J. Plasma Fusion Res. SERIES, Vol.2 (1999) 138-140

Generation of Magnetic Fields near Kelvin-Helmholtz
Instability Region in Pair Plasmas
NAKAYAMA Takuya, SAKAI Jun-ichi, NEUBERT Torstenr and NAMBU Mitsuhiro2
Laboratory for Plasma Astrophysics, Faculty of Engineering,Toyama University, jl90 Gofuku, Toyama 930-8555 Japant

Danish Meteorological Institute, Copenhagen, Denmark

2Tolryo Metropolitan Institute of Technology, Hino, Tolcyo 191-0065 Japan(Received: 8 December 1998 / Accepted: 6 May 1999)

Abstract
We present the simulation results by using a two-dimensional electromagnetic and relativistic
particle-in-cell (PIC) code that quasi-static magnetic field can be generated in the Kelvin-Helmholtz
instability region in unmagnetized pair plasmas.

Keywords:
Pair plasma, Kelvin-Helmholtz instability, magnetic field generation

1. Introduction

2. Simulation Model

The Kelvin-Helmholtz instability is a macroscopic
instability that grows in a velocity shear layer, causing

To set up the situation where the Kelvin-Helmholtz
instability occurs in the simulation domain, we impose
counter-streaming regions; vo = -0."1c in two regions of
0 < r < 50 and 150 < x < 200, and v6 = 0.7c in theregion of 50 <.x < 150.
We compare two cases: one is sharp change of theflow veocity on.x = 50 and.r = 150, and the other issmooth change of the flow velocity as follow,

momentum exchange through vortex motions between

two velocity layers.

It was reported recently [] that the high-frequency
electromagnetic waves can be excited from the KelvinHelmholtz instability region in pair plasmas. And it wasreported [2] that the whistler waves and the highfrequency electromagnetic waves can be excited fromthe Kelvin-Helmholtz instability and the whistler waves
are excited stronger than the electromagnetic waves inx- 100
x-300
LL )-tanh( . )-l}vo=0.3c{tanh( .

where Z = 5 is about the electron skin depth. In thesecond case, we use the same parameters as the firstcase, except for the velocity profile.

magnetized electron-ion plasmas.

Using a two-dimensional electromagetic andrelativistic PIC code, we show that quasi-static magnetic
fields are generated from a region where the KelvinHelmholtz instability develops in pair plasmas from an
unmagnetized state (Bs = 0). Then we investigate the3. Simulation Results
In the simulation, the background magnetic field Bs
is zero. But there is a fluctuating magnetic field (seedfield) initially owing to the electric cunent arising fromthe unbalanced motions ofoositrons and electrons. Thenenergy conversion rate that the initial flow energy isconverted to the quasi-static magnetic field energy, bychanging the flow velocity.

@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research138

Nakayama T. et al., Generation of Magnetic Fields near Kelvin-Helmholtz Instability Region in Pair Plasmas(a)the magnetic field is amplified from the fluctuating level(B? = 2.5 x 10-2) to form a quasi-static magnetic field.

1 150

Figures 1 and 2 show the simulation results for the$rroo{Isharp velocity boundary. Fig. I shows the spacial
distribution of the magnetic field B. from ttto.t = 14.9 to@o"t = 74.5. In the initial stage, the quasi-periodic
magnetic field structure with a size of the electron skindepth can be generated in the Kelvin-Helmholtz

1050

E roooMY

950

80

unstable regions, and it grows to a large structure. Aftero^-tar.t = 59.6, four pairs with positive and negative
magnetic polarity are generated and the structrureremains to be unchanged.
Figures 2(a) and 2(b) show the time history of theflow kinetic energy in the x and y-direction, respectively. Figure 2(c) shows the time history of themagnetic field energy n! in tbe whole system. In the
final stage,,B.2 is almost a constant as shown in Fig. 2(c).
0n801200)t0.3 0.4 0.5 0.6 0.7 0.8
flow velocity / cFig. 2 The time history of the kinetic energy in (a) x-direction, (b) y-direction, and the magnetic field
150

energy (c) 8,2 and (d) the energy conversion rate.

100150 200 0100150

100150As the result, about 2.5Va of the initial flow energy isconverted to the magnetic field energy B.2. But in thefinal stage, all flow energy are shifted the kinetic energy
in the y-direction.
We calculate the energy conversion rate that flowenergy is converted to the quasi-static magnetic fieldenergy, by changing the flow velocity as shown in Fig.
2(d). As seen this figure, the conversion rate varies
rapidly from flow velocity = 0.5c.
Figures 3 and 4 show the simulation results for thesmooth velocity boundary. Figure 3 shows the spacial
distribution of the magnetic field B. from a*t = 14.9 to{.oo"t = 74.5. Compared with Fig. I, we find that twopairs with positive and negative magnetic polarity are

x200
150

10050150 200 0xY

50200200

generated after oo"t = 59.6.

Figs. 4(a) and 4(b) show the time history of theflow kinetic energy in the x and y-direction, respectively. The kinetic energy in the x-direction is still
increasing and the kinetic energy in the y-direction isstill decreasing beyond the final stage. Fig. 4(c) shows
the time history of the magnetic field energy 8,2 in the150-0.075 -0.025 0.025 0.07s

0whole system. An increase of the magnetic field energy
8.2 is noise level until @ret = 25. And the KelvinHelmholtz instability grows from about op"t = 25. About
4.lVo of the initial flow energy is converted to the50 100 150 200 x

Fig. 1 The time development of the magentic field Q:(a) a4"t= 14.9, (b) 29.8. (c) M.7, (dl 59.6, (e) 74.5.

r39

Nakayama T. et al., Generation of Magnetic Fields near Kelvin-Helmholtz Instability Region in Pair Plasmas
(b)
2200

150.0

X 600

g0o

100.0o4oo
3ou 2o0og2t00
2000
1900
1800

100200300 400 0x0n"t100I;do60100 200 300
op"tt00
Yt00

300 400 0X

Fig. 4 The time history of the kinetic energy in (a) x-direction, (b) y-direction, and the magnetic field?00

energy (c) 8.'z.
150
10050

4. Conclusion
Using a two-dimensional electromagnetic and.0.10 ,0.05 0.00 0.05 0 10

relativistic PIC code, we have shown that the quasistatic magnetic fields in the z-direction can be strongly
generated from a region where the Kelvin-Helmholtz
instbility develops. The conversion rate from the initial
flow energy to the magnetic field energy is about a few
percent. The process of quasi-static magnetic field

0 l(n ?00 300 4ffi x
Fig. 3 The time development of the magentic field Q:(a) a4"r = 14.9, (b) 29.8, (c) M.7, (dl 59.6. (e) 74.5.

generation near the region with the velocity shear would

be important for understanding magnetic field generation in astrophysical plasmas.

magnetic field energy. The conversion rate of the secondcase is about 1.6 times effective than the first case.

We compare Fig. 2(c) with Fig. 4(c). In the sharp

velocity boundary case (Fig. 2(c)), the KelvinHelmholtz instability grows rapidly. That is, theAcknowlegment
We thank the Cosel compony for the support.

instability is excited strong in the non-linear stage. Thenit results in the rebound of the magnetic field energy B!
and then the magnetic field energy decreases a little.
The time history of the electric field energy E2 shows

References[1] J.I. Sakai, M. Eda, H. Miyazaki and T. Neubert: J.
Phys. Soc. Ipn.67 No.7, 2170 (1998).

t2l J.l. Sakai, T. Nakayama, M. Nambu and T.

the same development with the magnetic field energy.
While in the smooth velocity boundary case (Fig. 4(c)),
the Kelvin-Helmholtz instability grows slowly. Then it
doesn't cause the rebound. Therefore the energy
conversion rate of the smooth velocity boudary case isstronger than the sharp velocity boundary case.

Neubert: submitted to Phvs. Lett.

140