*
GUO Fan ( 郭帆 )
1,2 ， LU Quan-Ming ( 陆全明 )
1,2
1
School of Earth and Space Sciences, University of Science and Technology of China,
Hefei, Anhui 230026
2
Key Laboratory of Space Weather, Center for Space Science and Applied Research,
Chinese Academy of Sciences, Beijing 100080
Corresponding Author:
Quanming Lu
School of Earth and Space Sciences
University of Science and Technology of China
Hefei, Anhui 230026
Email:qmlu@ustc.edu.cn
* Supported by National Natural Science Foundation of China under Grant No. 40674093 and
Chinese Academy of Sciences.

Abstract
2-dimensional (2-D) particle-in-cell simulations are performed to investigate electron dynamics in the lower hybrid drift instability (LHDI) of Harris current sheet.
The results show that the drift motions have great importance on electron dynamics.
At first, the LHDI is excited at the edge of the current sheet and the motions of the electrons in the LHDI are controlled by E B drift. In the nonlinear evolution stage, the electromagnetic component of lower-hybrid drift waves penetrates into the center of the current sheet. The motions of the electrons in the center of the current sheet are dominated by gradient B drift, while their motions at the edge of the current sheet are still governed by E B drift. The consequence of these drift motions on the evolution of current sheet is also discussed.
PACS : 94.30 cp, 94.20.wf, 52.65.Rr

Magnetic reconnection is a fundamental plasma physical process in solar corona,
Earth’s magnetosphere and laboratory plasma experiments [1]. It leads to topological changes of magnetic field and rapidly converts magnetic energy into plasma kinetic energy [2-3]. On the other hand, current sheets with thickness on the order of ion scale are unstable to a variety of instabilities, which are thought to have central importance on the onset or nonlinear development of magnetic reconnection. One of such instabilities is the lower-hybrid drift instability (LHDI), which is driven by the diamagnetic current in the presence of inhomogeneities in the density and magnetic field [4]. According to the linear theory of the LHDI, the fastest growing modes are primarily electrostatic with k B 0 and wavelength on electron scale. In the past the LHDI has been generally considered as a possible candidate to provide anomalous resistivity needed in classic reconnection model. Unfortunately, while enhanced fluctuations are required in the central region of current sheet to produce enough anomalous resistivity, the linear theory predicts the fastest growing modes are localized at the edge of the current sheet due to the finite beta effect [5]. Thus it is impossible for the linear LHDI to provide any significant anomalous resistivity in the current sheet.
Recently some new results begin to challenge previous conclusion. First, although the fastest growing modes are on the electron scale, the LHDI is unstable over a broad range of wavelengths and frequencies, in which electromagnetic modes with longer wavelength can penetrate into the central region of the current sheet even though the

fastest growing modes are confined to the edge [6]. Second, LHDI can strongly modify the structure of current sheet, and causes anisotropic heating of electrons [7-8].
These effects will efficiently enhance the growth rate of collisionless tearing mode [9], therefore may play an important role on the onset of magnetic reconnection. In this paper, we examine electron dynamics in the LHDI of Harris current sheet with
2-dimensional (2-D) particle-in-cell simulations. Its significance on the evolution of the current sheet is also discussed.
In particle-in-cell simulations, the electromagnetic fields are defined on the grids and both the ions and electrons are advanced self-consistently in the electromagnetic fields. In our 2-D simulation code [10], the electromagnetic fields are updated by solving the Maxwell equations with a full explicit algorithm, and the simulations are performed in y
 z plane. Harris current sheet equilibrium is considered initially, in which an initial magnetic field is given by
B ( )
0 z

B
0 z L x
(1) where B
0
is the initial magnetic field on the edge of the simulation box and L is the half-width of current sheet. The plasma number density is given by n z
0
 n
0 sec
2
( / ) (2) where n is initial plasma number density in the center of current sheet. The
0 thickness of the current sheet thickness is set to be L

0 5c /
 pi
, where c /
 pi
is the ion inertial length defined using the peak Harris density n . In the simulations we
0 employ about three million particles to represent the ion or electron component. The

initial velocity distributions of the ions and electrons are Maxwellian, and their drift speeds in the y direction satisfy v i0
/ v e0
 
T i0
/ T e0
, in which the temperature ratio is chosen as T i0
/ T e0

5 (where the subscripts i and e stand for the ion and electron, respectively). The diamagnetic current is J z
0

0
( )( i0
 v e0
) . The mass ratio is taken to be m m i
/ e

180 and light speed in vacuum is set to be c=15v
A
, where v
A
is the Alfven speed defined based on B
0
and n
0
. The dimension of the simulation box is
L x

L y

( 12 8c /
  pi
6 4c /
 pi
) , where L and x
L y
are the spatial length in x direction and in y direction, respectively. The time step is t 0 001 , in which
 ci
is ion gyro-frequency defined using B
0
, and the spatial resolution is
Δy=Δz=0.025c ω pi
. Periodic boundary conditions are used along the y direction. The ideal conducting boundary conditions for electromagnetic fields are employed in the z direction, and particles are reflected if they reach the boundaries.
The evolution of the LHDI in the current sheet can be divided into linear and nonlinear stages, and in this paper we choose
  i t 2 0 and
  i t 6 0 to represent the two stages. Fig. 1 shows the fluctuating magnetic field in the x direction
(B x

B
0 tan( / )) / B
0
and electric field in the y direction (c/v )E /B
A y 0
at
(a) t 2 0 and (b) t 6 0 . At t 2 0 it is observed that the excited lower hybrid drift waves are localized on the edge of the current sheet, and they propagate along the y direction. The waves are mainly electrostatic and the wavelength is approximately 0 7c /
 pi
. These features of the excited waves are consistent well with the linear theory [5] and previous simulations [6] . At
  i t 6 0 , we can find that the

electromagnetic component of the lower hybrid drift waves with longer wavelength has already penetrated into the centre of the current sheet. Meanwhile the electrostatic component of the waves is strongly suppressed in the center of the current sheet and localized at the edge of the current sheet.
Fig. 2(a) illustrates the average velocity of electrons in the z direction v v ez A
at
 
.
i t 2 0 . Wave-like structures can be clearly found at the edge of the current sheet.
Such structures are formed by the electron drift motions due to the excited electric field of the lower hybrid drift waves. In the linear stage, because the electric field of the lower hybrid waves mainly point to the y direction, its corresponding drift velocity is mainly along the z direction. Fig. 2 (b) shows the value of the drift velocity in the z direction (c/v )(-E /B )
A y x
at
 
.
i t 2 0 . Compared with Fig. 2(a), it can be clearly found that the average velocity of electrons in the z direction can be well approximated by drift motions due to the electric field of the excited lower hybrid waves, and such motions generates electron vortices moving in the y direction.
In the nonlinear stages of the LHDI, the electron dynamics in the current sheet is more complicated. Fig. 3(a) shows the average velocity of electrons in the z direction v v ez A
at
  i t 6 0 . Fig. 3(b) describes the drift motions of electrons due to the excited electric field (c/v )(-E /B )
A y x
, while Fig. 3(c) shows the drift motion due to

B , which is along the z direction and its value is (

B /
 y c
 pi
) / B
0
. We can find that at the edge of the current sheet, the motions of the electrons are still governed by

electric field drift, while at the center of the current sheet the motions of the electrons are determined by the

B drift.
In summary, with 2-D particle-in-cell simulation we investigate the lower hybrid drift waves in Harris current sheet and its influence on electron dynamics. Our results indicate that drift motions are very important in the effect of LHDI on the dynamics of electrons. In the linear growth stage, the waves are localized at the edge of the current sheet, and the motions of electrons are determined by E B drift, in which the electric field is produced by the excited lower hybrid waves. In the nonlinear evolution stage, the electromagnetic lower hybrid waves with long wavelength can penetrate into the center region of current sheet, the electron motion in the centre is determined by

B drift while that of the outer region is still can be demonstrated by the electric field drift. Obviously, the motions of electrons will significantly distort the current profile, and may have great importance on the onset of collisionless magnetic reconnection. How it will influence the magnetic reconnection is the scope of our future study.

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Fig. 1

Fig. 2

Fig. 3

Fig. 1.
Contours of magnetic field in the x direction (B x

B
0 tan( / )) / B
0
and electric field in the y direction (c/v )E /B
A y 0
at (a)
  i t 2 0 and (b)
  i t 6 0 , respectively.
Fig. 2.
Contours of (a) electron average velocity in z direction v v ez A
and (b) estimated E B drift velocity in z direction (c/v )(-E /B )
A y x at
  i t 2 0
Fig. 3.
Contours of (a) electron average velocity in z direction v ez v
A
, (b) the estimated E B drift velocity in z direction (c/v )(-E /B x
) , and (c) drift velocity due to grad B drift in z direction (

B /
 y c
 pi
) / B
0 at t 6 0 .