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The Lower-Hybrid Drift Instability in Harris Current Sheet:
Particle dynamics
GUO Fan (郭帆)，LU Quan-Ming (陆全明)
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. On the other hand, current sheets with thickness of the order of ion scale are
unstable to a variety of instabilities, e.g. lower-hybrid drift instability [2]. These
instabilities are long thought to be central important in the onset and nonlinear
development of magnetic reconnection. However, while this topic has been studied
for several decades, its influence to the onset and development of magnetic
reconnection is still unclear.
The lower-hybrid drift instability (LHDI) is driven by the diamagnetic current in
the presence of inhomogeneities in the density and magnetic field [4]. According to
linear theory, 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, linear theory predicts the
fastest growing modes are localized on the edge region due to finite beta effects [3].
Thus it is impossible for the LHDI provide any significant anomalous resistivity
inside the current sheet. This conclusion is also confirmed by observations in the
magnetosphere [5], and laboratory experiments [6].
Although previous theory considers LHDI as an unimportant mode, recently some
new results begin to challenge the former conclusion. First, although the fastest
growing modes are on the electron scale, the LHDI is unstable over a broad range of
wavelengths and frequencies, the electromagnetic modes with longer wavelength can
penetrate into the central region even though the fastest growing modes are confined
to the edge [7]. Second, LHDI can strongly modify the structure of current sheet, and
causes anisotropic heating of electrons [8-9]. These effects will efficiently enhance the
growth rate of collisionless tearing mode [10], therefore may play an important role in
the onset of and nonlinear development of magnetic reconnection. However, the effect
of LHDI on the dynamics of individual species of particles and its importance in
evolution of current sheet is still unclear. In this paper, we discuss the dynamics of
electrons and protons in Harris current sheets with 2-D full particle simulation. Our
results show that due to the effect of LHDI, the dynamical features of electrons and
ions are strongly distinct and depend on different physical mechanism. While
electrons perform drift motion in the influence of excited LH waves, the protons are
mainly governed by a direct electrostatic field created by accumulation of ions at the
edge of current sheet. Their possible effects on the evolution of current sheet are also
discussed.
A two-dimensional full particle code is used to study the evolution of the Harris
current sheet [11]. In this code, the electromagnetic fields are defined on the grids and
are updated by solving the Maxwell equations with a full explicit algorithm. Both the
ions and electrons are advanced self-consistently in the electromagnetic fields.
The simulation box (y, z) is 12.8c / pi  6.4c / pi and the time step is tce  0.1,
the spatial grid is 512×256. Harris current sheet equilibrium is considered initially, in
which an initial magnetic field is given by
B0 ( z)  B0 tanh( z / L)ex (1)
and a plasma density is given by
n 0 ( z )  n 0 sec h 2 ( z / L)
(2)
In the simulation we employ about three million particles to represent the ions and
electrons individually in Harris current sheet. The distribution functions for the ions
and electrons are Maxwellian, and their drift speed in the direction of y
satisfy Vi0 / Ve0  Ti0 / Te0 , in which the temperature ratio is set to be Ti0 / Te0  5 , ‘i’
and ‘e’ for ions and electrons respectively. The diamagnetic current is
J 0 (z)  en 0 (z)(Vi0  Ve0 ) . The mass ratio is mi / me  180 , which is large enough to
suppress the drift-kink instability (DKI) [12]. The current sheet thicknesses
L  0.5c / pi is considered, where c / pi is the ion inertial length defined using the
peak Harris density n 0 , and c is the light speed in vacuum and is set to be c=15VA,
where VA is the Alfven speed defined based on B0 and n0. No background plasma is
introduced, since the presence of background plasma is strongly stabilizing to the
LHDI and adds complexity to the simulation results.
The 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.
Fig. 1 (a) shows the fluctuations in the magnetic field B x excited the growth of
LHDI at time i t  1.8 . In this figure, the initial magnetic field in Eq. (1) has been
subtracted to show the magnetic fluctuation. It is observed that the magnetic
fluctuations are mainly localized on the edge of the current sheet meanwhile
propagate in the direction of y, with a wavelength approximately 0.7c / pi . Fig. 1 (b)
displays the electric field E y , the fluctuation of electric field is excited in a larger
region than that of magnetic field. These features of excited waves are correspondent
well with linear theory [3] and previous simulations [7]. The excited waves will make
electrons perform E B drift motion and generate electron vortices which move in
the direction of y. These electron vortices will strongly reduce cross-field current and
dissipate the magnetic field [13]. The electric field E z is plotted in Fig. 1 (c). It is
shown that there is a large scale electric field point out of the current sheet in
simulation plane. This electric field is primarily electrostatic and formed due to the
accumulation of ions on the edge of the current sheet.
Fig. 2 (a) and Fig. 2 (b) illustrate the z-component velocities of electrons and ions,
respectively. In Fig. 2 (a), clear wave-like structures are formed in the region of
1  z  1 , which is consistent with the electric field E y . This result demonstrates the
electrons perform E B drift motion in the influence of LH waves. The y-component
(current direction) electron velocity can also be interpreted as drift motion due to the
influence of z-component electric field [8]. The electron drift motion will strongly
modify the current profile and may have significant roles in nonlinear development of
Harris current sheet [9]. On the other hand, the proton motion shown in Fig. 2 (b) is
consistent well with E z in Fig. 1 (c). This indicates that protons are accelerated in
the direction of z due to direct electric field. Thus the motions of electrons and
protons perform very distinct feature and are dependent on different physical
mechanism.
Because of the large separation between protons scale and electrons scale due to
large mass ratio, the dynamics of protons and electrons usually show very different
features in current sheet with thickness on the order of ion inertial scale. In this
condition, ions decouple from the magnetic field and electrons still perform a rotation
motion in the current sheet. In this work we report the effect of LHDI on the
dynamics of ions and electrons, the results show that due to the different motions of
ions and electrons, the dynamics of individual particles depends on two different
physical mechanisms and shows distinct features. First, the acceleration of electrons
in current sheet is primarily E  B drift, in which the electric field is from the LH
waves excited by LHDI. Second, the acceleration of protons in Harris current sheet is
mostly direct electric field acceleration. This electric field is created by accumulation
of ions on the edge of Harris current sheet due to the nonlinear effect of LHDI. Last,
when the electromagnetic component of LHDI penetrates into the current sheet, the
gradient magnetic drift will dominate motion of electrons, and may result in the kink
of current sheet. This effect may result in important nonlinear consequences.
It is needed to note that our simulation results are only 2-D and can not reveal the
full pictures of the motion of particles parallel to the magnetic field. In order to reveal
the full features of LHDI and its importance in reconnection physics, one should
implement a full 3-D simulation with high spatial resolution, which beyond the
computation current available.
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[12] Daughton W., J. Geophys. Res. 103, A12, 29429-29444
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