VOLUME 87, NUMBER 9

PHYSICAL REVIEW LETTERS

27 AUGUST 2001

Rapid Large-Scale Magnetic-Field Dissipation in a Collisionless Current Sheet via Coupling
between Kelvin-Helmholtz and Lower-Hybrid Drift Instabilities
I. Shinohara,1, * H. Suzuki,2 M. Fujimoto,2 and M. Hoshino3
1

2Max Planck Institute for Extraterrestrial Physics, Garching 85748, Germany
Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Tokyo 152-8551, Japan3
Department of Earth and Planetary Science, University of Tokyo, Tokyo 113-0033, Japan(Received 29 January 2001; published 13 August 2001)

Rapid large-scale magnetic-field dissipation is observed in a full kinetic simulation of cross-field current
instabilities in a current sheet even when the thickness of the current sheet is at ion scale. The KelvinHelmholtz instability caused by the velocity shear between the current-carrying ions and the cold background ions excites the lower-hybrid drift instability at the edges of the undulated current sheet. We showthat the nonlinear coupling between these two instabilities is responsible for the observed rapid dissipation. The simulation result presents a new route for magnetic-field dissipation in an ion-scale current
sheet and demonstrates the general significance of nonlinear cross-scale coupling in collisionless plasmas.
DOI: 10.1103/PhysRevLett.87.095001

PACS numbers: 52.35.Mw, 52.65. –yUnderstanding cross-scale coupling between MHD anddissipation scales in collisionless plasmas is challenging.
Magnetic reconnection is an impressive example of thiscross-scale coupling. Although magnetic reconnection itself is a MHD scale phenomenon, its dynamics is closely
related to the dissipation process in the diffusion region.
Many theorists believe that the turbulence in an electronscale current sheet or a strongly enhanced longitudinal current layer is important to provide sufficient dissipation totrigger fast reconnection in space plasmas, e.g., the magnetospheric substorm, the solar flare, etc. [1–4]. While
there is observational evidence that the magnetotail current
sheet thickness becomes as thin as comparable to the relevant ion inertia length prior to the substorm onset, whether
it thins further down to electron scale to initiate reconnection is an open question [5]. The solar corona situation is more serious, where even the ion inertia lengthis much smaller than the current sheet scales [6]. Recent
results of the laboratory experiments of magnetic reconnection also suggest that the thickness of the current layer
is much broader than the electron inertia length [7]. It istherefore an interesting question to ask if some dissipation mechanisms set in already at ion-scale current sheets.
Some among a number of instabilities associated with acurrent sheet can produce effective resistivity. However,no dissipation mechanism that causes the explosive onsetof magnetic reconnection in an ion-scale current sheet hasbeen reported. An aim of this work is to search dissipation
processes associated with cross-field current instabilities inan ion-scale current sheet which can lead to the fast onsetof magnetic reconnection. It is an important point to understand the explosive nature of magnetic reconnection
which is still poorly understood [8]. In this Letter, we show
a magnetic-field dissipation mechanism for an ion-scalecurrent sheet via nonlinear coupling between the KelvinHelmholtz instability (KHI) [9] and the lower hybrid drift
instability (LHDI) [10,11]. If a stationary backgroundplasma is distributed outside the current sheet, the KHI

caused by the shear flow between the background ions andthe current-carrying ions can grow [9]. Little attention has
been paid to this type of KHI because the KHI itself does
not provide anomalous resistivity. However, since the KHI
can significantly modulate the current sheet structure, there
is a possibility that other dissipation scale instabilities, suchas the LHDI, are newly excited associated with the KHI.
In order to examine the nonlinear coupling between theKHI and dissipation scale instabilities, we carry out a twodimensional electromagnetic full kinetic simulation. For
details of the scheme of our code, see Hoshino [12]. Concerning the study on cross-field current instabilities, it isknown that an artificial instability called the drift kink instability (DKI) has large growth rates at a low ion-electronmass ratio mi 兾me , 200 [13]. Since the DKI smears out
what we expect to operate in a real plasma, the growth of
the DKI must be suppressed. Note that the KHI can growwithin a few tens of ion gyroperiods even for the real mass
ratio, while the DKI masks the KHI in the low mass ratio
regime. High capability of recent supercomputers enables
us to perform full kinetic simulations with the mass ratio
as large as 400 even for ion-scale phenomena as we do inthis study. As the initial condition for Run 1, we makeuse of the Harris current sheet for simplicity; Bz 共 y兲 苷
B0 tanh共 y兾D兲, ncs 共 y兲 苷 Ncs 兾 cosh2共 y兾D兲, Jx 共 y兲 苷encs 共Ui 1 Ue 兲, and Ui 兾Ue 苷 2Ti,cs 兾Te,cs . D represents
the thickness of the current sheet, which we set D 苷 li(li 苷 c兾vpi , the ion inertia length). The simulation box
size is 512 grids 3 1024 grids corresponding to 8li 316li , and ⬃8.5 3 107 particles for both electrons and ions
are used. Conducting walls are assumed at the y boundaries, y 苷 68li , while a periodic boundary condition isused for the x boundaries. In the present case, the ionelectron temperature ratio is taken to be Ti,cs 兾Te,cs 苷 8so that the ions carry most of the cross-field current. A
stationary cold background plasma is distributed outside
the current sheet; nbk 共 y兲 苷 Nbk tanh2共 y兾D兲, Nbk 兾Ncs 苷0.1, and Te,bk 苷 Ti,bk 苷 Te,cs . (Different D and Lx are

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0031-9007兾01兾 87(9)兾095001(4)$15.00

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TABLE I. Simulation parameters and LHD amplitudes observed after the KHI growth.

1234

113兾41兾2095001-2

Lx 兾li 3 Ły 兾li

cjdElhd j兾B0heff8 3 162 3 166 3 164 3 16

⬃0.1···
⬃0.15
⬃0.2

21
1.4 3 1023 vpe···
211.7 3 1023 vpe
23 21
2.2 3 10 vpe

gradient at the current sheet edges and is propagating
along the density equicontour. To estimate the phasevelocity of the wave, a stack plot of Ex at y 苷 3li fort 苷 共62 67兲Vi21 is presented in Fig. 3. The phase velocity of the wave, ⬃0.3Ui , is about a half of the local ion
drift velocity, ⬃0.6Ui . The electron Larmor radius in thewave region is rL,e ⬃ 共0.2 0.3兲li , which is comparable
to the wavelength. Since all these features of the wave
meet the characteristics of the LHDI [11], it is reasonable
to conclude that the nonlinear evolution of the KHI excites
the LHDI at the edges of the undulated current sheet.
The temperature gradient in the region of wave activity isalmost parallel to the density gradient (antiparallel to themagnetic-field gradient) so that the growing condition ofthe LHDI is satisfied. The excitation of these LHD waves
continues until the KHI is saturated.
We expect that the LHDI newly activated by the nonlinear evolution of the KHI provides substantial magnetic-fielddissipation since the excited LHD wave is of considerable(a) Ω i t =5.1+4Niy/λ i

Ex0-4(b) Ω i t =35.2+4y/λ iadopted for the other three runs. See Table I.) If thecold background is uniformly introduced, the ion-ion kinkinstability (IIKI) [14] can grow even when the high mass
ratio is taken. To avoid the growth of the IIKI, we remove
the background component from around the neutral sheet.
Although there appears a weak pressure imbalance withthis background profile, it is quickly justified without any
significant modification of the current sheet structure.
In our simulation three instabilities are excited, and eachhas different scale lengths; (i) the LHDI (electron scale),(ii) the current sheet kink instability (CSKI) [15] [hybridscale, ⬃lh ⬅ 共li le 兲1兾2 ], and (iii) the KHI (MHD scale).
Figure 1 displays typical snapshots of each instability.
First, the LHDI grows at the boundary region of the current sheet. As shown in Fig. 1a, the excited waves makeelectrons E 3 B drift and generate electron vortices.
Since these electron vortices move in the same direction
as the ion diamagnetic drift, the cross-field current in theboundary region is strongly reduced and the magnetic
field is dissipated. The saturation amplitude is consistent
with a previous simulation study of the LHDI [16], andthe saturation is due to the electron dissipation process(details are shown later). After the saturation of the LHDI,
which takes place before the appearance of the CSKI, theLHD waves are damped and the magnetic-field dissipation
is much smaller than that before saturation since thecross-field current is reduced. At t ⬃ 30Vi21 , the neutral
sheet region starts to be kinked by the CSKI (Fig. 1b).
The CSKI is a drift mode driven by the cross-field current,i.e., the phase velocity is nearly equal to the ion drift velocity, and its wavelength is klh ⬃ 1 [15]. The kinkstructure is generated by the perturbed electron current
within the current sheet. This stage lasts until the KHI
starts to evolve [t 苷 共30 50兲Vi21 ] . The CSKI is presumably a short-wavelength extension of the DKI. However,
the kink structure of the CSKI does not enhance the dissipation process. Finally, at t ⬃ 50Vi21 , the wholecurrent sheet starts to warp with much longer wavelengthl 苷 8li 苷 Lx (Fig. 1c). At this time, the ion flow patternin the current sheet is globally modulated by the KHI,
while it is hardly modified by the earlier two instabilities.
In the KHI stage, a strong quasimonochromatic wave activity of the short wavelength appears at the edges of the undulated current sheet producing electron vortices similar
to those of the LHDI at earlier time. A region of the wave
activity in Fig. 1c is magnified in Fig. 2. Figure 2 clearlyshows that the wave is located around the largest density

Run D兾li

27 AUGUST 2001

PHYSICAL REVIEW LETTERS

0-4+4y/λ i

VOLUME 87, NUMBER 9(c) Ω i t =65.2

0-40x/λ i

8 0x/λ i

8FIG. 1 (color). Left three panels present color contours of Ex ,which is the dominant component of the electric field, at t 苷
5.1Vi21 , 35.2Vi21 , and 65.2Vi21 . In each contour the electron
flow vectors are also plotted. Right three panels show color
contours of the ion density and the ion flow vectors at the sametime. The white lines represent the magnetic neutral lines.

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VOLUME 87, NUMBER 9(c) Ω i t =65.2y/λ i62

24ω/k=

60.

Ui
ω/k=UiE x ( y =3 λ i )0x/λ i4FIG. 3 (color). A stack plot of Ex at y 苷 3li for t 苷 共6267兲Vi21 . The lines indicate the measure of the phase velocities.

Niy/λ i2k=

Ωi tEx0.3

Ui67
ω/427 AUGUST 2001

PHYSICAL REVIEW LETTERS

0FIG. 2 (color).

x/λ i3Magnification of Figure 1c.

amplitude and the wave activity is widely distributed. Indeed, the amplitude of the LHD wave observed in the KHI
stage is almost equal to that observed in the first LHDI
stage (cjdElhd j兾B0 ⬃ 0.1). To examine how much of themagnetic-field energy (⬅ EB ) is dissipated in the simulation run, the time evolution of EB is shown in Fig. 4a.
Three stages are recognized in the solid curve of Fig. 4a,and each stage corresponds to the three instability stages
noted above. In the first stage, about 1% of the initial
magnetic-field energy (⬅ EB0 ) is lost within 10Vi21 by
the LHDI. After the saturation of the LHDI, the dissipation rate (the gradient of the EB curve) is considerably reduced for t 苷 共10 50兲Vi21 . After the KHI starts to grow,the dissipation rate increases and a rapid magnetic-fielddissipation is observed. Since the LHDI is continuously
driven by the KHI, the high dissipation rate is kept during
the KHI stage. As a result, about 10% of EB0 is dissipated within about 50V 21i . To confirm the effect of thecoupling process to the dissipation, we carry out anothersimulation run in which the KHI is forbidden by setting
Lx 苷 2li (Run 2 in Table I). Consistent with a linear
analysis, we do not observe growth of the KHI in thisrun. The dashed curve in Fig. 4a presents the EB hysteresis
obtained from this case without the KHI. Rapid drop ofcannot be found without the KHI.
EB after t 苷 50V 21iFigure 4a well supports our suggestion that the nonlinear
coupling between the KHI and the LHDI indeed provides
the substantial magnetic-field dissipation. We should notethat this rapid magnetic field dissipation process may occur
before magnetic reconnection starts. Indeed, the tearing
instability is observed at a much later time in a twodimensional full kinetic simulation with the same parameter if no initial perturbation (such as the driven flow fromthe boundaries, etc.) is added to the Harris equilibrium.
However, the observed rapid dissipation process should
be a transient process since the drive for the KHI comesentirely from the overlap of the cold stationary ions andthe current-carrying ions. Thus, the dissipation process

FIG. 4. (a) Solid and dashed curves represent evolution curves of EB 兾EB0 for Run 1 (with the KHI) and 2 (without the KHI).
(b) Evolution curves of 共EB 2 EB0 兲兾E0 for Run 1 (solid), 3 (dashed), and 4 (dotted).

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VOLUME 87, NUMBER 9PHYSICAL REVIEW LETTERS

reported here is unlikely to be observed in the steady-state
reconnection.
We have also examined the effects of the current sheet
thickness D (Runs 1, 3, 4 in Table I). Figure 4b presents
the evolution curves of the diminution of the magnetic fieldenergy for each run. [The magnetic-field energy is normalized by E0 ⬅ Lx li 共B02 兾8p兲.] As predicted by linear
theory of the KHI [15], the start time of the KHI growththat marks the onset of rapid large-scale dissipation scales
as ⬃D 2 (t 苷 15, 34, and 60 for D 苷 1兾2, 3兾4, and 1).
In the case of D兾li 苷 1兾2, the KHI starts to grow soquickly that the CSKI stage is deleted. The LHD amplitudes in the KHI stage are shown in Table I. Similar tothe result of Run 1, the typical wave amplitudes are almost equal to those observed in the first LHDI stage andthe amplitude shows a dependence of ⬃D 21 . This dependence is consistent with a previous study [16]. We alsofind that the duration of the rapid large-scale dissipation
scales as ⬃D. Since the dissipation rates observed in thefirst LHDI stage and the KHI stage are almost equal, thefact that the KHI continuously drives the LHDI is an important issue for the rapid large-scale magnetic dissipation. For the reader’s convenience, the effective resistivity
for the rapid dissipation process is also shown in Table I.
The effective resistivity, heff , is estimated from Ohm’s law,
具E 1 Ve 3 B兾c典 苷 heff 具J典. E 1 Ve 3 B兾c and J are
averaged over the entire current sheet. The estimated values are the same order with, but smaller than, that of theDKI estimated by Horiuchi et al. [17].
Finally, we should comment on the electric field at themagnetic neutral line. An electric field along the current
at the neutral line is necessary for magnetic reconnection.
As shown in Fig. 1c, however, no electric field right at theneutral can be found. What is taking place inside the current sheet is the emergence of an intense current layer thatcompensates the reduced current at the edges. This compensation is required from the conservation of the total current integrated over the current sheet. (The magnetic fieldat the y boundary is invariant as long as the perturbationfrom the current sheet does not reach the boundary.) The
compensative current is embedded in a relatively narrow
region around the neutral sheet and is carried by accelerated meandering electrons. The electron acceleration takes
place at the edges of the embedded current sheet (but notat the neutral sheet) and is the very origin of the dissipation
process [Shinohara et al. (to be published)]. While recent
results of three-dimensional full kinetic simulations witha low mass ratio (mi 兾me , 200) [17,18] show that reconnection can be triggered by the DKI, there is a crucial difference that the DKI can directly produce the reconnection
electric field at the neutral sheet but the KHI does not. The
relation between the KHI and the magnetic reconnection
still needs investigation and we plan a three-dimensional
simulation for this end. Since the dissipation process re-095001-4

27 AUGUST 2001

ported here drastically modifies the current sheet structure,even if reconnection is not driven directly by the present
process, it is quite unlikely that its effects on a subsequent
process leading to reconnection may be ignored. For example, the newly accelerated meandering electrons can affect the growth of the electron tearing mode. Recently,
Rogers et al. [19] reported that the LHD fluctuations causea substantial relaxation of the density gradient and reduce
the reconnection rate. While the reconnection electric field
is supported by the electron shear flow mode associatedwith the sharp gradient (turbulence in the electron inertiascale) in their simulation, such a sharp gradient does notappear in our model due to the LHD mode. Therefore, themechanism that is responsible for the reconnection electricfield might be different even if our model can successfully
trigger magnetic reconnection.
In summary, we find a new route for magnetic-field dissipation in an ion-scale current sheet. Within several tensof ion gyroperiod, considerable magnetic-field energy is
dissipated through the nonlinear coupling between the KHI
and the LHDI. This dissipation time scale is so short thatit may be applicable to the magnetospheric substorm onset. We believe that the process reported here is a gooddemonstration for the general significance of nonlinear
cross-scale coupling in collisionless plasmas.
I. S. thanks Dr. M. Scholer and Dr. R. Treumann for
their fruitful discussions. This work has been supported bySTE Laboratory of Nagoya University via a collaborativeresearch program. Numerical simulations are performed
on Fujitsu VPP at ISAS and Nagoya University.

*Permanent address: Institute of Space and Astronautical
Science, Kanagawa 229-8510, Japan.
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