New measure of the dissipation region in collisionless magnetic reconnection.

Abstract

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.

Full text

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A 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, MD 20771, USA
(Dated: Submitted 7 February 2011; accepted 19 April 2011)
Abstract
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.
PACS numbers: 52.35.Vd, 94.30.cp, 95.30.Qd, 52.27.Ny
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arXiv:1104.3846v1 [astro-ph.SR] 19 Apr 2011

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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 + v × B ?= 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 collisionless 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 problem 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?= 0, where
E∗= E + ve× B = −1
neq∇ ·← →
Pe−me
q
?dve
dt
?
,(1)
and← →
Pe the electron pressure tensor. In particular, it is known that the divergence of
the pressure tensor sustains a finite Ey= E∗
yat 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?= 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-Alfv´ enic electron jet [6, 7].
Satellite observations found similar signatures far downstream 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-Alfv´ enic jet and E∗
y?= 0 condition stem from projections of
the diamagnetic electron current in a suitably rotated frame [10].
Meanwhile, a serious question has been raised by numerical 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/outer EDRs and the puzzling results
in asymmetric reconnection, it does not seem that E∗?=0 is a good identifier of the critical
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region.
In this Letter, we propose a new measure to identify a small, physically significant region
surrounding the reconnection site. We construct our measure based on the following 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 insensitive 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 candidates, we choose the rest frame of 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
requirements. It is a Lorentz-invariant (frame-independent scalar) and is related to the
nonideal energy transfer.
We follow the space-like convention (−,+,+,+). Let us start from the electromagnetic
tensor Fµν,



Using a 4-velocity (uµ) = γ(c,v), where γ is the Lorentz factor γ = [1 − (v/c)2]−1/2, we
obtain a 4-vector of the rest-frame electric field eµ[14],
Fµν=




0Ex/c Ey/c Ez/c
−Ex/c
−Ey/c −Bz
−Ez/c
0Bz
−By
Bx
0
By
−Bx
0







(2)
eµ= Fµνuν=¯Λµ
νe?ν.(3)
Here the prime sign?denotes the properties in the rest frame of an arbitrary motion uµ,
and¯Λ is the inverse Lorentz transformation from the moving frame. The components of eµ
and e?µare given by,
(eµ) =
?γv · E
c
,γ(E + v × B)
?
, (e?µ) = (0,E?).(4)
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We also use the 4-current (Jµ) = (ρcc,j), where ρcis the charge density. The current can be
split into the conduction current jµand the convection current, a projection of the motion
of the non-neutral frame [15],
Jµ= jµ+ ρ?
cuµ= jµ+ c−2(−Jνuν)uµ, (5)
such that (j?µ) = (0,j?) is purely space-like.
Let us define a dissipation measure D, the energy conversion rate in the moving frame.
The contraction of the covariant and contravariant vectors gives us a Lorentz-invariant scalar,
D(u) = j?· E?= j?
= Jµeµ+ c−2Jαuα(uµFµνuν) = JµFµνuν
= γ?j · (E + v × B) − ρc(v · E)?.
µe?µ≡ jµeµ
(6)
Choosing the frame of electron bulk motion (the number density’s flow), we obtain the
electron-frame dissipation measure,
De= γe
?j · (E + ve× B) − ρc(ve· E)?.(7)
In the nonrelativistic limit, one can simplify Eq. (7) by setting γe→ 1. One can confirm
this by multiplying j?= qniv?
In ion-electron plasmas, since Jµ= q(niuµ
i= (j − ρcve) and E?= E∗= (E + ve× B).
i− neuµ
e) where n is the proper density, we
obtain the following relation between the electron-frame and ion-frame measures,
neDe= niDi.(8)
Such a symmetric relation is reasonable, as ions are the current carrier in the electron’s frame
and vice versa. If ions consist of multiple species, neDe=?
species and Z is the charge number.
sZsnsDs, where s denotes ion
To see how our measure characterizes the reconnection region, we have carried out 2D
nonrelativistic PIC simulations. The length, time, and velocity are normalized by the ion
inertial length di= c/ωpi, the ion cyclotron frequency Ω−1
ci, and the ion Alfv´ en speed cAi,
respectively. The mass ratio is mi/me= 25, and electron/ion temperature ratio is Te/Ti=
0.2. Periodic (x) and conductive wall (z) boundaries are used. Four runs (1-4) are carried
out. Runs 1 and 2 employ a Harris-like configuration, B(z) = B0tanh(2z)ˆ x and n(z) =
0.2+n0cosh−2(2z). The domain of [0,102.4]×[−25.6,25.6] is resolved by 16002cells. 2.6×109
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particles are used. The speed of light is c = 10. In run 2, we impose a uniform guide field
By = B0. Runs 3 and 4 employ asymmetric configuration. Since no kinetic equilibrium
is known, we employ the following fluid equilibrium proposed by Ref. [16], B(z) = B0[1
?1−1
fields and the density vary from −B0/2 and n0to 3B0/2 and n0/3. The domain of [0,64] ×
[−12.8,12.8] is resolved by 1000×800 grid points. 9×108particles are used. The speed of
light is c = 20. In run 4, a guide field By= B0is added. In all runs, reconnection is triggered
2+
tanh(2z)] ˆ x and n(z) = n0
3tanh(2z)−1
3tanh2(2z)?. Across the current sheet, magnetic
by a small flux perturbation.
(a)
(b)
x
z
De
E*y
z
FIG. 1. (Color online) Snapshots of run 1 at t = 60, averaged over Ω−1
ci. (a) The nonideal electric
field E∗
yand (b) the electron-frame dissipation measure De(Eq. (7)).
The panels in Figure 1 present the popular measure E∗
yand the electron-frame dissipation
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