Nonlinear Evolution of the Lower-Hybrid Drift Instability in Harris Current Sheet
Magnetic reconnection is a fundamental plasma physical process which is important
in solar corona, Earth’s magnetosphere and laboratory plasma experiments [1]. It
leads to a topological change of magnetic field and rapidly converts magnetic energy
into plasma kinetic energy and thermal energy. On the other hand, current sheets with
thickness of the order of ion scale are unstable to a variety of instabilities, e.g.
collisionless tearing [2] and lower-hybrid drift instability [3]. 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, the influence of these instabilities 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]. The fastest
growing modes are primarily electrostatic with k B  0 and the wavelength on
electron scale. In the past the LHDI has been generally considered as a possible
candidate to provide anomalous resistivity needed in reconnection physics.
Unfortunately, while enhanced fluctuations are required in the central region of
current sheet to produce significant anomalous resistivity, theory predicts the fastest
growing modes are localized on the edge region [3]. Thus it is impossible for the
LHDI provide any significant anomalous resistivity. This conclusion is also confirmed
by observations in the magnetosphere [5], and laboratory experiments [6].
Although previous theories consider LHDI as an unimportant mode, recently some
new results begin to challenge the elder conclusion. First, although the fastest growing
modes are on the electron gyro-scale, the LHDI is unstable over a broad range of
wavelengths and frequencies, longer wavelength LHDI with k y ie
1 can
penetrate into the central region even though the fastest growing modes with
k y e
1 are confined to the edge [7]. The required thickness for this penetration is
approximately i / L  1.5 . 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, former simulations are mainly focused on the early phase of the LHDI.
Shinohara et al. [11] discussed the late time nonlinear evolution of an ion-scale
current sheet, but they focus on K-H instability excited by velocity shear between
background plasma and ions in current sheet. The effects of thickness of current sheet
on the nonlinear evolution of LHDI are still controversial.
In this paper, we revisit the nonlinear development of LHDI in Harris current sheets
with different thickness ( L  0.5c / pi and 1.0c / pi ) with 2-D full particle simulation.
We pay more attention on the late time evolution of the LHDI.
Numerical method and simulation code
A two-dimensional full particle code is used to study the LHDI in the Harris current
sheet. 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.
Harris current sheet equilibrium is considered initially, in which an initial magnetic
field given by
B0 ( z)  B0 tanh(z / L)ex (1)
and a plasma density given by
n 0 ( z )  n 0 sec h 2 ( z / L)
(2)
In our simulations, 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 given by 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)
[Daughton
1998].
Two
different
current
sheet
thicknesses
L  0.5c / pi and 1.0c / pi are considered, where c / pi is the ion inertial length
defined using the peak Harris density n 0 , and c=15VA, where VA is the Alfven speed
based on B0 and n0. No background plasma being introduced, since the presence of
background plasma is strongly stabilizing to the LHDI and adds complexity to the
simulation results.
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. In all simulation cases we employ more than two million
particles to represent the plasma in Harris current sheet. 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.
Simulation Results
We employ 2-D full particle code to simulate the evolution of Harris current sheets,
with no initial perturbation or background plasma introduced. The main points we
addressed are on the waves and their nonlinear evolution in the current sheets.
Figure 1 shows the magnetic field variations of current sheets at different time, with
the thicknesses of current sheets are 0.5c / pi and 1.0c / pi , respectively. The initial
equilibrium magnetic field has been subtracted to show the magnetic fluctuations. It is
found that both the evolution of the two current sheets can be divided in to three
phases. In the case of L  0.5c / pi , at the time of ci t  2.0 , the fastest growing
LHDI with wavelength at the electron scale have saturated. The magnetic fluctuations
are highly localized in the edge of current sheet. At the same time these fluctuations
can penetrate directly into the centre region of the current sheet. This result is
consistent with the linear Vlasov simulation [Daughton 2003]. At later times in this
simulation, significant growth of longer wavelength modes with k y di d e
1 is
observed at the center of the current sheet. Finally, at ci t  13.0 , it converts into a
kink mode, with a wavelength on ion scale. Similarly, as shown in Fig 1 (b), the
evolution of current sheet with the thickness of 1.0c / pi also can be divided into
three phases. At the time of ci t  6.0 , it can be observed that the LH waves are well
localized at the edge of the current sheet. No obvious fluctuation appears in the centre
region because of the strong stable effect of thick current sheet. At the later time of
this simulation, the current sheet is thinning due to the effect of LHDI. Meantime, the
active wave region can extend out of current sheet. Finally, at ci t  33.0 , these
waves can extend into the centre region of current sheet, form a long wavelength
mode. The main region of waves are confined in the current sheet region, and no
strong kink is observed. The properties of these modes, will be discussed below.
Because the simulation is periodic in the y direction, we perform Fourier
transformation in the direction of y to show the amplitudes of wave modes with
different wave vector. At the different position along z axis, the Fourier
transformations are carried out to show the amplitudes of modes at the three phases.
Fig. 2 shows these modes in the case of L  0.5c / pi . At the time of ci t  1.6 , it
can be shown that the short wavelength waves with kdi  5.4 (k di d e  1.4) is mostly
confined at the edge. Only longer wavelength waves can penetrate into the centre
region. At the time of ci t  5.2 , it is found that the waves with peak at
kdi
5.4 ( k di d e
1.4) can excited in the centre region of the current sheet. This
mode decreases as it extend out of the current sheet. As the current sheet evolutes, at
the late time the whole wave spectrum drifts to long wavelength waves. In the end of
the simulation ( ci t  13.0 ), because the current sheet evolutes into a kink mode,
therefore the waves of the inner and outer regions show similar wave spectrums,
which are just different in amplitudes.
Fig. 3 shows the modes of different positions at different times, in the case of
L  1.0c / pi . At first, while the main waves are on the edge of the sheet, there are no
significant waves in the centre region. In the later time there are also waves excited in
the centre regions. However, these modes show great differences with that of thinner
current sheet. First, the wave spectrum in the thinner sheet can extend to
kdi
7.0 (k di d e
1.91) , in the case of L  1.0c / pi , it is mainly confined below
kdi
4.5 ( k di d e
1.23) . Second, in the case of 1.0, the modes with wavelength
kdi  4.5 (k di d e  1.23) decrease from the edge of current sheet to centre.
Oppositely, in the thinner case, the wave modes between kdi  4 (k di d e  1.1) and
kdi  7 (k di d e  1.9) decrease from the centre to the edge. In the late time, the
thinner current sheet is founded to kink significantly, whereas the waves are mostly
confined in a thin layer in the thicker current sheet case.
In this paper we report the new features of Harris current sheet with thickness on the
order of ion inertia length. Two different thicknesses ( L  0.5c / pi and 1.0c / pi ) and
long-term evolution are considered. First, in both the two cases, the development of
Harris current sheets can be divided into three phases. Second, the two current sheets
show distinct features as they evolute.
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