Dust acoustic shock waves have been studied in six component cometary plasma by deriving the Korteweg-deVries-Burgers
(KdVB) equation. The constituents of the plasma include two components of electrons described by kappa distributions with
different temperatures and spectral indices, lighter (hydrogen) ions and a pair of oppositely charged, heavier ions (positively and
negatively charged oxygen ions), all of them described by Maxwellian distributions with different temperatures. Charge varying
negatively charged dust grains is the sixth component.
Shock waves solutions of the KdVB equation, studied for typical parameters of comet Halley, show that the shock amplitudes are
consistently larger when charge fluctuations on the dust grains are taken into consideration. The superthermal, second component
of electrons affects the phase velocity and the shock velocity as well as it’s width. The amplitude of the shock wave is also affected
by the densities and temperatures of all the ions: it increases with increasing positively charged oxygen ion densities and decreases
with increasing temperatures of these ions. The amplitude, however, decreases with increasing densities and increases with
increasing temperatures of the other two types of ions.
Keywords: Cometary plasma; Shock waves; Dusty plasma; Charge variation; 6 components;
The presence of charged dust grains introduces new features to
the non-linear structures in space environments such as planetary
rings, planetary magnetospheres, comet environments, interstellar
media and earth-space environments [1-4]. The non-linearities in
dusty plasmas give rise to localization of waves, generating different
types of fascinating coherent structures, namely, solitary structures
, double layers , shock waves , vortices [8,9] etc. In dusty
plasmas, the interaction of dust grains with energetic particles
such as electrons and ions lead to the charging of dust grains. The
charging currents to the dust grains carried by the plasma particles
can be calculated using the Orbital Motion Limited (OML) approach
[10,11]. Extensive studies on the effect of dust charge variation on
solitary [12-15] and shock structures [16-18] have been carried
There is a strong current interest to understand the relevance
of the dust charge variation on the formation of dust acoustic shock
waves [19-21]. Popel et al  discussed the possibility of the
observation of shock waves related to the dust charging process
in the presence of electromagnetic radiation in active rocket
experiments which involved the release of some gaseous substances
in the near-earth space. Further, Duha et al  investigated the
dust-ion-acoustic solitary and shock waves associated with the
dynamics of negative ions, Maxwellian positive ions, trapped
electrons and charge fluctuating stationary dust by employing
the reductive perturbation method. Several authors [24-26] have studied Dust-Acoustic (DA) shock waves in a charge varying nonextensive dusty plasma. Hadjaz and Tribeche  obtained a dusty
plasma model that supported solitary as well as shock waves, for
which the main properties such as phase velocity, amplitude and
width were drastically influenced by trapping, nonthermality and
charge variation. El-Shewy et al  highlighted the effects of dust
viscosity and electron-ion nonthermality fraction on DA shock
waves in inhomogeneous dusty plasmas with nonadiabatic dust
charge fluctuation. Later, Wang et al  studied the effects of dust
size distribution and dust charge fluctuation of dust grains on small
but finite amplitude nonlinear dust ion-acoustic shock waves, in an
unmagnetized multi-ion dusty plasma.
In a large number of space environments, velocity distributions
have commonly been reported that are Maxwellian-like in the
low-energy range, but possesses a power-law tail at superthermal
particle energies [30-34]. Such distributions are modeled by a
generalized form of Lorentzian or kappa distribution first proposed
by Vasyliunas . The Maxwellian distribution is a special case of
the kappa function; in the limit of the spectral index κ → ∞ . In
a recent study, Broiles et al  evaluated the observations made
by Rosetta spacecraft of the superthermal electron environments
near comet 67P/Churyumov-Gerosimenko and fitted the observed
electron velocity space densities with a combination of two threedimensional kappa distributions. Hence there has been a great deal
of interest in studies related to dust charge fluctuation on shock
waves in a suprathermal dusty plasma [37-39]. Recently, Ferdousi
et al  analysed the properties of low frequency dust-acoustic
shock waves in a plasma medium in the presence of superthermal
kappa distributed electrons.
A cometary plasma has been observed to be a genuine multi-ion
plasma as it is composed of solar wind protons and electrons;the
dissociation of water molecules contributes positively charged
hydrogen (H+) and oxygen (O+) and photo-electrons . In addition to
the existence of positive ions, negative ions have also been observed
in cometary plasmas . Besides, the above mentioned ion pairs,
dust of opposite polarities has also been observed in cometary
environments [43,44]. Hence, Manesh et al  investigated the
existence of Ion-Acoustic (IA) shock waves in a five component
cometary plasma consisting of positively and negatively charged
oxygen ions, kappa described hydrogen ions, hot solar electrons,
and slightly colder cometary electrons. Very recently, Sijo et al 
studied the effect of the drift velocity of lighter ions on shock waves
in the above five component cometary plasma. Further, Mahmoud
 analyzed the effects of the non-extensive parameters on the
structure of the envelope ion acoustic waves in five-component
cometary plasma system containing positively and negatively
charged oxygen ions, non-extensive hot electrons from solar origin,
colder electrons of cometary origin and positive hydrogen ions.
Thus for reasons given above, we investigate the effect of dust
charge variation on dust acoustic shock waves in a six component
cometary plasma composed of lighter hydrogen ions, positively
and negatively charged oxygen ions, charge varying dust grains of
negative polarity, cometary photo-electrons and hot, solar wind
We consider a six component, unmagnetized plasma containing
negatively charged dust (denoted by subscript ’d’), negatively and
positively charged oxygen ions (denoted by subscripts ’1’ and
’2’ respectively), positive hydrogen ions (denoted by subscript ’H’)
and superthermal electrons described by kappa distributions (hot
electrons from solar origin and colder electrons of cometary origin
denoted by subscripts ’he’ and ’ce’, respectively).
At equilibrium, charge neutrality requires:
where nhe0 and nce0 represent the equilibrium densities
of cometary electrons and solar electrons respectively; n10 , n20, nH 0 and nd 0 are the equilibrium densities of negatively charged
oxygen (O− ) ions, positively charged oxygen (O+ ) ions, hydrogen
ions and negatively charged dust respectively. z10 , z20 and zd 0
are the equilibrium charge numbers of O− , O+ ions and dust
The dynamics of the negatively charged dust particles can be
described by the following hydrodynamic equations:
where Vd and nd are the fluid velocity and kinematic viscosity of the dust grains, md , the mass and e, the electronic charge.
The above equations, in their dimensionless forms, are:
where the kappa distributions model the number densities
for cometary ( ce ) and solar ( he ) electrons; while the hydrogen
ions (denoted by H , negatively charged oxygen ions (identified by
'1') and positively charged oxygen ions ('2') obey a Maxwellian
distribution. The dust grain density is normalized by nd 0 and
zd by zd 0 ; the densities of electrons and ions are normalized by zd 0 nd 0 . The space 'x' and 't' time coordinates are normalized
by the Debye length
and the inverse of dust
respectively. The dust fluid
velocity d v and electostatic potential are normalized by the DA
respectively, in which the effective temperature is :
where Td , Tce , The , TH , T
and T2 are the temperatures of dust,
cometary electrons, solar electrons, hydrogen ions, negative and
positive oxygen ions, respectively
The variable dust charge Zd is obtained from the current balance
where IH , I1 , I2 , Ice and Ihe are hydrogen ion, negatively
charged oxygen, positively charged oxygen, colder and hotter
electron currents, respectively
The ion currents are :
where IH and I2 are the charging currents for attractive
potentials (qj φd < 0 ) due to hydrogen and positively charged
oxygen ions respectively. Here φd denotes the dust grain surface
potential relative to the plasma potential φ ; r is the radius of the
dust grain and the other notations are standard. Also
where I1 is the charging current for a repulsive potential
(qj φd >0 ) due to the negatively charged oxygen ions.
The currents due to the cometary and solar, kappa distributed
electrons to the negatively charged dust grains are ,
and , the effective thermal speed for the electron, is
Here mj is the mass, nj is the electron number density, Tj is the temperature and Kj, the spectral index of the jth species.
Using equations (10) - (14) in the current balance equation (9),
we arrive at
The number of charges residing on the dust grains, Zd, is defined as
is the dust surface floating potential corresponding to the unperturbed plasma potential and is determined from the following expression,
Zd can be expanded in terms of φ as follows
come from expanding ψ
Taking the derivative of equation (15) with respect to φ
and finding its value at φ = 0 gives,
Taking the second derivative of (15), with respect to φ , and
finding its value at φ = 0 further gives,
Dust acoustic shock waves - Derivation of KdVB equation:
We shall derive the KdVB equation by employing the reductive
perturbation method . The stretched co-ordinates are
introduced in the following form ,
where ϵ is a small parameter that measures the size of the
perturbation amplitude and λ is the velocity of the shock wave
normalized by Cd . In a weak damping situation, the kinematic
viscosity of dust ions can be considered small but finite. Here, ηd 0
is a finite parameter.
The various parameters are expanded as a power series in ϵ as
Also from equation (17),
Using transformation equations in equations (5)-(7) and
equating different powers of ∈ , the lowest order of ∈ leads to:
Equating terms of power of order ϵ in Poisson’s equation, the
linear dispersion relation of the wave can be expressed as:
Next, equating terms of order in
and using equations (28),
Simiarly, equating coefficients of terms of order 2 ∈ from
Poisson’s equation, we get
Taking the derivative of (31) and using equations (28)-(30), the
KdVB equation is derived as
where the non-linear coefficient
the dispersion coefficient
and the dissipation coefficient
, of which
Shocks in dusty plasmas have been generally studied using either the Burger’s equation [46, 52 – 58] or the Korteweg-deVries-Burgers (KdVB) equation [45, 46, 59 – 64]. Both Manesh et al  and Sijo et al  had considered five component plasmas; this study can thus be considered as extending and complementing these studies as we have an additional component namely, negatively charged dust grains.
Solution of KdVB equation
The "tanh method" can be used to obtain the shock-like solution of KdVB equation (34) [67 - 69]. We used a transformed co-ordinate
moving with the shock speed and employed boundary conditions
For a localized solution, we can write equation (34) as
Again using the transformation α χ = tanh and assuming a
series solution of the form
, the shock solution of
KdVB equation is determined as
is the shock speed and
is the inverse of
the shock width.
Results and Discussions
The KdVB equation (34) can be applied to any multi-ion/
dusty plasma environment. However, the parameters relevant to
plasma environment of comet Halley are used to plot the figures. The hydrogen ion density was set at 4.95 cm−3 with a temperature
of TH= 8 × 104 K; the solar electron temperature The = 2 × 105 K
. The temperature of the colder, cometary electrons was set at
Tce = 2 × 104 K. The negatively charged oxygen ion was set at
n10=.05cm −3 . The density of the positively charged heavier oxygen
n20 = 0.5cm−3 at a temperature of 1.16 × 10K4 [41, 42]. The
density of dust of negative polarity was set at
nd0= 0.1cm-3 and
its equilibrium charge at . Also,
The values of normalized kinematic
dust ion viscosity used was ηd0 = 0.5 .
(Figure 1) shows a plot of the shock profiles, with and without dust charge variation. The dotted (blue) curve is for γ1 = 0
, γ2 = 0 ; while the continuous (red) curve is with dust charge
variation; obtained for n10=.05cm −3, md=200 a.m.u and zd0 = 50. The other parameters for the figure are: The =2 x 105 K, Tce= 2 x 104, TH= 8 x 104K , Td= T1= T2= 1.16x10K; the densities are nH0= 4.95cm-3, n10= 0.05cm-3, n20=0.5cm-3, nd0=0.1cm-3, nhe=nce=0.5ne. From the figure, it is clear that with the inclusion
of dust charge variation, the amplitude of the shock wave increases.
Figure 1: Plot of shock profiles with and without dust
In a study related to comet Halley, Tribeche and Bacha 
concluded that nonlinear damping due to dust charge variation
would lead to the formation of shock waves. A similar conclusion
was arrived at more recently by Naeem et al . Also the pickup of
heavy ions could lead to the formation of shock waves . Also the
charge exchange reaction which leads to the formation of negatively
charged oxygen ions could also contribute to the formation of shock
(Figure 2) depicts the variation of (a) phase velocity, λ , (b)
shock speed, V , and (c) solitary width, W , versus the spectral
index kappa, . Again, the dotted (blue) curves are for the cases
where the charges on the dust particles are a constant; the
continuous (red) curves depict the other case. The parameters are
the same as in figure 1. It is obvious that as the specral index kappa
for cometary electron increases, the phase velocity and the shock
width increase whereas the shock speed decreases with κ ce . However,
with dust charge variation included, the values of phase velocity, and
width of the shock wave get reduced but the shock speed increases.
Figure 2a: Variation of phase velocity, λ with spectral index κce with and without dust charge variation.
Figure 2b: Variation of shock speed, V with spectral index κce with and without dust charge variation.
Figure 2c: Variation of the shock width W with spectral index κce with and without dust charge variation.
(Figure 3) shows the variation of the phase velocity, λ, versus equilibrium charge number zd0. The dotted (blue) curve depicts the case where the charges on the dust particles are a constant; the continuous (red) curve is for the case where the charge varies. The other parameters are the same as in (figure 1). We find that λincreases with an increase in the equilibrium charge number zd0; also λ is lower when the charges on the dust particles vary.
Figure 3: Plot of phase velocity, λ, versus dust charge number, zd0, with and without dust charge variation.
It may be noted that in a study of dust acoustic shock waves in four component plasma of charged mobile dust, kappa distributed electrons, positively charged lighter ions and negatively charged heavier ions; the variation of the phase velocity with κ was similar . Again, in a study of low frequency shocks in a magnetized dusty plasma of negatively charged dust and kappa distributed electrons and ions, the variation of the shock velocity with the spectral index of the ions is similar .
Figure 4 depicts the variation of (a) the nonlinear coefficient, A, and (b) the dispersion coefficient, B , with κce with and without dust charge variation. Here too, the dotted (blue) curves denote the cases where the charges are on the dust particles are a constant; the continuous (red) curves are for the cases where there is charge variation. The other parameters are the same as in (figure 1). From the figures, it is clear that as the values of κce increases, coefficient A and B increases. Also, with dust charge variation included, the value of coefficient A increases whereas B reduces.
Figure 4a: Variation of the nonlinear coefficient, A with κce with and without dust charge variation.
Figure 4b: Variation of the dispersion coefficient, B with κce with and without dust charge variation.
(Figure 5) illustrates the variation of the shock amplitudes for different values of the kinematic viscosities of the dust particles with and without dust charge variation. Here too, the dotted (blue) curve represents the case of constant dust charge with ηd0=0.3 and the dotted (red) curve depicts constant dust charge with ηd0=0.4 . Also the continuous (blue) curve is for the case where the charge varies with ηd0=0.3 and the continuous (red) curve is for charge variation with ηd0=0.4. The other parameters are the same as in figure 1. From the plots it is seen that as the kinematic viscosity of the dust particles increase, the amplitude of the shock profile increases. In addition, with the inclusion of dust charge variation, the shock amplitude increases.
Figure 5: Variation of shock profiles for different values of dust ion kinematic viscosity with and without dust charge variation; the other parameters being the same as in figure 1.
(Figure 6) shows the variation of the shock profiles with (a) negative oxygen ion densities, (b) positive oxygen ion densities and (c) hydrogen ion densities, with and without dust charge variation. The dotted curves represent the cases where the charges on the dust particles are a constant; the continuous curves are for the cases where the charges vary. The blue colour denotes n10=0.01cm-3 in figure (a), n20=0.04cm-3 in figure (b) and nH0=4.95cm-3 in figure (c). The red colour represents n10=0.05cm-3 in figure (a), n20=0.45cm-3 in figure (b) and nH0=0.45cm-3 in figure (c). The other parameters are the same as in figure 1. The shock amplitude decreases with increasing negative oxygen and hydrogen ion densities (in figures 6(a) and 6(c) respectively). However, as the positive oxygen ion densities increase, the shock amplitude increases (figure 6(b)). Further, with inclusion of the variation of dust charge, the shock amplitudes increase as in (figure 1).
Figure 6a: Variation of shock profile with negative oxygen ion density with and without dust charge variation.
Figure 6b: Variation of shock profile with positive oxygen ion density with and without dust charge variation.
Figure 6c: Variation of shock profile with hydrogen ion density with and without dust charge variation.
(Figure 7) shows the variation of the shock profiles with (a) negative oxygen ion temperatures (b) positive oxygen ion temperatures and (c) hydrogen ion temperatures, with and without dust charge variation. Here too, the dotted curves represent the cases where the charges on the dust particles are a constant; the continuous ones are for the cases where the charges vary. The blue colour denotes
in figure (a),
in figure (b) and
in figure (c).The red colour represents
in figure (a),
in figure (b) and
in figure (c). The other parameters are the same as in figure 1. The shock amplitude increases with increasing negative oxygen and hydrogen ion temperatures, in figures 8(a) and 8(c) respectively. As the positive oxygen ion temperature increases, the shock amplitude decreases in figure 8(b). Further, with inclusion of the variation of dust charge, the shock amplitude shows the same variation as in figure 1.
We have, in this paper, studied dust acoustic shock waves in a six component cometary plasma by deriving the Korteweg-deVries-Burgers (KdVB) equation. The constituents of the plasma include two components of electrons described by kappa distributions with different temperatures and spectral indices, lighter (hydrogen) ions and a pair of oppositely charged, heavier ions (positively and negatively charged oxygen ions), all of them described by Maxwellian distributions with different temperatures. Charge varying negatively charged dust grains is the sixth component.
Shock waves solutions of the KdVB equation, studied for typical parameters of comet Halley, show that the shock amplitudes are consistently larger when charge fluctuations on the dust grains are taken into consideration. The superthermal, second component of electrons affects the phase velocity and the shock velocity as well as it’s width. The amplitude of the shock wave is also affected by the densities and temperatures of all the ions: it increases with increasing positively charged oxygen ion densities and decreases with increasing temperatures of these ions. The amplitude, however, decreases with increasing densities and increases with increasing temperatures of the other two types of ions.
Positively and negatively charged nano-grains have been observed at comet 67P / Churyumov – Gerasimenko . Cometary heavy ions (oxygen in our case) picked up by the solar wind produce mass loading resulting in a decrease in the solar wind speed. And these heavy ions play a key role in cometary shocks. Such pick up ions have been observed by Giotto at comet Halley and almost all other comets; such shocks are called “mass loading shocks”. An very recently Rosetta was able to cross a newly formed infant bow shock at comet 67P / Churyumov – Gerasimenko. Thus we feel that the results of this paper would contribute to an understanding of shocks observed at comets.
Conflict of Interest
The authors declare that there is no conflict of interest.
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