Acta Polytechnica


Acta Polytechnica 53(2):174–178, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

INSTABILITY GROWTH RATE DEPENDENCE ON INPUT
PARAMETERS DURING THE BEAM–TARGET PLASMA

INTERACTION

Miroslav Horký∗

Department of Physics, Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic
∗ corresponding author: horkymi1@fel.cvut.cz

Abstract. The two-stream instability without magnetic field is described by the well-known Buneman
dispersion relation. For more complicated situations we need to use the Generalized Buneman Dispersion
Relation derived by Kulhánek, Břeň, and Bohata in 2011, which is a polynomial equation of 8th order.
The maximal value of the imaginary part of the individual dispersion branches ωn(k) is very interesting
from the physical point of view. It represents the instability growth rate which is responsible for the
turbulence mode onset and subsequent reconnection on the ion radius scale accompanied by strong
plasma thermalization. The paper presented here is focused on the instability growth rate dependence
on various input parameters, such as magnitude of a magnetic field and sound velocity. The results are
presented in well-arranged plots and can be used for a survey of the plasma parameters close to which
the strong energy transfer and thermalization between the beam and the target occurs.

Keywords: Buneman instability, numerical simulations, plasma, dispersion relation.

1. Introduction
Two-stream instabilities are the most common insta-
bilities in plasmas which originate on the microscopic
scale and which can develop to macroscopic phenom-
ena like a thermal radiaton from strong thermalization
or non-thermal radiaton from reconnections. If we con-
sider that both streams have parallel direction of their
velocities, we talk about Buneman instability [1] and
if we consider intersecting directions of velocities and
anisotropy of temperatures, we talk about Weibel in-
stability [8]. The dispersion relation for two-stream
instability without magnetic field in cold plasma is de-
scribed by the relation

2∑
α=1

ωpα

(ω − k · u0α)
2 = 1, (1)

where ω is the wave frequency, ωpα is the plasma fre-
quency of the first and second stream respectively, k is
the wave vector, and u0α is the vector of the velocity
for the first and the second stream respectively.

The most simple situation, in which we can use this
relation, is the interaction of two identical streams
moving in opposite directions. Equation 1 has simple
one dimensional form [4]

ω2p
(ω −ku0)2

+
ω2p

(ω + ku0)2
= 1. (2)

The two-stream instabilities are usually used for
the study of the origin of the observed macroscopic
phenomena (e.g. particle acceleration in relativistic
plasma shocks [6]). This paper is focused on the gen-
eral study of the plasma jet interaction on the mi-
croscopic scale (not only on the study of one partic-

ular phenomenon origin) and for this case the gen-
eral dispersion relation is needed. Two generaliza-
tions of the two-stream instability dispersion relation
were done in last years, first was done by Kulhánek,
Břeň and Bohata [5] in 2011 and second was done
by Pokhotelov and Balikhin [7] in 2012. In this paper
we do all the calculations from [5], because the gener-
alization is more rigorous and precise. The authors
called it Generalized Buneman Dispersion Relation
(GBDR) and it is described by equation

2∏
α=1

{
Ω4α −Ω

2
α

[
i
F(0)α · k
mα

+ c2sαk
2 + ω2pα + ω

2
cα

]

−
Ωαωcα
mα

(
F(0)α × k

)
· eB

+ ω2cα(k · eB)
[

i
F(0)α · eB
mα

+
(
c2sαk

2 + ω2pα
) k · eB

k2

]}

−
2∏

α=1

ω2pα
k2

[
Ω2αk

2 −ω2cα (eB · k)
2
]

= 0,

(3)

where
Ωα = ω − k · u(0)α , (4)

ωcα is the cyclotron frequency, F
(0)
α is the Lorentz

magnetic force, eB is the unit vector in the direc-
tion of magnetic field and csα is the sound velocity.
For B = 0 and cold plasma limit csα = 0, the general-
ized relation becomes the Eq. 1.
For its analysis it is useful to convert this relation

to a non-dimensional form to ensure the scale invari-
ance of the results. The system of coordinates used
in the solution is in Fig. 1.

174

http://ctn.cvut.cz/ap/


vol. 53 no. 2/2013 Instability Growth Rate Dependence

The directions of the vectors uα, B and k are pre-
sented in Fig. 1 in the case of our coordinates system.
The wavevector can have any direction, the magnetic
field is only in the (x–z) plane and the velocity vec-
tors of the streams are only along the x-axis. These
vectors have coordinates

uα = (uα, 0, 0),
B = (B sin θB, 0,B cos θB),
k = (k cos ϕ sin θk,k sin ϕ sin θk,k cos θk).

1.1. The non-dimensional form
Non-dimensional variables are defined by relations [2]

cs1 ≡
cs1
u1
, cs2 ≡

cs2
u1
,

ωc1 ≡
ωc1
ωp2

, ωc2 ≡
ωc2
ωp2

,

ωp1 ≡
ωp1
ωp2

, ωp2 ≡
ωp2
ωp2

= 1,

u2 ≡
u2
u1
, u1 ≡

u1
u1

= 1,

k ≡
ku1
ωp2

, ω ≡
ω

ωp2
,

where index 1 denotes a jet and index 2 denotes a back-
ground. Under these definitions, we can convert Eq. 3
into a non-dimensional form [2] which will be

[
Ω

4
1 + iΩ

2
1ωc1k(G1) −Ω

2
1

(
c2s1k

2
+ ω2p1

)
−Ω

2
1ω

2
c1

− Ω1ω2c1k(G3) +
(
ω2c1k

2
c2s1 + ω

2
c1ω

2
p1

)
(G2)2

]
·
[
Ω

4
2 + iΩ

2
2ωc2ku2(G1) −Ω

2
2

(
c2s2k

2
+ 1
)
−Ω

2
2ω

2
c2

− Ω2ω2c2ku2(G3) +
(
ω2c2k

2
c2s2 + ω

2
c2

)
(G2)2

]
−
[
ω2p1

(
Ω

2
1 −ω

2
c1(G2)

2
)]
·
[
Ω

2
2 −ω

2
c2(G2)

2
]

= 0,
(5)

where we denoted

G1 = (cos θB sin ϕ sin θk),
G2 = (cos ϕ sin θk sin θB + cos θk cos θB),
G3 = (cos2 θB cos ϕ sin θk − cos θB cos θk sin θB),
Ω1 = ω −k cos ϕ sin θk,
Ω2 = ω −ku2 cos ϕ sin θk.

The main goal is to find the solution for the ω de-
pendence on k. Equation 5 is a polynomial equation
of 8th order.

2. Numerical Solution
A classical Newton’s algorithm for finding the roots
of polynomial equations has one big disadvantage.
It does not specify the initial points (points where an
algorithm starts the iterations) so it does not guar-
antee the finding of all the roots. In 2001 Hubbard,

Figure 1. System of coordinates used in the sim-
ulations.

Schleicher and Sutherland published the article “How
to Find All Roots of Complex Polynomials With New-
ton’s Method”, where they demonstrated how to de-
termine the initial points to find all the roots of poly-
nomial equation [3].

2.1. Principle of the algorithm
fundamentals

Basic principles are described in [4]. For each k we
have a polynomial equation of a type

c0 + c1ω + c2ω2 + c3ω3 + c4ω4 + c5ω5 + c6ω6

+c7ω7 + c8ω8 = 0.
(6)

At first we must rescale the polynom, so we have
to find

Amax = 1 + max
k

{∣∣∣∣ ckcN
∣∣∣∣
}
. (7)

From now we will work with the polynomial

Q(z) ≡
N∑
k=0

ckz
k, (8)

where
z ≡

ω

Amax
, ck ≡ ckAkmax. (9)

The second step is to determine the initial points
where the algorithm will start the iterations. The net
of initial points is determined by radii and angles
in the complex plane:

rl ≡
(

1 +
√

2
)(N − 1

N

)(2l−1)/4L
, (10)

l = 1, . . . ,L, (11)
L ≡d0.26632 ln Ne , (12)

175



Miroslav Horký Acta Polytechnica

ξm ≡
2πm
M

, (13)

m = 0, . . . ,M − 1, (14)
M ≡d8.32547N ln Ne . (15)

Then the net of initial points is

zlm = rl exp(iξm), (16)
l = 1, . . . ,L, (17)
m = 0, . . . ,M − 1. (18)

The initial net of points has definitely LM numbers.
From these numbers the algorithm starts the iterations.
A number of iterations O is defined by accuracy ε by
the definition

O ≡
⌈

ln(1 +
√

2) − ln ε
ln N − ln(N − 1)

⌉
. (19)

The bracket dxe means the ceiling function (first inte-
ger number which is higher or equal to x). Solutions
which do not accomplish |Q(zo) < ε| are not the roots
of the polynomial.
After finding all the roots in the rescaled polynomial
we have to do the backscaling

ωo = Amaxzo. (20)

2.2. Example of the solution

The first numerical solution was made in [5] for the sit-
uation of two identical opposite plasma beams in
a magnetic field. This example of dispersion branches
is for more complicated situation – one plasma beam
penetrates into the plasma background and magnetic
field has both perpendicular and parallel components.
The parameters of this simulation are in Tab. 1 and
graphical result is in Fig. 2.
The result is depicted in well arranged plot where

blue dots represent real branches and red dots imag-
inary branches of the solution. Also maximal value
of the imaginary branch which is so called Plasma
Instability Growth Rate (PIGR) is depicted with sign
“Max”.

Parameter Value
ωc1 = ωc2 0.5
cs1 = cs2 0.1

u2 0
ωp1 1
θk π/2
ϕ 0
θB π/4

Table 1. Parameters used in example of the solution.

Figure 2. Example of solution of the GBDR with
marked maximal value of imaginary part.

3. PIGR dependence on various
input parameters

3.1. Dependence on cyclotron
frequencies

At first the PIGR dependence on both jet and the back-
ground cyclotron frequencies was found.

3.1.1. Results for ωc1
The parameters used in the simulations are presented
in Tab. 2 and the results are depicted in Fig. 3. It is
obvious that the PIGR grows linearly from the value
ωc1 = 0.6.

Parameter Value
ωc1 〈0.5, 3〉
ωc2 0.5

cs1 = cs2 0.1
u2 0
ωp1 1
θk π/2
ϕ 0
θB π/4

Table 2. Parameters used in the simulations with
various parameter ωc1.

Figure 3. The PIGR dependence on ωc1.

176



vol. 53 no. 2/2013 Instability Growth Rate Dependence

3.1.2. Results for ωc2
The parameters used in the simulations are presented
in Tab. 3 and the results are shown in Fig. 4.
From these results we can see the local minimum

of PIGR which origins due to the bifurcation of the so-
lution. The bifurcation is depicted in three dimen-
sional plot where the first axis is k, second is ω and
third is ωc2 (see the Fig. 5).

Parameter Value
ωc1 0.5
ωc2 〈0.5, 3〉

cs1 = cs2 0.1
u2 0
ωp1 1
θk π/2
ϕ 0
θB π/4

Table 3. Parameters used in the simulations with
various parameter ωc2.

Figure 4. The PIGR dependence on ωc2.

Figure 5. Imaginary part of the solution in three
dimensions with an observable bifurcation.

3.2. Dependence on sound velocities
Subsequently the PIGR dependence on both jet and
the background sound velocities was found.

3.2.1. Results for cs1
The parameters used in the simulations are presented
in Tab. 4 and the results are depicted in Fig. 6.

Parameter Value
ωc1 0.5
ωc2 0.5
cs1 〈0.1, 2〉
cs2 0.1
u2 0
ωp1 1
θk π/2
ϕ 0
θB π/4

Table 4. Parameters used in the simulations with
various parameter cs1.

Figure 6. The PIGR dependence on cs1.

It is obvious that after value cs1 = 1, there is no
imaginary branch of the solution, so there are not any
instabilities.

3.2.2. Results for cs2
The parameters used in the simulations are presented
in Tab. 5 and the results are depicted in Fig. 7.
Figure 7 presents a similiar bifurcation point like

in Fig. 4.

Parameter Value
ωc1 0.5
ωc2 0.5
cs1 0.1
cs2 〈0.1, 1.5〉
u2 0
ωp1 1
θk π/2
ϕ 0
θB π/4

Table 5. Parameters used in the simulations with
various parameter cs2.

177



Miroslav Horký Acta Polytechnica

Figure 7. The PIGR dependence on cs2.

3.3. Results overview
We found the PIGR dependence on four parameters
ωc1, ωc2, cs1, and cs2. The main dissimilarity be-
tween the dependencies on the cyclotron frequencies
is caused by zero velocity of the background. Since
the jet has non-zero velocity with a component per-
pendicular to the magnetic field, the jet particles react
to the change of the magnetic field more strongly than
the background particles. The dissimilarity between
the dependencies on the sound velocities has the same
origin. Beause of the non-zero velocity of the jet,
the jet could be subsonic and therefore it could be
in the state with no instabilities.

4. Conclusions and future work
First of all, the GBDR was converted into a non-
dimensional form which ensures the scale invariance
of the problem, which means that the results can be
used both for the laboratory and astrophysical plas-
mas. Afterwards the dispersion relation had been
solved for the angular frequency via the algorithm
suggested by Hubbard, Schleicher, and Sutherland.
In every solution branch there were separated real
and imaginary parts and subsequently found plasma
instability growth rate numerically. Finally, the PIGR
dependence on four input parameters ωc1, ωc2, cs1
and cs2 was found. All these numerical calculations

were done on microscopic scale and in the linear ap-
proximation. These results can be used for lookup
of the plasma parameters close to which the strong
energy transfer and thermalization between the beam
and the target occurs which will be the first part of the
future work. Another part will be Particle In Cell sim-
ulations of plasma turbulences origin in the vicinity
of PIGR maximum.

Acknowledgements
Research described in the paper was supervised
by Prof. P. Kulhánek from the FEE CTU in Prague and
supported by the CTU grants SGS10/266/OHK3/3T/13,
SGS12/181/OHK3/3T/13.

References
[1] O. Buneman. Dissipation of currents in ionized media.

Phys Rev 115(3):503–517, 1959.
[2] M. Horky. Numerical solution of the generalized
Buneman dispersion relation. In Proceedings of Poster
2012. Prague, 2012.

[3] J. Hubbard, D. Schleicher, S. Sutherland. How to find
all roots of complex polynomials with Newton’s method.
Inventiones Mathematicae 146:1–33, 2001.

[4] P. Kulhanek. Uvod to teorie plazmatu. AGA, Prague,
1st edn., 2011. (in Czech).

[5] P. Kulhanek, D. Bren, M. Bohata. Generalized
Buneman dispersion relation in longitudinally
dominated magnetic field. ISRN Condensed Matter
Physics 2011, 2011. Article id 896321.

[6] I. Nishikawa, K., P. Hardee, B. Hededal, C., et al.
Particle acceleration, magnetic field generation, and
emission in relativistic shocks. Advances in Space
Research 38:1316–1319, 2006.

[7] A. Pokhotelov, O., A. Balikhin, M. Weibel instability
in a plasma with nonzero external magnetic field. Ann
Geophys 30:1051–1054, 2012.

[8] E. S. Weibel. Spontaneously growing transverse waves
in a plasma due to an anisotropic velocity distribution.
Phys Rev Lett 2(3):83–84, 1959.

178


	Acta Polytechnica 53(2):174–178, 2013
	1 Introduction
	1.1 The non-dimensional form

	2 Numerical Solution
	2.1 Principle of the algorithm fundamentals
	2.2 Example of the solution

	3 PIGR dependence on various input parameters
	3.1 Dependence on cyclotron frequencies
	3.1.1 Results for bold0mu mumu c1c12c1c1c1c1
	3.1.2 Results for bold0mu mumu c2c22c2c2c2c2

	3.2 Dependence on sound velocities
	3.2.1 Results for bold0mu mumu cs1cs12cs1cs1cs1cs1
	3.2.2 Results for bold0mu mumu cs2cs22cs2cs2cs2cs2

	3.3 Results overview

	4 Conclusions and future work
	Acknowledgements
	References