Dust acoustic solitary waves in a charge varying pair- ion dusty plasma

Abstract

A numerical simulation has been made to analyze the propagation of dust acoustic solitary waves in unmagnetized dusty plasma which consists of a negatively charged cold dust fluid, Boltzmann distributed electrons, positrons and ions. A system of highly nonlinear equations is integrated numerically using a scheme suitable for stiff problems and assuming a hydrogen plasma. This system can be solved either as a boundary-value problem or as an initial-value problem. For simplicity, we have solved it as an initial value problem. It is found that the properties of these solitary waves may change depending on the number of positrons present in our plasma model. The effect of dust charge variation on these dust acoustic solitons is investigated using the full charging equation. Reduction in the width of the soliton profile occurs in the presence of charge variation. Moreover, it is shown that the dust charge fluctuations produce a dissipation which is responsible for non collisional shock wave formation.

Full text

Proceedings of the International Conference on
Modelling and Simulation (MS’07 Algeria)
July 2 - 4, 2007, Algiers, Algeria
Dust acoustic solitary waves in a charge
varying pair- ion dusty plasma
M. Tribeche and A. Berbri
Faculty of Sciences-
Physics, Theoretical
Physics Laboratory, University of Bab-
Ezzouar, USTHB, B.P. 32, El Alia, Algiers
16111, Algeria.
Abstract
A numerical simulation has been made to analyze the propagation of dust acoustic solitary waves in unmagnetized dusty plasma
which consists of a negatively charged cold dust fluid, Boltzmann distributed electrons, positrons and ions. A system of highly
nonlinear equations is integrated numerically using a scheme suitable for stiff problems and assuming a hydrogen plasma. This system
can be solved either as a boundary-value problem or as an initial-value problem. For simplicity, we have solved it as an initial value
problem. It is found that the properties of these solitary waves may change depending on the number of positrons present in our
plasma model. The effect of dust charge variation on these dust acoustic solitons is investigated using the full charging equation.
Reduction in the width of the soliton profile occurs in the presence of charge variation. Moreover, it is shown that the dust charge
fluctuations produce a dissipation which is responsible for non collisional shock wave formation.
Keywords: Dusty Plasma, wave, soliton, shock wave, dust, dust charge variation.
1. Introduction
Charged dust grains constitute an important
component of matter in space and astrophysical systems
such as the planetary rings, asteroid zone, cometary tails,
interstellar clouds, as well as the lower parts of the
Earth's ionosphere [1,3]. Dust grains are also found in
the laboratory environments [4]. Dust grains immersed
in ambient plasmas and radiative environments are
electrically charged due to various processes, such as
plasma (electron and ion) current, photo-emission,
secondary emission and field emission. Charged dust
grains could react with electromagnetic and gravitational
fields, and give rise to a host of low-frequency collective
phenomena in dusty plasmas. Over the last decade, the
field of dusty plasmas has grown rapidly with
applications to space and laboratory systems. In recent
years, a large number of authors have investigated
theoretically various aspects of linear and nonlinear
wave propagation in homogeneous dusty plasmas. In
practical situations, the grain sizes are usually in the
micron range, and hence the dust particles are many
orders of magnitude heavier than the plasma ions.
Consequently, the characteristic time scales associated
with the dust and the ions are very much different from
each other and, therefore, it is possible to separate the
various plasma normal modes arising due to their
dynamics. In fact, dusty plasmas open up an ultra-low-
frequency regime for the existence of novel types of
wave modes, which other wise are not possible in the
usual electron {ion plasmas, but with different ion
species. For example, the existence of an ultra-low-
frequency electrostatic acoustic-like mode, called the
dust acoustic wave (DAWs), was firstly predicted by
Rao et al [5] for demagnetized dusty plasma and
D'Angelo [6] for magnetized dusty plasma
independently. Their model consists of Boltzmann-
distributed thermal electrons and ions which provide the
restoring force, while the inertia arises due to the heavier
dust component. On the other hand, at a higher
frequency, Shukla and Silin showed the existence of
dust-ion-acoustic waves (DIAWs) where the electrons
are Boltzmann-distributed while the inertia is provided
by the ions in the presence of stationary dust grains [7].
Recent laboratory experiments on dusty plasmas have
confirmed the existence of DAWs and DIAWs [8,9].
Nonlinear propagation of the low-frequency DAWs
leads to the formation of coherent structure such as
solitons. Recently, Nejoh [10] pointed out that the dust
charge variation with some parameters would affect the
characteristic collection motion of the plasma. Ma and
Liu [11] and Xie and co-workers [12,13] have
considered this effect and showed the existence of dust
acoustic solitons. We note that laboratory observations
of low phase velocity DAWs are associated with a
significant depletion of the electron number density [8],
MS’07 - 1

suggesting that the wave dynamics is governed by the
inertia of the dust fluid and the pressure of inertialess
ions only [14]. In this paper, we investigate the
properties of nonlinear dust acoustic wave (DAWs) by
incorporating the effects of the dust charge fluctuation
and positrons.
2. Basic equation
We consider the formation of soliton potential
structures in dusty plasma whose constituents are
Boltzmann distributed electrons, ions, positrons and
heavy (compared with the ions) dust grains, which carry
a negative charge. In the presence of the dust grains, the
electrons, ions and positrons may be considered as point
particles. Moreover, the electrons, ions and positrons
are in a local thermodynamic equilibrium and their
densities are given by the Maxwell-Boltzmann
distribution
( )
0exp /
ss s s
nn q T= −Φ
(1)
where
e
T
,
i
T
and
p
T
are the electron, ion and positron
temperature respectively,
Φ
is the electrostatic
potential,
0e
n
,
0i
n
and
0p
n
are the unperturbed number
densities of electrons , ions and positrons respectively.
Let us begin to look at a very simple case, where each
cold species is a beam of particles, each particle of
species s having the same speed at a given position.
Thus, we choose[15-16]
( ) ( )
00
,2
x dd
f xv n v v v
α αα
δ
= −
(2)
where
1/2
020
0
2
1d
dd dd
v v qd
mv
Φ


=−Φ
∫



(3)
Integrating the dust distribution function fd over all
velocity space, we find
( )
01/2
0
1
1
dd
ddd
nv
Nnv
αχ
= = = −

(4)
where
2
22
0
2de
dd
rT
emv
σ
α
=
(5)
and
0d
Qd
χ
Ψ
= Ψ
∫
(6)
Poisson’s equation can be expressed as
2
/
11 0
2(1)
e e p dd d
dNfNfN ff NQQ
dX ++
Ψ=−− −
(7)
where
( )
exp
e
N
σ
Ψ=
( )
exp
i
N−Ψ=
( )
1
exp
p
N
σ
Ψ
=
and
( )
00
/
ie
f nn=
,
( )
/
ie
TT
σ
=
,
( )
00
1/
pe
f nn=
,
()
1/
ie
TT
σ
=
. The following normalized quantities
( )
i
eT
Ψ= Φ ,
( )
Dm
Xx
λ
=
,
( )
d de
d
Q eq r T=
are introduced and
1/2
2
0
4
ie
Dm T ne
λπ


=
. Charge
neutrality at equilibrium
( )
0Ψ=
requires
( )
0
1 00
1/
de d de
f f rTZ e n n=−+
(8)
In the standard orbit-limited probe model for the dust
grain [17], the latter is charged by the plasma currents at
the grain surface. The charging currents originate from
electrons and ions hitting the grain surface.
Accordingly, the variable dust charge
dd
q eZ= −
is
determined self-consistently by (note that we consider a
one –dimensional stationary plasma)
d
d eip
q
v III
x+
∂= +
∂
(9)
where
e
I
,
i
I
and
p
I
are the average microscopic electron
and ion currents entering the dust grains, as governed
by the potential difference between the grain surface
and the local plasma. The grain current from the
electron, ions and positrons are [18]
1/2
2
4 exp
2
ed
e de
ee
Te
I rne mT
ππ
Φ
= −
  
  
  
(10)
1/2
2
41
2
id
i di
ii
Te
I r ne mT
ππ
Φ
= −
  
  
  
(11)
1/2
2
41
2
pd
p dp
pp
Te
I rne mT
ππ
Φ
= −
  
  
  
(12)
with
( )
/
d dd
Ze rΦ=−
, where
d
r
is the radius of the
dust grain and
,
ei
mm
and
p
m
are the electron , ion and
positron mass.
After remaining the terms in Eq. (9), one can obtain the
following normalized charging equation:
11
1
1
exp( )
exp( ) 1 exp( ) 1
dd
dd
d
dQ KN Q
d
QQ
ff
µ
µ
σ
σ
σ
ξ
σσσ





 
+
 


 
= − Ψ+ +
−Ψ− −Ψ−
(13)
With
K
given by
22
0
2
0
2ed
id
v
n re
Km
σ
=
(14)
MS’07- 2

and
ie
mm
µ
=
. Equation (13) is the additional
dynamical equation that is coupled to the plasma
equations through the plasma currents. The dust electric
charge becomes a dynamical variable which is coupled
self consistently to the other dynamical variables such
as the density and potential . Initially, in the absence of
any perturbation
( )
0Ψ=
, eq (13) yields
1
00
0
111
1 exp( ) 1
dd
d
QQ
f Qf
µµ
σσ σ σ
−
  
 
  
= − −−
 
 
  
 
  
(15)
It may be useful to note that the Eq. (15) does not
represent the condition for balancing the electron, ion
and positron current which determines the floating
potential. In the following numerical simulation, the
value of f1 is deduced from the above relation while the
remaining other parameters are given first.
3. Numerical results
We now proceed with the presentation of our numerical
results. Eqs. (6), (7) and (13) are solved numerically as
an intial value problem using a suitable algorithm. To
start the numerical integration, a small edge (
0x=
)
electric field
( )
12
010E d dx −
=−Ψ =−
is chosen. In
figure 1, we study the effect of dust charge fluctuation
on the soliton profiles
()
ξ
Ψ
. We plot the two cases
mentioned above, with the dashed and solid curves
representing the solution obtained by numerically
solving Eqs. (6), (7) and (13). The following
parameters
0
1.4
d
Q= −
,
0.5
σ
=
,
15
σ
=
,
00.8
d
v=
cm/s,
and
4
10
d
r−
=
µm .have been chosen so that the
conditions for the existence of soliton solutions are
fulfilled. It is seen that the variation of the dust charge
produces a reduction of the dust acoustic soliton width in
comparison to the case
d
Z=
constant (ignoring the
charging equation) as well as an increase of its
amplitude. In figure 2, we show the wave profile for a
larger value of
0
i
n
(K), it can be seen that as
0
i
n
decreased (K decreased), the wave amplitude suffers the
well known anomalous damping, the importance of
which is roughly proportional to the ion density. The
dissipation causes the wave amplitude to decay
algebraically and the conservation of soliton mass leads
to the development of a noise tail. The leading edge of
the localized structure moves away from
x
axis and will
never return to
0x=
. The nature of wave structure
depend to the parameter
0i
n
, so that, at
2
010
i
n=
cm-3 and
12
010
i
n=
cm-3 we obtain the wave shock structure and
solitary wave structure respectively. It indicates that an
decrease of
0i
n
gives rise to shocks for which structure
is monotonic instead of being oscillatory. In figure 3a,
we study the effect of the parameter
1
f
(the number of
positrons in our model plasma) on the dust acoustic
soliton for
4
3.6610K=
,
12
010
i
n=
cm-3 when the charge of
the dust grain is variable. It is seen that when
1
f
increases, the amplitude of the solitary structure
increases as well as its width increases. In figure 3b, for
2
010
i
n=
cm-3,
0.36K=
, we found that smaller values
of
1
f
favor the development of dissipative structure.
4. Summary
To conclude we have addressed the problem of
nonlinear electrostatic solitary waves representing
saturated states of three stream unstable collisionless
charge varying dusty plasma. Our results show that in
such a plasma (which is, to our knowledge, yet
unexplored and not yet described in the dusty plasma
literature) spatially localized structures can exist. Their
spatial patterns are significantly modified by the
presence positron component. In particular, it may be
noted that an addition of a concentration positron
abruptly modifies the dust acoustic soliton properties.
Furthermore, our results show that the dust charge
fluctuation reduces the soliton width and provides an
alternate physical mechanism causing dissipation and,
as consequence, causes the wave amplitude to decay
and transfer to the so-called noise tail. However,
considering the wide relevance of nonlinear oscillations
in laboratory and space dusty plasmas, our investigation
may be taken as a prerequisite for the understanding of
the large, but finite amplitude solitary waves that may
occur in space and laboratory plasmas.
Fig.1: Localized solution for the electrostatic potential
Ψ
with
0
1.4
d
Q= −
,
0.5
σ
=
,
15
σ
=
,
0.8f=
,
00.8
d
v=
cm/s,
4
10
d
r−
=
µm,
4
3.6610K=
and
12
010
i
n=
cm-3. The value of
1
f
deduced from Eq. (15) is
10.48f=
.
10 12 14 16 18 20 22
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
ξ
Φ
constant dust charge
variable dust charge
MS’07- 3

-10 010 20 30 40 50
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
ξ
Φ
n
i0
=10
2
,K=0.36
n
i0
=10
12
,K=3.6610
4
Fig.1: Soliton solution for the electrostatic potential
Ψ
for
various value of
0i
n
(K) in the case of variable dust charge.
The values of relevant parameters are those used for fig.1.
20 25 30 35
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
ξ
Φ
f
1
=0.48
f
1
=0.45
f
1
=0.42
Fig.3a: Soliton solution for the electrostatic potential
Ψ
for
various value of
1
f
in the case of variable dust charge with .
The values of relevant parameters are those used for fig.1.
020 40 60 80 100
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
ξ
Φ
f
1
=0.48
f
1
=0.42
Fig.3b: Soliton solution for the electrostatic potential
Ψ
for
various value of
1
f
in the case of variable dust charge, with
0.36
K=
,
2
010
i
n=
cm-3. The values of relevant parameters
are those used for fig.1.
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MS’07- 4