back and forth along a field line, while slowly drifting

with the magnetic field towards the X line.

(a) Illustration of the magnetic geometry and

−200

0

200

(a)

−200

0

200

(b)

−200

0

200

(c)

(d)

(e)

(f)

(g)

−5

0

5

0

5

10

−5

0

5

07:55t [UT]08:0008:05

0

0.1

0.2

Flow reversal

Hall magnetic fields

vx [km s−1]

vy [km s−1]

vz [km s−1]

Bx [nT]

By [nT]

Bz [nT]

Density [cm−3]

FIG. 1.

components, and density during the encounter with an active

reconnection region. Further away from the X line vyand vz

remain small, while the ion outflow speed, jvxj, approaches the

Alfve ´n speed of 400 km=s.

Wind measurements of the ion flow, the magnetic field

090180

10−22

10−20

10−18

0 90 1800 90180

0 90 180

10−22

10−20

10−18

090 1800 90180

0 90180

10−22

10−20

10−18

0 901800 90180

Wind,

Theory,

Theory,

215 eV

377 eV

639 eV

1063 eV

1289 eV

1902 eV

2799 eV

4111 eV

8824eV

115 eV

15 eV

53 eV

6027eV

f [s3m−3]

f [s3m−3]

f [s3m−3]

15 eV

53 eV

115 eV

215 eV

377 eV

639 eV

1063 eV

1289 eV

1902 eV

2799 eV

4111 eV

6027eV

8824eV

8824eV

6027eV

4111 eV

2799 eV

1902 eV

1289 eV

1063 eV

639 eV

377 eV

215 eV

53 eV

115 eV

15 eV

Pitch Angle (degrees)

08:00:22 UT

08:03:38 UT 08:06:53 UT

(a)

(b)

(c)

(d) (e) (f)

(g)

(h)

(i)

08:00:22 UT: Φ0 = 0 V

Φ0 = 300 V

Φ0 = 900 V

Φ0 = 1150 V: 08:00:22 UT

08:03:38 UT

08:06:53 UT

FIG. 3 (color).

sured by the 3D plasma and energetic particle instrument [25] on

the Wind spacecraft at locations indicated in Fig. 2. Each line

represents the phase-space density for a given energy; the re-

spective energies are given on the right of the figure. In (a) the

blue parts correspond to trapped electrons, and the red parts

correspond to passing electrons. At sufficiently high energies the

electron orbits become unmagnetized, leading to pitch angle

scattering; this effect reduces the level of anisotropy observed

above 6 keV [the green parts in (a)]. (d)–(i) Theoretical distri-

butions calculated for locations [inferred from the measured

values of ?Bx;Bz?] corresponding to the data in (a)–(c). The

distributions in (d)–(g) correspond to the center of the diffusion

region (t ? 08:00:22UT) calculated with increasing values of

the trapping potential (?02 f0;300;900;1150g V). The best

match to the data in Fig. 3(a) is obtained with ?0? 1150 V.

The distributions in (h) and (i), also obtained with ?0? 1150 V,

correspond to the Wind observations in the outflow region [see

(b) and (c)].

(a)–(c) Electron pitch angle distributions mea-

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The measured distributions are therefore closely related to

changes in kinetic energy that the particles undergo along

their trajectories into the X-line region.

The electrons can reach the X-line region via two dis-

tinct types of trajectories: passing and trapped. Anexample

of a passing trajectory is shown in Fig. 2(b). The passing

electron reaches the location of the measurement along a

field line. Electrons of this type (if not intercepted by the

spacecraft) continue to follow the magnetic field line out of

the region, without experiencing any significant changes in

energy.

An example of a trapped electron is given in Fig. 2(c). In

their rapid bounce motion back and forth along field lines,

the total energy is conserved. Consequently, the variations

in the kinetic energy during the bounce motion are linked

to thevariations in the electrostatic potential along the field

lines. Thus, the trapping of electrons is a consequence of

the electric field structure in the reconnection region as

well as the local minimum in the magnetic field strength.

The evolution of the kinetic energy in the slower motion

where the trapped electrons drift with the magnetic field

towards the X line is governed by adiabatic invariance. For

thermal electrons the magnetic moment, ? ? mv2

and the second adiabatic invariant of the bounce motion,

J ?Hvkdl, are conserved [17]. The conservation of ? and

the decrease in B as the X line is approached causes a

decrease in v2

served in vk, which is due to the conservation of J and

the decreasing orbit bounce length (Hdl) in the inflow

region. The combination gives particles with vk* v?

(vk& v?) a net gain (loss) in Ekin the inflow region. By

is large enough that ? is conserved for thermal electrons at

all locations, and no pitch angle diffusion is observed at the

X line. In the outflow region the changes in Ekare reversed.

The trapped orbit motion also implies that f is nearly

symmetricin pitchangle

f?v;180?? ?? [16].

For sufficiently high energies (Ek> 1 keV) only the

magnetic forces are important in determining if an electron

is trapped or passing. The flat parts of the distribution in

Fig. 3(a), indicated by the red lines, correspond to passing

orbits, whose energies do not change on the path from the

ambient plasma. The blue parts are associated with trapped

electrons; the dip in phase-space density as a function of ?

is caused by the cooling of these electrons in the inflow

region. This interval of trapped electrons is determined by

the magnetic field strength, By? 6 nT, along the X line

and the asymptotic value, B1? 10:5 nT, away from the X

line. From the finite ratio By=B1, standard magnetic mirror

trapping arguments predict a phase-space boundary be-

tween passing and trapped particles (a loss-cone boundary)

at the pitch angles, ?b? 49?and ?b? 131?. These

boundaries agree with the interval covered by the dips in f.

For the bulk portion of the distribution (Ek? 639 eV)

the dips in phase-space density extend from ? ? 0?to ? ?

?=?2B?,

?[11,13]. Meanwhile, an increase is ob-

around90?:

f?v;?? ?

180?. This is evidence that the bulk electrons, at all pitch

angles, enter the X-line region along trapped trajectories.

The broadening of the interval for trapping implies the

presence of an electrostatic potential, ?trap, that traps all

thermal electrons.

In the remaining part of this Letter, we describe our

theoretical modeling of the measured distributions. The

magnetic geometry close to the X line is approximated

by B ? b0??z=??^ x ? ?x^ z ? l0^ y?. Consistent with the

Wind magnetic field measurements (see Fig. 1) we apply

By? b0l0? 6 nT, and l0? 65 km. The case with ? ? 1

corresponds to a currentless cusp for which the angle of the

separatrix is 90?(the equation of the separatrix is jzj ?

j?2xj). For other values of ? the separatrices meet at some

other angle. Several values of ? have been applied in our

modeling scheme; the results prove to be independent of

this parameter. For simplicity, here we consider a configu-

ration with ? ? 1. The electric fields are approximated by

E ? ?r??x;z? ? Ey^ y. Here ? ? ?rec? ?trap, where

?trap? ?0exp???x2? z2?=l2

rated to trap the thermal electrons. The reconnection rate

is estimated by assuming that vinflow? 0:1vA, from which

wefind

Ey? 0:1 ? 400 km=s ? 6 nT ? 0:24 mV=m

(note that our results are insensitive to Ey). The role of

?rec?1

E ? B ? 0 (required from E ? v ? B ? 0) outside the

electron diffusion region [17]. We apply a recent experi-

mental result that the size ? of the diffusion region scales

with the drift orbit width of the trapped electrons, ? ?

?c?

also been found applicable to configurations without a

guide magnetic field [20]). The potential ?rec? l0Ey?

15 V is small compared to the electron temperature

(400 eV), and the theoretical distributions are therefore

nearly independent of ?recand ?.

We assume that the electrons have reached a point x1

outside the X-line region when B?x1? ? B1. Hence, orbits

are followed back in time from their initial locations

?x0;v0?

b0

x2

0

? B1, where x1? ?x1;y1;z1?. The am-

bient plasma sheet is approximated by an isotropic distri-

bution f1?jv1j?, with a temperature of Te? 400 eV for the

bulk electrons and a power law dependence, f / E?4:5, for

the high energy tail, Ek> 2 keV, consistent with Wind

measurements away from the diffusion region. Additional

simulations (not included here) show that the fluctuation

level in the magnetic field observed in Fig. 1 does not

influence the results of our analysis.

Figures 3(d)–3(g) represent theoretical distributions

corresponding to the Wind measurements at time

08:00:22 where Wind was near the center of the diffu-

sion region. These theoretical distributions are obtained

with increasing values of the trapping potential (?02

f0;300;900;1150g V). As seen, the measured distributions

0? is the potential incorpo-

4l0Eylog??x2? ?2?=?z2? ?2?? is to ensure that

????????????????????????

mvth=?qb0?

p

? 10 km [18] (the length scale ?chas

tothefirstlocation,

?x1;v1?,forwhich

??????????????????????????

1? z2

1? l2

q

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are best matched for ?0? 1150 V. The value ?0?

1150 V was also applied in calculating the distribu-

tions in Figs. 3(h) and 3(i) corresponding to the location

of Wind for times 08:03:38 and 08:06:53. The agree-

ment between observations [Figs. 3(a)–3(c)] and model

[Figs. 3(g)–3(i)] covers several points which support the

present interpretation:

(i) For Ek> 1 keV,thewidth and the depth of the dipsin

the theoretical phase-space density are highly sensitive to

the ratio of By=B1; agreement between the measured f and

the theoretical f is obtained only when the measured

values of Byand B1are used in the simulation.

(ii) The theoretical anisotropy agrees with the measure-

ments not only near the center of the diffusion region, but

also at locations in the outflow region; compare Figs. 3(b)

and 3(c) with Figs. 3(h) and 3(i).

(iii) For Ek< 1 keV, agreement with the measured dis-

tributions are obtained by including an electrostatic poten-

tial, ?trap, whose strength is uniquely determined by the

anisotropy in the electrons. Furthermore, the energy gained

by the unmagnetized ions in their acceleration in the out-

flow region agree with ?trap. For the applied electrostatic

geometry, the potential drop from the center to the edge of

the simulation region is about 1000 V. The reported out-

flow speed of the ions, vion;out? 400 km=s [10], corre-

sponds to acceleration in a potential drop of ?835 V.

This suggests that ?trapis responsible (and adequate) for

accelerating the unmagnetized ions out of the X-line re-

gion. Such electrostatic structures have also been observed

in self-consistent particle simulations [21] (but were not

associated with trapping because vth? vout).

(iv) For electrons with energies above 6 keV, the Larmor

radii of the electrons are larger than the characteristic

length scale of the magnetic field curvature, l0. Thus, these

electrons see significant changes in B over a single Larmor

period which causes pitch angle diffusion [22]. Such non-

adiabatic effects are evident in both the measured and the

theoretical distributions: the levels of anisotropy decrease

rapidly above 4 keV.

(v) Inside the ion diffusion region the trapped electrons

bounce back and forth along field lines while drifting with

the magnetic field. They remain ‘‘frozen in’’ to the mag-

netic field to within the length scale ?cfrom the X line.

Thus, the electron dynamics are consistent with the Hall

magnetic field evident in Fig. 1 and discussed in Ref. [9].

Simulations with vth? vout[7,21,23,24] do not capture

the electron dynamics evident in the Wind measurements:

the electrons do not have time to bounce back and forth

along the field lines, so theoretical distributions, in general,

will not display the observed symmetry (typically in these

simulations jk? qnvth). However, the regime vth? voutis

likely to be relevant for reconnection sites closer to Earth

(characterized by higher values of B and lower ninflow

producing a higher vA).

In summary, we have applied the electron phase-space

distributions f, measured by Wind within a reconnection

region in Earth’s magnetotail, to infer properties of the

electrostatic geometry inside the ion diffusion region. For

the first time, the presence of a strong electrostatic poten-

tial, trapping all thermal electrons, is revealed. The sym-

metric properties of f associated with trapping can be

expressed as f?vk;v?? ? f??vk;v??. It follows that jk?

qRvkf?vk;v??dv3? qnvth, which allows the plasma to

support an electric field along the X line, mediating fast

reconnection [17,19]. Additional investigations in the re-

gime vth? vout, numerical as well as experimental, are

needed to determine how the trapped orbit dynamics trans-

late into dissipation free r ? P and v ? rj terms in the

generalized Ohm’s law.

This work is partly funded by NSF/DOE Grant No. DE-

FG02-03ER54712 at MIT and NASA Grants No. NAG5-

12941 and No. NAG5-10428 at U.C. Berkeley.

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MagneticReconnection

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