J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Eﬀects of the ion mass ratio on the relativistic Weibel
instability
Chang-Mo Ryu and Helen H. Kaang
Department of Physics, POSTECH, Pohang 790-784, Korea(Received: 28 October 2009 / Accepted: 12 February 2010)

The Weibel instability has received renewed attention for its applicability in various astrophysicaland laboratory plasmas. The Weibel instability is a plausible mechanism for the magnetic ﬁeld
generation in the gamma ray burst. It is also considered as the source of the cosmological magnetic
ﬁeld in the early universe. This instability also seems to play an important role in a high-intensity
laser produced plasma. In this paper, we investigate the ion mass eﬀect on the relativistic Weibel
instability by using a particle-in-cell simulation technique, considering two diﬀerent ion-to-electron mass
ratios. The study extends to the nonlinear physics regime. It is found that when the ion-to-electronmass is smaller, the magnetic ﬁeld energy grows stronger and is more quickly damped, manifesting
clearer inverse cascade process following the quasilinear saturation stage. We ascribe this mass eﬀect tothe increased electrostatic ﬂuctuation associated with the ion acoustic wave by the enhanced ion mobility.
Keywords: Weibel instability, dynamo mechanism, magnetic ﬁeld ampliﬁcation, Particle-in-cell simulation, ion acoustic wave
requires a relativistic treatment [20, 21, 22, 23, 24].
The Weibel instability is excited by the localized
charged particle motions. The classical study of theWeibel instability has focused mainly on the electrons
with an assumption that the ions are immobile [25].
Of many studies into the Weibel instability, only a few
on the electron-positron plasmas in gamma ray bursts
have dealt with the ion eﬀects, and the ion eﬀects on
the Weibel instability have been neglected in general.
Even in the case that the ion eﬀects are included, they
are merely limited to satisfying the charge neutrality
condition. However, with better understanding, realization of the importance of the ion eﬀects starts toemerge. For instance, Medvedev and Loeb have shownthat the magnetic ﬁeld strengths shown in previous
simulations are too small compared to actual observations, and thus that the ion eﬀects may have to beconsidered [16]. In the nonlinear regime where strong
coupling to various modes take place, ion eﬀects might
be important.
In the present paper, we investigate the ion-toelectron mass ratio eﬀects on the Weibel instability which is driven by temperature anisotropy in a
relativistic regime, by employing a one-dimensional
PIC(Particle-in-cell) simulation method. We consider
two diﬀerent cases of ion-to-electron mass ratios.
The organization of the paper is as follows: In
Sec. 2 the simulation conditions are described. Sec. 3,
and Sec. 4 presents the simulation results, and Sec. 5
presents conclusions.

1. Introduction
Mechanisms of magnetic ﬁeld generation in plasmas have been of great interest since it has been realized that the magnetic ﬁeld plays a crucial role indetermining the physical properties of various plasmas. Ampliﬁcation of the magnetic ﬁeld in a highly
conducting medium has been observed not only inastrophysical bodies such as the Earth, the Sun,galaxies, interstellar media, and even in black holes[1, 2, 3, 4, 5, 6, 7], but also in many laboratory plasmas
such as the laser evaporated plasma [8, 9], z-pinch [10],reversed ﬁeld pinch [11, 12], and ﬁeld reversed conﬁguration [13, 14, 15]. A variety of interesting physicalphenomena are found to occur via the interplay between the magnetic ﬁeld and these plasmas. However,the processes leading to the large scale magnetic ﬁelds
are largely unknown.
The Weibel instability has recently drawn attention for its possibility to occur in various astrophysical situations. Medvedev and Loeb suggest that theWeibel instability can be a candidate mechanism forgenerating magnetic ﬁelds in the gamma ray burst[16, 17]. Schlickeiser proposes that the Weibel instability might be responsible for the generation of cosmological magnetic ﬁelds in the early universe [18].
The role of the Weibel instability in generating large
scale cosmic magnetic ﬁelds might also be relevant indark energy [19]. Plasmas in these astrophysical andcosmic bodies, as well as laser plasmas where generation of strong magnetic ﬁelds are often observed arehighly energetic. As a consequence, the discussion of
the Weibel instability in the energetic plasma systems
author’s e-mail: ryu201@postech.ac.kr

491©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

C.-M. Ryu and H.H. Kaang, Effects of the Ion Mass Ratio on the Relativistic Weibel Instability

Fig. 1 Schematic of the Weibel instability.
Fig. 2 Evolution of the energy. Magnetic ﬁeld energy(solidline) and kinetic energy(dashed line) of the electrons are plotted together with the kinetic energy(dotted line) of the ions for (a) M/m = 1833 and(b) M/m =10, respectively.

2. Simulation Conditions
The initial electron distribution is taken as a fully
relativistic bi-Maxwellian distribution, which is given
in Ref. [21],fAα⊥ e−α⊥ (γ−γ� )−α� γ�
,4π(mc)3 K2 (α� ) A��α�K1 (α� )
,
= 1+
−1
α⊥
α� K2 (α� )=perpendicular thermal anisotropic parameters, andamong these, we have chosen α⊥ = 5 and α� = 30
as the representative parameters.

(1)

3. Growth of the Magnetic Field Energy

where γ = (1 + p2 /m2 c2 + p2� /m2 c2 )1/2 is the relativistic Lorentz factor with m and c being the electron
rest mass and the speed of light in vacuo, respectively.
The direction of the wave propagation is taken as thex axis, and the parallel and perpendicular directions
are deﬁned along and perpendicular to it, respectively.
Then, we deﬁne γ� = (1 + p2� /m2 c2 )1/2 . Kn (x) is
the modiﬁed Bessel function of the second kind of order n. Dimensionless parameters α� = mc2 /T� andα⊥ = mc2 /T⊥ characterize the plasma temperaturedeﬁned along the parallel and perpendicular directions
to the wave propagation direction, respectively.
The computer code used in this study is arelativistic electromagnetic particle-in-cell(PIC) codewith one spatial and three velocity dimensions(1D3V). Particles are initially loaded according to thebi-Maxwellian distribution function described above.
The number of grids Nx is 4096, and for each cell, 4000
particles are allocated. The plasma density is initially
uniform. A periodic boundary condition is used atboth ends. The grid size is given by Δx = 0.1c/ωpe
and the time step by Δt = 0.001/ωpe , where ωpe =(ne e2 /�0 m)1/2 is the plasma frequency. Here, �0 is the−1
vacuum permittivity. We take ωpeas the unit time
and the electron skin depth c/ωpe as the unit length.
We considered two cases of the ion-electron plasma
with mass ratios of M/m = 1833 and M/m = 10.
When the mass ratio is small, the ion mobility increases, and thus we expect that the ion eﬀects would
be pronounced. The ions are taken to have the same
temperature as the electrons.
The simulation was performed until 400ωpe t. The
Weibel instability occurs when the perpendicular temperature is higher than the parallel temperature. The
schematic of its operating mechanism is illustrated
in Fig.1. The simulations are performed for various

In Figure 2, time evolution of the magnetic ﬁeld
energy and the kinetic energy of the electrons areshown together with the kinetic energy of the ions.
The ﬁrst two energies are normalized by the initial
electron thermal energy and the ion kinetic energy isnormalized by the initial ion energy. The increaseof the magnetic ﬁeld energy is about the same asthat of the decrease of electron energy. Although notshown, the electrostatic ﬁeld energy does not show appreciable change, indicating that the unstable modeis the electromagnetic one. However, in the case of
M/m = 10, the ion kinetic energy is shown to continuously increase in time.
In Figure 2, it is shown that the increase of themagnetic ﬁeld energy is accompanied by a decreasein the electron thermal energy of equal magnitude forthe time up to ωpe t ∼ 40. Since the ions are muchheavier than electrons, the ion eﬀects can appear only
in a later phase, and, as a result, the simulations forthe two diﬀerent ion masses show almost the same behavior in the initial phase when the Weibel instability
grows linearly. This observation is in agreement withthe theoretical expectation.
After passing the linear stage, the plasmas enter a quasilinear and then a nonlinear state. For
M/m = 1833, the B ﬁeld energy ﬁrst undergoes asmall overshoot before settling down to a quasi-steadyasymptotic state, until the end of the simulation. The
saturated B ﬁeld energy is at about 3% of the initial
kinetic energy of the electrons. The B ﬁeld energy fora smaller ion mass increases up to about 5% of the total initial kinetic energy of the electrons taking longer
time ωpe t ∼ 80. As the nonlinear eﬀect involving theions sets in, the electron distribution also shows satu-

492

C.-M. Ryu and H.H. Kaang, Effects of the Ion Mass Ratio on the Relativistic Weibel Instability

ration.
With the assumption that the ions are immobile,the solution of the Vlasov equation for the electrons
alone can give rise to the dispersion relation for the
Weibel instability as [24],c2 k 2
ωp2

=

2
α⊥
α�
α� K2 (α� ) + ΔK1 (α� )���Δ K0 (α� )×+ K1 (α� )
2
α⊥
α⊥
�
�
K0 (α� )Γ ∞ dτ
−3Δ2
2
ck 0 ξ⊥
ξ⊥
�
�
ξ�
ξ�
α� K1 (ζ)
+ 3Δ
+
−
ξ⊥
ξ⊥
α⊥
ζ
��
α
K
(ζ)
�
2
,
(2)
+ ξ�2
α⊥ ζ 2

Fig. 3 Linear growth rate obtained from Eq. (2) vs. wave
number for α⊥ = 5 and α� = 30.

where τ is the normalized time, and
α�
Δ =
− 1,
α⊥
Γ
τ,
ξ⊥ = α ⊥ +ckΓτ,ξ� = α� +
ck
�
ζ

=

ξ�2 + τ 2 .

Figure 3 shows the growth rate Γ/ωp e obtained
from the solution for Eq. (2) with α⊥ = 5, α� = 30. In
this graph, Γ/ωp e has a maximum at kc/ωpe ≈ 0.916
with the value 0.151ωpe , and thus, the most unstablemode appears at kc/ωpe = 0.916.
Simulation results are shown in Fig. 4 for thetwo diﬀerent mass ratios. A few modes growing most
rapidly are plotted. The vertical scale is on a logarithmic amplitude scale. It can be seen that theslopes of the mode amplitudes are about the same,
indicating that the linear growth rates are almost
the same, but the mode with kc/ωpe = 0.92 has thehighest growth rate in agreement with the theoretical prediction. Therefore, we select the mode withkc/ωpe = 0.92 in the PIC simulation as the representative case of the study. By estimating the growthrate from the linear ﬁtting, we get Γ = 0.144ωpe for
M/m = 1833, 0.137ωpe for M/m = 10 [Fig. 5]. The
growth rate estimated from the simulation shows an
excellent agreement with the one obtained from Eq.
(2), which considered the electrons alone. The resultthat almost the same growth rates are obtained fortwo diﬀerent ion-to-electron mass ratios, in agreementwith the theoretical prediction conﬁrms that there is
no eﬀect of ions on the Weibel instability in the early,
linearly growing phase.
It is shown in case of the smaller ion-to-electronmass ratio that the increase of the magnetic ﬁeld occurs simultaneously with the reduction of the ion tem-

Fig. 4 (Color online) Time evolution of several diﬀerent
modes in the linear regime, for (a) M/m = 1833and (b) M/m = 10.

perature anisotropy. Time evolutions of the electron
and ion momentum distribution are plotted in Figs. 6
and 7. The initial distribution (shown in Figs.6 and
7 (a) and (c)) has an elliptical shape in the px − pymomentum space. Figures 6 and 7 (b) show that the
distribution broadens in time in both the x and y directions. There is no diﬀerence in the electron distributions for the two diﬀerent mass ratios. However, thediﬀerence in the ion distribution is noticeable. Unlike
the case of M/m = 1833 where there is no change,in the case of M/m = 10, ion distribution becomes
broadened to a rectangular shape. The ion momentum values do not go beyond the boundary values,
similarly to the case of the electrons. The rectangularboundary remains the same after the particle distributions are settled, and the magnetic ﬁeld energy has
stopped its increase.
On the other hand, as the ion temperature
anisotropy is reduced, the ion thermal energy does notconvert into the magnetic ﬁeld energy, in contrast to
the fact that the electron thermal energy converts to493

C.-M. Ryu and H.H. Kaang, Effects of the Ion Mass Ratio on the Relativistic Weibel Instability(a) ωpet = 0.00(b) ωpet = 81.92

3 electron

electron

105

10

py/mc

2
1
0

104

-1
-2
-3
-3 -2 -1

0

1

px/mc

2

3

-3 -2 -1(c) ωpet = 0.00

py/mc(b) ωpet = 81.92

electron

105

10

2

3

103

ion

0.5

Fig. 5 Time evolution of the mode with kc/ωpe = 0.920
for (a) M/m = 1833 and (b) M/m = 10.

(a) ωpet = 0.00

1(d) ωpet = 81.92

ion

3 electron

0

px/mc

102

0.0
-0.5

py/mc

2
10
0

1

-0.5

0

-0.5

0.0

px/mc

0.5

2

101

10

-2

Fig. 7 (Color online)Electron and ion distributions in the
momentum space at diﬀerent times for M/m =10.

-3
-3 -2 -1

0

1

px/mc

2

3

(c) ωpet = 0.00

ion

-3 -2 -1

0

1

px/mc

2

3

103

(d) ωpet = 81.92

200

ion

0.05

102

0.00

ωpe t

py/mc

px/mc

0.5

4

-1

0.10

0.0

-0.05
-0.10
10
0

-0.10 -0.05 0.00 0.05 0.10

px/mc

-0.10 -0.05 0.00 0.05 0.10

2

101

px/mc

(a) By10-21.200000•10

150

10-3

100

10-4

50

10-5

0

0.00

1.200000•10 0

0.50

1.00

ckx / ωpe

Fig. 6 (Color online)Electron and ion distributions in
the momentum space at diﬀerent times for M/m=1833.

(b) ni

1.50

0.00

0.50

1.00

2

1.50

ckx / ωpe

Fig. 8 (Color online) Power spectra of (a) magnetic ﬁeld
By and (b)ion density ni for M/m=1833.

the magnetic ﬁeld energy. It is shown that as the ion
distribution become isotropic, the thermal energy increases, whereas as the electron distribution becomes
more isotropic, more magnetic ﬁelds are generated[Fig. 2(b)]. It can also be seen that for M/m = 10,
around 50 ∼ 80ωpe the electron thermal energy is reduced and the magnetic ﬁeld energy increases by thesame amount. The increase of the electron isotropy
for a lower ion-to-electron mass ratio plasma has also
been observed in other PIC simulations [26].

early linear growth phase, the strongest wave appears
at kc/ωpe ∼ 0.9 as mentioned before. As the time
evolves, the k numbers of the dominant waves become
smaller. The inverse cascade features of B ﬁeld andion density show signiﬁcant diﬀerence for the two different mass ratios. In case of the magnetic ﬁeld, theinverse cascade is independent of the ion-to-electronmass ratio, but for M/m = 10 merging of modes occurs more actively. The ion density ﬂuctuation showsa manifest diﬀerence for the two diﬀerent mass ratios.
Figure 10 shows the plot of the electrostatic ﬂuctuations of ω − k spectrum. We have drawn this ﬁgure to identify the modes in Fig.9(b)of ni ﬂuctuation.
The red line corresponds to the dispersion curve for
the IA(Ion Acoustic) wave. On this curve, the existence of the IA wave can be identiﬁed by a strong

4. Nonlinear regime
After the linear phase and passing the quasilinear phase, merging of small-scale structures of ﬂuctuations into a larger scale one shows up revealing an
inverse cascade process. Figs. 8 and 9 show the time
evolution of the By and ni spectra. The two spectra are normalized by the electron rest energy. In the494

C.-M. Ryu and H.H. Kaang, Effects of the Ion Mass Ratio on the Relativistic Weibel Instability(a) Byωpe t200(b) ni

150

10-3

100

10-4

50

10-5

0

1.200000•10 0

0.00

0.50

1.00

1.50

0.00

ckx / ωpe

0.50

1.00

The Weibel instability is one of those candidate
mechanisms to explain such a large scale magnetic
ﬁeld in a plasma. The interesting aspect of theWeibel instability is that excess energy stored in theanisotropic distribution of the electrons in a plasma
could be released as magnetic ﬁeld energy by the instability, producing the magnetic ﬁeld in the electrostatic environment of the null magnetic ﬁeld. Thisinstability could provide a seed magnetic ﬁeld such as
the primordial ﬁeld in the early universe.
Classical study of the Weibel instability has
mainly focused on the electron motion with an assumption that ions are immobile. In the nonlinear
regime where strong coupling to various modes could
take place, this assumption may not be valid. We haveinvestigated the ion mobility eﬀect on the Weibel instability by using a 1D PIC simulation. In the early
growth phase, when the ions do not have time to contribute, the magnetic ﬁeld energy increases exponentially, independent of the mass ratio. However, the ion
eﬀect becomes evident in the nonlinear regime. Whenthe mass ratio is small compared to the realistic one,the maximum magnetic ﬁeld energy attained becomesmuch higher. In this case, the ion distribution expands as the electron distribution does. When thisexpansion stops, the magnetic ﬁeld energy does notincrease further. For a small mass ratio case, the ion
thermal energy is shown to increase continuously.
The inverse cascade process is clearly seen for themagnetic ﬁeld energy in both cases. In the inverse cascade process, coupling to the electrostatic ﬂuctuations
like the ion acoustic mode is strongly indicated.
This work was supported by the grants from BK21
and by the Korea Science and Engineering Foundation (KOSEF) grants R11-2008-072-01003-0 and R012003-000-10402-0 from the Korean Ministry of Education, Science and Technology (MEST).

10-2

1.200000•10

2

1.50

ckx / ωpe

Fig. 9 (Color online) Power spectra of (a) magnetic ﬁeld
By and (b)ion density ni for M/m=10.

ω / ωpe t

0.20

8.000000•10

0.15

10-2

0.10
10-3

0.05
0.00
-1.0

8.000000•10 0

-0.5

0.0

0.5

2

10-4

1.0

kx c / ωpeFig. 10(Color online) ω − k spectrum of the ion density ni
for M/m=10 during the time 0.0 ∼ 204.8ωpe t. The
red line is the dispersion curve for the ion-acoustic
wave.

intensity area around kx c/ωpe t ∼ 0.2. Below this IA
mode, many ﬂuctuations are shown to exist near zerofrequency.
The inverse cascade process seems to involve anelectrostatic mode induced by the ion motion via thewave-wave coupling. The result that the electrostatic
ﬂuctuation shows an inverse cascade pattern similar tothat of the magnetic ﬁeld, and that ni ﬂuctuation intensity increases during this process indicate they are
not independent of each other. Electrostatic modes
like the IA wave can participate in the inverse cascade
process of the Weibel instability.

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5. Discussion and conclusions
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495

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