New Measure of the Dissipation Region in Collisionless Magnetic Reconnection
Seiji Zenitani, Michael Hesse, Alex Klimas, and Masha Kuznetsova
NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
(Received 7 February 2011; published 11 May 2011)
A new measure to identify a small-scale dissipation region in collisionless magnetic reconnection is
proposed. The energy transfer from the electromagnetic field to plasmas in the electron’s rest frame is
formulated as a Lorentz-invariant scalar quantity. The measure is tested by two-dimensional particle-in-
cell simulations in typical configurations: symmetric and asymmetric reconnection, with and without the
guide field. The innermost region surrounding the reconnection site is accurately located in all cases.
We further discuss implications for nonideal MHD dissipation.
DOI: 10.1103/PhysRevLett.106.195003 PACS numbers: 52.35.Vd, 52.27.Ny, 94.30.cp, 95.30.Qd
Magnetic reconnection [1] is a fundamental process in
many plasma systems, ranging from laboratory and solar-
terrestrial environments to extreme astrophysical settings.
The violation of the ideal condition, E þ v B Þ 0,is
essential to allow the magnetic flux transport across the
reconnection point. The critical ‘‘diffusion region’’ (DR)
where the ideal condition is violated is of strong interest for
understanding the key mechanism of reconnection. In col-
lisionless plasmas, since ions decouple first from the
magnetic fields, it is thought that the DR consists of an
ion-scale outer region and an electron-scale inner region.
In two-dimensional (2D) reconnection problems in the
x-z plane, a popular criterion to identify the innermost
‘‘electron diffusion region’’ (EDR) is the out-of-plane
component of the electron nonideal condition, E
y
Þ 0,
where
E
¼ E þ v
e
B ¼
1
n
e
q
rP
$
e
m
e
q
dv
e
dt
; (1)
and P
$
e
the electron pressure tensor. In particular, it is
known that the divergence of the pressure tensor sustains
a finite E
y
¼ E
y
at the reconnection point, arising from
local electron dynamics [2,3].
Recent large-scale particle-in-cell (PIC) simulations
have shed light on the electron-scale structures around
the reconnection site. Earlier investigations [4,5] found
that the EDR identified by E
y
Þ 0 [4] or the out-of-plane
electron velocity [5] extends toward the outflow directions.
Previous research has suggested that the EDR has a two-
scale substructure: the inner EDR of E
y
> 0 and the outer
EDR of E
y
< 0 with a super-Alfve
´
nic electron jet [6,7].
Satellite observations found similar signatures far down-
stream of the reconnection site [8]. The roles of these
EDRs are still under debate; however, there is a growing
consensus that only the inner EDR or a similar small-scale
region should control the reconnection rate [7,9,10].
Importantly, it was recently argued that the outer EDR is
non- or only weakly dissipative, because the super-
Alfve
´
nic jet and E
y
Þ 0 condition stem from projections
of the diamagnetic electron current in a suitably rotated
frame [10].
Meanwhile, a serious question has been raised by nu-
merical investigations on asymmetric reconnection, whose
two inflow regions have different properties such as in
reconnection at the magnetopause [11,12]. It was found
that various quantities including E
fail to locate the
reconnection site in asymmetric reconnection, especially
in the presence of an out-of-plane guide field [11].
Considering the debate on inner or outer EDRs and the
puzzling results in asymmetric reconnection, it does not
seem that E
Þ 0 is a good identifier of the critical region.
In this Letter, we propose a new measure to identify a
small, physically significant region surrounding the recon-
nection site. We construct our measure based on the fol-
lowing three theoretical requirements. First, we are guided
by the notion that dissipation should be related to nonideal
energy conversion. Second, we desire a scalar quantity. If
we use a specific component of a vector, we have to choose
an appropriately rotated frame [10]. Using a scalar quantity
instead, we do not need to find the right rotation in a
complicated magnetic geometry. Third, it should be insen-
sitive to the relative motion between the observer and the
reconnection site. For example, the reconnection site can
retreat away [13], or, for example, the entire reconnection
system may flap over a satellite due to the magnetospheric
motion.
Our strategy is as follows. We choose a frame that can be
uniquely specified by the observer. Among several candi-
dates, we choose the rest frame of an electron’s bulk
motion because it would be the best one to characterize
electron-scale structures. Next we consider the energy
transfer from the field to plasmas in this frame, which is
a scalar quantity. We then expand it with observer-frame
quantities. The obtained measure meets all three require-
ments. It is a Lorentz invariant (frame-independent scalar)
and is related to the nonideal energy transfer.
We follow the spacelike convention ( , þ, þ, þ).
Let us start from the electromagnetic tensor F
,
PRL 106, 195003 (2011)
PHYSICAL REVIEW LETTERS
week ending
13 MAY 2011
0031-9007=11=106(19)=195003(4) 195003-1 Ó 2011 American Physical Society