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Modified Jeans instability of strongly coupled inhomogeneous magneto dusty plasma in the

presence of polarization force

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2014 EPL 107 15001

(http://iopscience.iop.org/0295-5075/107/1/15001)

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July 2014

EPL, 107 (2014) 15001 www.epljournal.org

doi: 10.1209/0295-5075/107/15001

Modified Jeans instability of strongly coupled inhomogeneousmagneto dusty plasma in the presence of polarization force

Prerana Sharma(a)

Physics Department, Ujjain Engineering College - Ujjain, M. P., 456010, India

received 2 March 2014; accepted in final form 10 June 2014published online 30 June 2014

PACS 52.27.Gr – Strongly-coupled plasmasPACS 52.27.Lw – Dusty or complex plasmas; plasma crystals

Abstract – In the present work, the influence of the magnetic field and dust polarization forceon the Jeans instability of self-gravitating strongly coupled inhomogeneous dusty plasma has beeninvestigated. The dusty plasma containing strongly correlated negatively charged dust grains andweakly correlated Maxwellian electrons and ions which are embedded in a uniform magnetic field isconsidered. The construction of the equations is done by employing the generalized hydrodynamic(GH) model for magnetized strongly coupled dusty plasma. In deriving a dispersion relation, theplane-wave solutions are used on the linearized perturbation equations. The analysis is done bynormal mode analysis theory. The dispersion relation is analyzed to obtain the Jeans criterion ofinstability. Numerical results are presented to show the effect of the polarization parameter, themagnetic field and the strong correlation effect of dust. The growth rates are further compared inthe kinetic and hydrodynamic regime of propagation. It is observed that the decay in the growthrate is faster in the kinetic mode of propagation in comparison to the hydrodynamic mode.

Copyright c© EPLA, 2014

Introduction. – The dusty plasma is an extension ofthe plasma theory and has become a source of new phe-nomena, applications [1] and challenging problems. It isnow widely recognized that the dust components in theseplasmas are strongly coupled to each other through theirmutual Coulomb interaction as dust particles are highlycharged. In recent years much attention has been devotedto studying the waves and instabilities in strongly coupleddusty plasmas (SCDPs) by both theoretical [2] and ex-perimental research. Several theoretical models viz. GHmodel [3], quasi-localized charge approximation [4], localfield corrections method [5], thermodynamic model [6] andfluid approach [7], have been proposed to include the ef-fects of strong correlation among highly charged dust inSCDP. Shukla and Mamun [8] have studied the dust acous-tic shocks in SCDP using Korteweg-de Vries (KdV) equa-tions derived by GH equations and Boltzmann equations.The nonlinear propagation of the dust acoustic waves ina SCDP has been investigated by Mamun and Cairns [9]employing the GH model and the reductive perturbationmethod. Low-frequency electrostatic dust-modes in SCDPare investigated by Mamun et al. [10] including dust graincharge fluctuation and for the equilibrium grain charge

(a)E-mail: preranaiitd@rediffmail.com

inhomogeneity. Kaw and Sen [3] have investigated thelow-frequency dusty modes in unmagnetized SCDP withGH model. A generalized hydrodynamic model is usedto examine the influence of the shear-flow–induced viscos-ity gradient in a SCDP by Banerjee et al. [11]. Janakiand Chakrabarti [12] have studied the electrostatic trans-verse shear waves in SCDP employing GH model equa-tions. Thus, it is found that waves and instabilities havebeen widely studied in the SCDP using the GH model inunmagnetized plasma.

Along with that, the presence of strong correlationsamong particles in the presence of magnetic field leads tounique properties in the system, so it is of interest to seethe response of strongly coupled plasma in the presenceof magnetic field. The presence of magnetic field mod-ifies the behavior of SCDP. The importance of magneticfield in SCDP has been recognized by many researchers us-ing different approaches. Viscoelastic modes in a stronglycoupled cold magnetized dusty plasma has been discussedby Banerjee et al. [13] using generalized magnetohydrody-namic equations. Xie and Chen [14] have investigated low-frequency modes in a strongly coupled magnetized dustyplasma by using the GH model, and found that a longitu-dinal hybrid-like mode and transverse shear mode exist insuch dusty plasma system. Xie [15] has used the GH model

15001-p1

Prerana Sharma

to study the instability of a longitudinal wave in mag-netized strongly coupled dusty plasma. Low-frequencywaves in strongly coupled magnetized SCDP have beenstudied by Xie [16] incorporating the GH model. A the-ory for the formation of Mach cones in magnetized dustyplasma with strongly correlated charged dust grains hasbeen presented using the GH model and quasi-localizedcharge approximation by Mamun et al. [17]. Linear andnonlinear propagation of dust acoustic waves in a mag-netized strongly coupled dusty plasma is theoretically in-vestigated by Shahmansouri and Mamun [18] using KdVequations. Therefore, it is established that the waves andinstabilities in magnetized SCDP are formulated using theGH model.

Furthermore, the dust polarization force is the phe-nomenon which modifies the collective effect in dustyplasma. The studies related to the polarization force ofdust have been considered both in the homogeneous andinhomogeneous system. Actually, in the dusty systemthe dust particles are highly negatively charged and theirDebye sheaths contain an excess of ions and less number ofelectrons. The ion density increases toward the dust grainsand these ions create ion pressure. In the uniform plasmathe resultant force on the dust due to this pressure is zerowhile in the non-uniform case the net force due to ionpressure is different on both ends of the dust and, hencedust polarizes. The effect of the dust polarization force isextensively considered in the study of waves and instabil-ities in weakly and strongly coupled dusty plasma. Theinfluence of the polarization force on the propagation oflow-frequency dust acoustic waves in weakly coupled dustyplasma has been studied by Khrapak et al. [19]. The roleof the polarization force and of the non-thermal electronon dust acoustic waves in inhomogeneous weakly coupleddusty plasma with positively charged dust has been stud-ied by Asaduzzaman and Mamun [20]. The effect of thepolarization force of dust has also been incorporated inthe studies of SCDP. The effect of the polarization forcewith effective dust temperature on dust acoustic solitarywaves has been discussed by Mamun et al. [21] consider-ing homogeneous SCDP. Time-dependent cylindrical andspherical dust acoustic solitary and shock waves propagat-ing in a homogeneous SCDP in the presence of polariza-tion force and dust temperature have been investigated byAshrafi et al. [22]. The effect of the polarization force inSCDPs with suprathermal electrons is observed by Zobaerand Mamun [23]. The effects of the polarization forceand fast electrons on DA shock waves in strongly coupleddusty plasma have been discussed by Pervin et al. [24].Dust acoustic solitary waves in a strongly coupled inho-mogeneous dusty plasma in the presence of polarizationforce and effective dust temperature have been discussedby Alinejad and Mamun [25]. Thus, we can say that thestudies including dust polarization force have been widelycarried out in SCDP. Apart from that, the dusty plasmastructure is formed, where the gravitational force of at-traction among the heavier grains is comparable to the

electromagnetic force in the plasma. The role of the dustpolarization force on Jeans instability in magneto weaklycoupled dusty plasma including radiative condensation hasbeen studied by Prajapati [26]. The Jeans instability of aviscoelastic fluid is investigated by Janaki et al. [27] usingthe GH model.

Thus, from the above-mentioned works the importanceof magnetic field, polarization force and self-gravitationeffects of dust in the SCDP has been established. Thepresent study is motivated by the work done by Xie andChen [14] and Khrapak et al. [19]. Thus, in the presentwork, the Jeans self-gravitational instability of inhomo-geneous SCDP has been examined considering the dustpolarization force following Khrapak et al. [19] and con-sidering a magnetic field with the GH model following Xieand Chen [14]. The paper is organized as follows: the nextsection is devoted to the formulation of the model. In thethird section, the modified dispersion relation is derivedand discussed analytically and numerically for differentregimes of propagation. In the last section some conclu-sive comments are given.

Governing equations. – A strongly coupled dustysystem is considered, whose constituents are negativelycharged dust, electrons and ions embedded in a uniformmagnetic field B0 (0, 0, B). We assume that electronsand ions are weakly coupled due to their higher temper-atures and smaller electric charges, and to the fact thatdust is strongly coupled because of its lower temperatureand larger electric charge. Therefore, there are differentpossible low-frequency dust modes in the presence of mag-netic field. The formulation of the present work is donefor an intermediate frequency of the order of the dust gyrofrequency. For that purpose the GH model [11,14–16] isused and it is based on the assumption that the dust po-larization force is aroused due to the electrostatic inter-actions between thermal ions and dust grains. Thus, inthe presence of the low-phase-velocity dust acoustic waves(in comparison with electron and ion thermal velocities),the electron and ion number densities obey the Maxwelliandistribution and their densities ne and ni are respectivelygiven by

ne = ne0 exp(

eφ

Te

), (1)

ni = ni0 exp(−eφ

Ti

), (2)

where φ is the electrostatic potential, Te(i)is the elec-tron (ion) temperature and ne0 and ni0 are the unper-turbed densities of electron and ion, in equilibrium we haveni0 = ne0 + qdnd0, where nd0 is the unperturbed densityof dust. The dynamics of the dust particles in such SCDPis governed by the well-known GH equations as follows:

the continuity equation

∂nd

∂t+ ∇ · (ndvd) = 0; (3)

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Modified Jeans instability of strongly coupled inhomogeneous etc.

the momentum transfer equation for dust grain usinglinear theory

(1 + τm

∂

∂t

) [mdnd

dvd

dt+ mdnd∇ψ + qdnd∇φ

− qdndR

(ni

ni0

)1/2

∇φ + qdnd (vd × B)

]= ηl

∂2vd

∂x2; (4)

Poisson’s equation for electrostatic potential

∂2φ

∂x2= 4πe (ne − ni) − 4πqdnd; (5)

Poisson’s equation for self-gravitational potential

∇2ψ = 4πGmdnd, (6)

where qd = −Zde, Zd is the number of the electrons resid-ing on the dust particulate, and e is the magnitude of theelectronic charge, R = (1/4) (|qd|e/λDTi) (1 − Ti/Te), nd

is the dust number density, vd is the dust fluid speed, τm isthe viscoelastic relaxation time, ψ is the gravitational po-tential and ηl is the longitudinal viscosity coefficient, λD isthe Debye length defined below. We note that the fourthterm on the left-hand side of eq. (4) is due to the polar-ization force and fifth term due to the external magneticfield B0.

Perturbation equations and dispersion relation.– We write the space- and time-dependent variablesnd,e,i vd, φ and ψ in the form of a sum of the equilib-rium and perturbed parts as

nd = nd0 + nd1, ni = ni0 + ni1, ne = ne0 + ne1,

φ = φ0 + φ1, ψ = ψ0 + ψ1, vd = vd0 + vd1.(7)

The subscript “0” indicates the equilibrium terms and“1” stands for the perturbed terms. However, we considerthat the dusty magnetized plasma is characterized by

φ0 = 0 and vd0 = 0. (8)

We assume that all equilibrium quantities are independentof space and time. Our aim is to investigate the instabilitydue to perturbation in such a system which initially hasno inhomogeneity in dust number density and magneticfield. Here, we can neglect inhomogeneity in dust numberdensity provided the scale length considered must be longenough in comparison to the inhomogeneity scale lengthof dust density. Therefore, in equilibrium the dust numberdensity and the value of the magnetic field in equilibriumwill be zero in our case. Using eqs. (7) and (8), we linearizethe basic equations (1)–(6) to a first-order approximationand employing a plane-wave analysis we consider that allthe perturbed quantities vary as exp (ik ·r+iωt) (where ωis the frequency of the harmonic disturbances and kx, z arethe wave numbers in perpendicular and parallel directionsto the magnetic field and related k2

x+k2z = k2). We obtain

that the linearized perturbation equations for electron andion fluids are

ne1 = ne0

(eφ1

Te

), (9)

ni1 = ni0

(−eφ1

Ti

). (10)

The perturbed momentum transfer equation for dust grainreads

(1 + τm iω)[mdnd0iω vd1 + mdnd0 ik ψ1

+ qdnd0 (1−R) ikφ1+qdnd0 (vd1×B)]=ηl (ik)2 vd1. (11)

The perturbed continuity equation is

−ωnd1

nd0= k · vd1. (12)

The Poisson equations for electrostatic potentials are

k2φ1 = 4π (e(ni1 − ne1) + qdnd1) . (13)

The Poisson equations for self-gravitational potential are

−k2ψ1 =ω2

Jd

nd0nd1, (14)

where ω2Jd = 4πGmdnd0 is the square of the Jeans dust

frequency. Further, from eq. (11) with the help of eqs. (12)and (14), we obtain a relation for perturbed dust densityas

nd1 =(1 + τmiω) qdnd0 (1 − R) k2/md

(1 + τmiω) (ω2 + ω2Jd − iωωcd) − ηliωτmk2

mdnd0

φ1,

(15)where ωcd = qdB/md is the dust cyclotron frequency. Sub-stituting the values of ni1, ne1 and nd1 from eqs. (9), (10)and (15), respectively, into Poisson’s equation given byeq. (13), then we obtain the following dispersion relation:

(1 + τmiω)(ω2 + ω2

Jd − iωωcd

)− ηliωk2

mdnd0−

ω2pdk

2λ2D (1 − R) (1 + τmiω)

1 + k2λ2D

= 0, (16)

where ω2pd = 4πq2

dnd0/md is the dust plasma fre-

quency, λD = λDi/

√1 + (λDi/λDe)

2 with λDi(e) =Ti(e)/4πe2ni(e). Equation (16) represents the required dis-persion relation for self-gravitating SCDP in the presenceof magnetic field when the polarization force of dust hasbeen taken into account. Further, we discuss the disper-sion relation given by eq. (16) for both the two ranges ofpropagation for the low-frequency (hydrodynamic) regimeand the relative high-frequency (kinetic) regime.

Hydrodynamic regime: The dispersion relation for low-frequency wave modes in SCDP under the limit ωτm � 1can be discussed within the hydrodynamic regime as

ω2 + ω2Jd − iωωcd − V 2

cdiωτmk2 −ω2

pdk2λ2

D (1 − R)1 + k2λ2

D

= 0,

(17)

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Prerana Sharma

where V 2cd = ηl/mdnd0τm. The dispersion relation given

by eq. (17) represents the self-gravitating magnetizedmode in the presence of polarization force and strong-coupling effects of the dust in the hydrodynamic regime.Moreover, in the absence of strongly coupled effects of thedust, we have from eq. (17)

ω2 + ω2Jd − iωωcd −

ω2pdk

2λ2D (1 − R)

1 + k2λ2D

= 0. (18)

This equation (18) represents the Jeans instability of tra-ditional weakly coupled dusty plasma modified due to dustpolarization force and magnetic field. This dispersion rela-tion resembles the result obtained by Prajapati [26]. Fur-thermore, if we ignore strong-coupling, magnetic-field andgravitational effects from the dispersion relation (17), weget

ω2

k2=

ω2pd λ2

D (1 − R)1 + k2λ2

D

. (19)

The above expression describes the usual dispersion re-lation for dust acoustic waves in weakly coupled plasmamodified due to the dust polarization force. Equation (19)resembles the result given by Khrapak et al. [19] neglectingthe dust temperature given in that case. If we neglect thepolarization force term, eq. (19) gives the expression fordust acoustic speed c2

da = ω2/k2 = ω2pd λ2

D /(1 + k2λ2D ).

Further incorporating the strong-coupling effects and ig-noring the gravitational, magnetic-field and polarizationeffects from eq. (17) we get

ω2 − V 2cdiωτmk2 −

ω2pdk

2λ2D

1 + k2λ2D

= 0. (20)

The above dispersion relation resembles the one given byKaw and Sen [3] with some limitations. Further, substi-tuting iω = σ, the dispersion relation given by eq. (17)will be

σ2 + σωcd + V 2cdτmk2σ +

ω2pdk

2λ2D (1 − R)

1 + k2λ2D

− ω2Jd = 0.

(21)Equation (21) is a quadratic equation and it has two roots.However, if the constant term is less than zero, then therewill be two real roots of the equation, in which one of theroot is essentially positive, which implies that the consid-ered plasma system is unstable. Therefore, the conditionof the Jeans instability for the above-considered systemcan be written using eq. (21) as

ω2pdk

2λ2D (1 − R)

1 + k2λ2D

< ω2Jd. (22)

The relation shows that the condition of the Jeans instabil-ity in the SCDP gets modified due to the polarization forceand it is not affected due to the presence of magnetic-fieldand strong-coupling effects of dust. From relation (22) itis clear that the configuration is unstable, if the value of

the wave number k becomes less than the critical valuekJ1 given by

kJ1 =

⎧⎨⎩ ω2

Jd(ω2

p d λ2D (1 − R)

)⎫⎬⎭

1/2

. (23)

In determining the above expression, we have assumedthat k λD � 1. If the size of the perturbation is largerthan some critical Jeans wavelength of the order of λJ1 =2π/kJ1, then the enhanced self-gravity can exceed thepressure to grow the perturbations. The correspondingcritical Jeans mass is MJ1 = (4/3) π λ3

J1 md nd0. Substi-tuting the value of λJ1 from (23), we get

MJ1 = (4/3) π (2π)3 mdnd0

{ω2

Jd

ω2pd λ2

D (1 − R)

}−3/2

.

(24)The above expression represents the critical Jeans massrelation modified due to the polarization force of dust. Ifthe growing gravitational perturbation makes the mass ofthe astrophysical objects larger than the Jeans mass, thegravitational collapse occurs. Equation (24) predicts themaximum mass that SCDP can sustain above which itcollapses. We numerically discuss the gravitating modeobtained in the hydrodynamic regime. The effect of eachparameter on the growth rate of the Jeans instability canbe observed solving eq. (21) numerically by normalizing itas

γ2+γ ω∗cd+γ η∗τ∗

mk∗2+Ω2dλ

∗2D k∗2 (1 − R)−1 = 0, (25)

where the following dimensionless parameters have beenused:

ω∗cd =

ωcd

ωJd, γ =

σ

ωJd, η∗ =

V 2cd

c2da

, τ∗m = τmωJd,

λ∗D =

λD ωJd

cda, Ωd =

ωpd

ωJd, k∗ =

kcd a

ωJd. (26)

In fig. 1(A), we show the growth rate of instability againstthe normalized wave number for various values of dimen-sionless magnetic field. It is clearly seen from the figurethat as the value of magnetic field increases, the growthrate decreases and the region of instability also decreases.From the curves it is evident that the growth rate is de-creasing with increasing wave number. Therefore, it is ap-parent that ω∗

cd has a stabilizing influence on the growthrate of the Jeans instability. In fig. 2(A), the effect of thepolarization parameter is shown on the growth rate of theJeans instability. It is evidently seen from the curves thatthe growth rate increases as the value of the polarizationparameter increases. Therefore, it is obvious that the po-larization force shows the destabilizing influence on thegrowth rate of the Jeans instability in the hydrodynamicmode of propagation.

Kinetic regime: In the kinetic regime the frequency ofthe perturbation comes in the range ω τm � 1 which is

15001-p4

Modified Jeans instability of strongly coupled inhomogeneous etc.

Fig. 1: (Colour on-line) The dimensionless growth rate ofthe Jeans instability Re (γ) vs. the dimensionless wave num-ber (k∗) for various values of the magnetic field ω∗

cd in thehydrodynamic case (A), and in the kinetic case (B). Solidblack line, red dashed and blue dotted lines are for ω∗

cd =0.1, 0.2 and 0.3, respectively. The constant parameters areτ∗

m = 0.2, R = 0.1, Ωd = 0.3.

a relatively high-frequency regime. In this situation thefeatures of the wave mode become different from those inthe case of the hydrodynamic regime and the dispersionrelation (16) simplifies to

ω2− iωωcd +ω2Jd−V 2

cdk2−

ω2pdk

2λ2D (1 − R)

1 + k2λ2D

= 0. (27)

This dispersion relation represents a gravitating mode inthe presence of strong-coupling effects and polarizationforce of dust with Maxwellian distributed electrons andions in the kinetic regime. Substituting iω = σ in eq. (27)we get

σ2 + σωcd + V 2cdk

2 +ω2

pdk2λ2

D (1 − R)1 + k2λ2

D

− ω2Jd = 0. (28)

Equation (28) is the quadratic equation and it is clearthat eq. (28) will have at least one positive root and thesystem described by the above equation will be unstable.Accordingly,

V 2cdk

2 +ω2

pdk2λ2

D (1 − R)1 + k2λ2

D

< ω2Jd. (29)

The above expression shows the condition of the Jeans in-stability in the SCDP in the kinetic regime. It is obviousthat in the kinetic regime the magnetic field does not af-fect the Jeans condition of instability, but strong-couplingand polarization effects of dust surely modify the Jeanscondition of instability. From expression (29) it is clearthat the configuration is unstable, if the value of the wavenumber k becomes less than the critical value, kJ2 is given

Fig. 2: (Colour on-line) The dimensionless growth rate ofthe Jeans instability Re (γ) vs. the dimensionless wave num-ber (k∗) for various values of polarization parameter R inthe hydrodynamic case (A) and in the kinetic case (B).Solid black line, red dashed and blue dotted lines are forR = 0.1, 0.2 and 0.3, respectively. The constant parametersare ω∗

cd = 0.1, η∗ = 0.25, τ∗m = 0.2, Ωd = 0.3.

as (under the limits kλD � 1)

kJ2 = ωJd

(V 2

cd + ω2pd λ2

D (1 − R))−1/2

. (30)

The system characterized by eq. (28) will be unstable forall the wave numbers k < kJ2. We solve eq. (28) numer-ically. The dispersion relation (28) is written in dimen-sionless form using the dimensionless parameters definedby eq. (26) as

γ2 + γω∗cd + η∗k∗2 + Ω2

d λ∗2D (1 − R) k∗2 − 1 = 0. (31)

In figs. 1(B), 2(B) we have described the dimensionlessgrowth rates against the dimensionless wave numbers forvarious values of magnetic field and polarization force inthe kinetic regime. In fig. 1(B) the influence of the mag-netic field is shown on the growth rate of the Jeans in-stability. This figure shows that with the increasing valueof the magnetic field, the growth rate decreases, whichmeans that the magnetic field has a stabilizing role on theJeans instability in the kinetic regime. Consequently, wecan state that the magnetic field has a stabilizing influenceon the Jeans instability. In fig. 2(B) we have observed theeffect of the polarization parameter by tracing the curvesbetween the growth rate of the Jeans instability vs. wavenumber. From the curves it is clear that the growth rateincreases as the value of polarization parameter increases.Therefore, it is evident that the polarization force showsthe destabilizing influence on the growth rate of the Jeansinstability in the kinetic regime.

In addition, with the help of eq. (25) and eq. (31), wehave compared the growth rate of the Jeans instability forthe hydrodynamic and kinetic regimes as shown in fig. 3.

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Prerana Sharma

Fig. 3: (Colour on-line) The dimensionless growth rate of theJeans instability Re (γ) vs. the dimensionless wave number(k∗) under hydrodynamic and kinetic limits for the differentvalues of the viscoelastic parameter. Solid and dotted linesare for the hydrodynamic and the kinetic case, respectively.Lines with dots are for η∗ = 0.45 and regular lines are forη∗ = 0.25. The constant parameters are ω∗

cd = 0.1, τ∗m =

0.2, Ωd = 0.3, R = 0.1.

The solid line shows the hydrodynamic case and the dot-ted line shows the kinetic case. From the curves it is clearthat the growth rate decreases with the increase of thewave number. It is clear that the growth rate is slowin the kinetic regime in comparison to the hydrodynamicregime. It is obvious that the decay in the growth of un-stable mode is faster under the kinetic limits rather thanunder hydrodynamic limits. Further, the growth rates arealso compared for the different values of the viscoelasticparameter. The curves with dots show the growth rate inboth the regimes for higher values of the viscoelastic pa-rameter. It is clear from the figure that for a larger valueof the viscoelastic parameter, the growth rate decreasesand hence it has a stabilizing influence on the growth rateof the Jeans instability in both the kinetic and the hydro-dynamic regimes.

Conclusions. – We have studied the effect of magneticfield and polarization force acting on the dust grains of theJeans instability of inhomogeneous SCDP. The Jeans con-dition of instability is found to be modified due to both thedust polarization force and strong-coupling effects of dustin the kinetic regime of propagation, while it is affectedonly due to the polarization force of the dust in the hydro-dynamic regime of propagation. The growth rate of theJeans instability is numerically observed and it is foundthat strong-coupling effects and magnetic field both havea stabilizing influence on the growth rate of the system andthe polarization force has a destabilizing influence on theJeans instability of the SCDP. The growth rates are alsocompared for both hydrodynamic and kinetic limits andit is found that the decay of the growth rate of unstable

Jeans modes is faster in the case of kinetic limits ratherthan of hydrodynamic limits.

∗ ∗ ∗

I would like to thank Prof. R. K. Chhajlani, VikramUniversity, Ujjain, for fruitful discussions. I also gratefullyacknowledge the anonymous referee for offering insightfulcomments. The work is partially supported by Universitygrant commission, New Delhi, India.

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