S8-6
Control of the Weibel instability and structure formation
by a strong magnetic field
強磁場によるワイベル不安定性の制御と構造形成	 Toshihiro Taguchi, Thomas M. Antonsen, Jr. 1) and Kunioki Mima 2)
田口俊弘, T. M. Antonsen, Jr. 1)，三間圀興2)
Setsunan University, University of Maryland 1), The Graduate School for the Creation of New Photonics Industries 2)
摂南大学理工学部, メリーランド大学 1), 光創成大学院大学 2)
The magnetized beam instability has been investigated in linear and nonlinear regimes. According to a linear analysis ofthe Vlasov equation in two dimensional wave number space, it is found that the transversely unstable mode (the Weibelinstability) is suppressed under a strong magnetic field. The nonlinear analysis is performed using a hybrid simulation
code. When a sufficiently strong magnetic field is applied along the beam direction, the instabilities are well suppressed
and the electron stream becomes laminar. When the magnetic field strength is not large enough, however, electron flow
stagnates and a shock-like structure is formed.

1. Introduction
When a ultra-intense laser is irradiated on asolid target, a large amount of relativistic
electrons are generated at the critical surface andpropagate into the overdense region. When sucha large electron flow penetrates the plasma, areturn electric current is induced in thebackground plasma in order to maintain current
neutrality. Such a two-stream state is unstable totwo kinds of perturbation. One is longitudinal,
in which the wave number is parallel to the beam
propagation direction. For this mode, anelectrostatic field grows as well as a density
modulation. The other mode is transverse, inwhich the wave number is perpendicular to thebeam propagation direction. For this mode,known as the Weibel instability, current
separation takes place and a quasi-static magnetic
field grows. The magnetic field deflects theelectrons broadening their spread in angles.
In order to suppress the broadening of theangular spread, a group at the Institute of Laser
Engineering [1] proposed the application of astrong magnetic field. In their experiments, alarge magnetic field, which is on order of 1 kT,
has been successfully generated. When such alarge magnetic field is applied along the beam
propagation direction, the angular spread can besuppressed in two ways. One is by guiding theelectrons along the magnetic field. The strong
magnetic field restricts transverse motion, and thebeam propagates along the magnetic field lines.
The other mechanism is suppression of theWeibel instability. The dispersion relation of a
two-stream plasma is modified by a strong
magnetic field, and unstable modes may bestabilized.
2. Linear dispersion relation of two stream
instability in a strong magnetic field
In order to investigate the stability of thetwo-stream state in a strong magnetic field, wecalculate the linear dispersion relation using theVlasov theory. In the weak instability regime,
kinetic effects are an important factor of theanalysis [2].
Figures 1 show the growth rates in
two-dimensional wave vector space including
both longitudinal and transverse components. In
these results, beam electron temperature (10 keV),background electron temperature (5 keV) and thedensity ratio between beam and backgroundelectron (1:9) are fixed. The drift velocity of thebeam is set to be 0.9c, so the initial beam energy
is 0.66MeV. Figures 1 (a), (b) and (c) are fordifferent external magnetic field strengths, which
are denoted by the normalized value ωc/ωpe(cyclotron frequency/plasma frequency). In thesefigures, kx is the longitudinal wave number
component and ky is the transverse wave number
component.
As shown in the figures, the growth rate of thetransverse modes depends on the strength of theexternal magnetic field, while the longitudinalmodes are not affected as much by the field.
When the external field is weak, the transverse
mode grows as well as the longitudinal modes. As
the strength of the field increases, the transverse
modes are stabilized but the longitudinal modes
are still unstable for the parameters considered.

γ/ω peγ/ω peγ/ω pe

0.20
0.20
0.20
0.15
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
0.00
0.00
3.0
0.0
3.0 0.0
3.0 0.0
0.5
2.5
0.5
2.5
0.5
2.5
1.0
2.0
1.0
2.0
1.0
2.0
1.5
1.5
1.5
1.5
1.5
1.5
cky 2.0
cky 2.0
cky 2.0
1.0 ckx
1.0 ckx
1.0 ckx
ω pe
ω pe
ω pe
2.5
0.5
2.5
0.5
2.5
0.5
ω pe
ω pe
ω pe3.0 0.0
3.0 0.0
3.0 0.0

ω(a) ω c = 0.01
pe

Figure 1.

ω(b) ω c = 0.5peω(c) ω c = 1
pe

Growth rate of magnetized two-stream instability

filament carries net electric current as the
background electrons evacuate the filaments tomaintain charge neutrality.
The current
filaments then attract each other due to themagnetic force and they tend to collide with eachother. As a result, the current filaments change
direction randomly, and this behavior results inthe large angular spread of the electrons.
On the other hand, when the magnetic field isstrong enough (Fig.2(c)), the beam flows
quasi-laminarly, although fine filament structures
still remain. One reason for the quasi-laminar
0.6
flow is the guiding effect on
60.5
the filament of the strong
0.4
40.3
magnetic field. As a result, a0.2
2
sufficiently strong magnetic
0.1
0field has the capability to0.0
0102030(a) ω c ω pe = 0
suppress the increase of the0.6
6angular spread.
0.5
0.4
However, electron flow
40.3
stagnates
for a medium0.2
20.1
strengthmagnetic
field. As
00.00102030
shown in Fig.2(b) (ωc/ωpe(b) ω c ω pe = 0.5
0.6=0.5), the electrons accumulate
60.5
near x=10µm and form a0.4
y 4
0.3
shock-like structure.
Our
0.2
2
analysisshows
that
this0.1
00.0
structure is sustained by an
0102030(c) ω c ω pe = 1
electromagnetic field. Morexdetails will be shown in theFigure 2. Beam electron density profiletalk.
The beam velocity is 0.99c and the initial beam
Referencestemperature is 150keV.
The left and right[1] S. Fujioka et al., Sci. Rep., 3, 01170 (2013).
boundaries are open, while the top and the bottom[2] B. Hao et al., Phys. Rev. E,80, 006402 (2009).
boundaries are periodic. The beam electrons arecontinuously injected in the vicinity of the left
boundary as their density is set to be 1/10 of the
background plasma density.
The length isscaled in µm and the background plasma density
is taken to be 1022 1/cm3.
When there is no magnetic field (Fig.2(a)), thegrowth rate is large and the beam breaks up into
filaments due to the Weibel instability. Each
2. Nonlinear hybrid simulation
In order to analyze the nonlinear evolution of
the beam instability, we performed several runsof a hybrid simulation, which treats thebackground plasma as a fluid and the beamelectrons as particles as in a PIC simulation.
Typical results are shown in Figs. 2, which are
two dimensional false color images displaying thebeam electron density. Figures 2 (a), (b) and (c)correspond to the cases of the normalized
magnetic field ωc/ωpe=0, 0.5 and 1, respectively.