Dust-acoustic solitary waves in an adiabatic hot dusty plasma

Physics Letters A 372 (2008) 884–887 www.elsevier.com/locate/pla

Dust-acoustic solitary waves in an adiabatic hot dusty plasma A.A. Mamun Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Received 22 June 2007; accepted 16 August 2007 Available online 13 October 2007 Communicated by F. Porcelli

Abstract An adiabatic hot dusty plasma (containing non-inertial adiabatic electron and ion fluids, and negatively charged inertial adiabatic dust fluid) is considered. The basic properties of arbitrary amplitude dust-acoustic (DA) solitary waves, which exist in such an adiabatic hot dusty plasma, are explicitly examined by the pseudo-potential approach. To compare the basic properties (critical Mach number, amplitude and width) of the DA solitary waves observed in a dusty plasma containing adiabatic electron, ion and dust fluids with those observed in a dusty plasma containing isothermal electron and ion fluids and adiabatic dust fluid, it has been found that the adiabatic effect of inertia-less electron and ion fluids has significantly modified the basic properties of the DA solitary waves, and that on the basic properties of the DA solitary waves, the adiabatic effect of electron and ion fluids is much more significant than that of the dust fluid. © 2007 Elsevier B.V. All rights reserved. PACS: 52.35.Fp; 52.35.Lv; 52.35.Ra; 52.35.Mw

The physics of charged dust particles, which are ubiquitous in space [1–3] and laboratory [3–5] plasmas, has received a great deal of interest in understanding electrostatic density perturbations and potential structures that are observed in space environments and laboratory devices. It has been shown both theoretically [6] and experimentally [7] that in an unmagnetized dusty plasma the dust charge dynamics introduces a new eigenmode, namely dust-acoustic (DA) waves [6,7], where the dust particle mass provides the inertia and the pressures of inertialess electrons and ions give rise to the restoring force. Mamun et al. [8] have investigated the DA solitary waves in a two-component unmagnetized dusty plasma consisting of a negatively charged cold dust fluid and an inertia-less isothermal ion fluid. The work of Mamun et al. [8] is only valid when a complete depletion of electrons onto the dust grain surface is possible. A number of theoretical investigations [9–12] have been made of the DA solitary waves in order to generalize the work of Mamun et al. [8] by assuming a three-component unmagnetized dusty plasma consisting of a negatively charged cold dust fluid and inertia-less isothermal electron and ion flu-

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ids. These works are only valid for a cold dust and isothermal electrons and ions. Recently, the effects of the dust fluid temperature on the DA solitary waves have been investigated by a number of authors [13–16]. Roychoudhury and Mukherjee [13] considered a two-component unmagnetized dusty plasma consisting of a negatively charged adiabatic dust fluid and an inertia-less isothermal ion fluid, and investigated the effects of dust fluid temperature on large amplitude solitary waves by the pseudo-potential approach [18]. Mendoza-Briceño et al. [14] assumed a two-component dusty plasma containing the adiabatic dust fluid and non-adiabatic ions following the nonthermal distribution of Cairns et al. [17], and studied the effect of the dust fluid temperature on the DA solitary waves by the pseudo-potential approach [18]. Gill et al. [15] assumed a dusty plasma containing the adiabatic dust fluid and non-adiabatic ions following the bi-Maxwellian distribution of Nishihara and Tajiri [19], and studied the effect of the dust fluid temperature on the DA solitary waves by the pseudo-potential approach [18]. Sayed and Mamun [16] assumed a dusty plasma containing the adiabatic dust fluid and non-adiabatic (isothermal) inertia-less electron and ion fluid, and studied the effect of the dust fluid temperature on the DA solitary waves by the reductive perturbation method [20]. It is obvious that all these investiga-

A.A. Mamun / Physics Letters A 372 (2008) 884–887

tions [13–16] are concerned with different dusty plasma models which are not consistent (appropriate) in general. The inconsistency of all these dusty plasma models arises from the consideration of one component (dust) being adiabatic, and other components (electrons or ions or both) being non-adiabatic. Therefore, in the present work a consistent dusty plasma model, which assumes a dusty plasma containing non-inertial adiabatic electron and ion fluids, and negatively charged inertial adiabatic dust fluid, has been considered in order to perform a proper investigation of the basic properties of arbitrary amplitude DA solitary waves by the pseudo-potential approach [18]. The dynamics of the DA waves in one-dimensional form in such an adiabatic hot dusty plasma is governed by ∂ ∂ns + (ns us ) = 0, ∂t ∂x ∂ps ∂us ∂ps + us + γps = 0, ∂t ∂x ∂x ∂Ψ ∂pe = ne α , ∂x ∂x ∂Ψ ∂pi = −ni , ∂x ∂x ∂ud ∂Ψ σ ∂pd ∂ud + ud = − , ∂t ∂x ∂x nd ∂x

(1) (2)

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at ξ → ±∞) in (1)–(5), one can express ne , ni , and nd as  1 2 2 ne = 1 + αΨ , 3  1 2 2 ni = 1 − Ψ , 3 1 Mσ nd = √ (Ψ1 − Ψ2 ) 2 , 6σ where Ψ1 = 1+2Ψ/Mσ2 , Ψ2 =

(7) (8) (9) 

Ψ12 − 12σ M 2 /Mσ4 , and Mσ =

(M 2 + 3σ )1/2 . Again, using the transformation ξ = x − Mt, substituting (7)–(9) into (6), multiplying the resulting equation by dΨ/dξ , and applying the boundary condition, dΦ/dξ → 0 at ξ → ±∞, one obtains   1 dΨ 2 + V (Ψ ) = 0, (10) 2 dξ

(3)

where V (Ψ ) is given by

(4)

 3 3  2 2 μe 2 2 V (Ψ ) = C − − μi 1 − Ψ 1 + αΨ α 3 3

(5)

∂ 2Ψ = μ e n e − μi n i + nd , (6) ∂x 2 where ns is the number density of species s (with s = e for the electron fluid, s = i for the ion fluid and s = d for the dust fluid) normalized by its equilibrium value ns0 , us is the fluid speed normalized by cd = (kB Ti0 /md )1/2 , Ψ is the wave potential normalized by kB Ti0 /e, ps is the fluid thermal pressure normalized by ns0 kB Ts0 , γ is the adiabatic index, α = Ti0 /Te0 , σ = Td0 /Zd Ti0 , μ = ne0 /ni0 , ns0 is the equilibrium fluid number density, Ts0 is the equilibrium fluid temperature, Zd is the number of electrons residing on a dust grain surface, md is the dust particle mass, kB is the Boltzmann constant, −e is the electronic charge, μe = μ/(1 − μ), and μi = 1/(1 − μ). The time and space variables t and x are normalized by −1 = (md /4πe2 Zd2 nd0 )1/2 and λD = (kB Ti0 /4πe2 Zd2 nd0 )1/2 , ωpd respectively. It is important to mention here that for an isothermal process γ = 1 and ps = ns kB Ts with constant Ts (i.e. Ts = Ts0 ), and hence (1) and (2) are identical. It is also important to note that for isothermal processes, (3) and (4) reduce to ne = exp(Ψ/α) and ni = exp(−Ψ ) which are used by Roychoudhury and Mukherjee [13], Mendoza-Briceño et al. [14], Gill et al. [15], and Sayed and Mamun [16]. To consider an adiabatic hot dusty plasma, one cannot not use γ = 1 and ps = ns kB Ts with constant Ts . Therefore, in the present work (1D problem) γ = 3 and Ts = const are used to study arbitrary amplitude DA solitary waves in an adiabatic hot dusty plasma by the pseudo-potential approach [18]. Now, using the transformation ξ = x − Mt (where M is the Mach number, solitary wave speed/cd ), the steady state condition (∂/∂t = 0), and the appropriate boundary conditions for localized perturbations (viz. ns → 1, us → 0, ps → 1 and Ψ → 0

1 3 3 M M3 − √ Mσ (Ψ1 + Ψ2 ) 2 − 2 2 σ 3 (Ψ1 + Ψ2 )− 2 , (11) Mσ 2

where the integration constant C = μe /α + μi + σ + M 2 is chosen in such a way that V (Ψ ) = 0 at Ψ = 0. Eq. (10) can be regarded as an “energy integral” of an oscillating particle of unit mass, with pseudo-speed dΨ /dξ , pseudo-position Ψ , pseudotime ξ , and pseudo-potential V (Ψ ). This equation is valid for arbitrary amplitude stationary DA solitary waves in an adiabatic hot dusty plasma. The expansion of V (Ψ ) around Ψ = 0 is V (Ψ ) = C2 Ψ 2 + C3 Ψ 3 + · · · ,

(12)

where

  1 + αμ 1 1 − , 2 M 2 − 3σ 3(1 − μ)   1 3(M 2 + σ ) 1 − α2μ + . C3 = − 6 (M 2 − 3σ )3 9(1 − μ)

C2 =

(13) (14)

First, let us consider small amplitude DA solitary waves for which V (Ψ ) = C2 Ψ 2 + C3 Ψ 3 holds good. This approximation allows us to write the small amplitude solitary wave solution of (10) as Ψ = Ψo sech2 (ξ/Δ),

(15)

where Ψo = −C2 /C3 and Δ = (−4/C2 are the amplitude and the width of the DA solitary waves, respectively. This means that when C2 < 0, i.e.    1 2 1−μ M > Mc = 3 (16) + 3σ , 1 + αμ )1/2

the DA solitary waves with negative potential exist since C3 (M = Mc ) is always negative [one can easily prove it from (14) and (16)] for all possible values of μ, α and σ which are

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A.A. Mamun / Physics Letters A 372 (2008) 884–887

Fig. 1. The variation of Mc with μ and α for a fixed value of σ = 10−4 .

positive and are always less than one. Therefore, Mc is the critical value of M above which the DA solitary waves exist. It is obvious from a comparison between (16) and the expression for critical Mach number given in the existing published literatures [13–16] that the factor 3 in front of the term (1 − μ)/(1 + αμ) arises just due to the adiabatic effect of electron and ion fluids. This factor 3 would not arise if one would consider isothermal electron and ion fluids. Since [(1 − μ)/(1 + αμ)]  σ for both space and laboratory dusty plasmas (μ = 0.1–0.9, α = 0.1–0.9, and σ = Td0 /Zd Ti0 = 10−5 –10−3 ), it is obvious that due to the adiabatic effect of electron and √ ion fluids the critical Mach number is increased by a factor 3, and that the effect of dust fluid temperature is almost negligible in comparison with that of the adiabatic electron and ion fluids. However, it is obvious from (16) that for fixed values of μ and α the critical Mach number (Mc ) increases with the increase of the dust fluid temperature. The variation of Mc with μ and α for a fixed value of σ = 10−4 , i.e. the combined effect of μ and α on Mc is shown in Fig. 1. It is observed here that the critical Mach number (Mc ) decreases with the increase of both μ and α. It may be noted that Mc  1.635 for μ = 0.1, α = 0.1, and σ = 10−5 , and Mc  0.411 for μ = 0.9, α = 0.9, and σ = 10−3 . The inspection of (13) and (14) implies that the factor 1/3 in front of the term (1 + αμ)/(1 − μ) in (13), and the factor 1/9 in front of the term (1 − α 2 μ)/(1 − μ) in (14), arise just due to the adiabatic effect of electron and ion fluids. These factors 1/3 and 1/9 would not arise if one would consider isothermal electron and ion fluids. Therefore, the adiabatic effect of electron and ion fluids on the amplitude and the width of the DA solitary waves is very significant. Since 3σ [= Td0 /Zd Ti0 ]  M 2 , on the DA solitary waves the adiabatic effect of the dust fluid is almost negligible in comparison with that of electron and ion fluids. To observe how the other basic properties (amplitude and width) of these DA solitary waves are modified by the combined effects of μ and α, the amplitude (Ψo ) and the width (Δ) are numerically analyzed. The results are displayed in Figs. 2 and 3, which indicate that the amplitude (width) increases (decreases) with the increase of μ and α. Next, let us investigate the basic properties of arbitrary amplitude DA solitary waves by the numerical analysis of the pseudo-potential V (Ψ ) given by (11). It is clear from (12) that V (Ψ ) = dV (Ψ )/dΨ = 0

Fig. 2. The variation of Ψo with μ and α for M = Mc + 0.1 and σ = 10−4 .

Fig. 3. The variation of Δ with μ and α for M = Mc + 0.1 and σ = 10−4 .

at Ψ = 0. Therefore, solitary wave solutions of (10) exist if (i) (d 2 V /dΨ 2 )Ψ =0 < 0, i.e. C2 < 0, so that the fixed point at the origin is unstable, and (ii) V (Ψ ) < 0 when 0 > |Ψ | > |Ψm |, where |Ψm | is a non-zero value of Ψ for which V (Ψm ) = 0, and Ψm is the amplitude of the solitary waves. The condition (i) is satisfied when M > Mc , where Mc is defined by (16) and its variation with μ and α is shown in Fig. 1. To examine whether the conditions (i) and (ii) are simultaneously satisfied, V (Ψ ) [given in (11)] is numerically analyzed using typical plasma parameters, namely μ = 0.1 − 0.9, α = 0.1–0.9, σ = 10−5 –10−3 , and M > Mc . One can easily show by the numerical analysis of V (Ψ ) [given in (11)] that the DA solitary waves exist only with negative potential (Ψ < 0), but not with positive potential (Ψ > 0). A part of the numerical analysis, showing the formation of the potential wells in the negative Ψ -axis, i.e. showing the existence of the DA solitary waves with negative potential, is displayed in Figs. 4 and 5. Fig. 4 shows the formation of the potential wells in the negative Ψ -axis, which corresponds to the formation of the DA solitary waves with negative potential, for μ = 0.1, α = 0.1, σ = 10−5 , M = 1.64 (solid curve), M = 1.67 (dotted curve), and M = 1.70 (dash curve). Fig. 5 shows the formation of the potential wells in the negative Ψ axis, which corresponds to the formation of solitary waves with

A.A. Mamun / Physics Letters A 372 (2008) 884–887

Fig. 4. Showing how the potential well starts to form in the negative Ψ -axis when M exceeds Mc = 1.635, where μ = 0.1, α = 0.1, σ = 10−5 , M = 1.64 (solid curve), M = 1.67 (dotted curve), and M = 1.70 (dash curve).

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easily show the existence of large, even extremely large amplitude DA solitary waves just by increasing the value of M. It is found from this visualization (after a more numerical analysis with different values of μ and α, which are not shown) that the variation of the amplitude and the width with μ and α in the case of arbitrary amplitude DA solitary waves are exactly the same as that in the case of small amplitude DA solitary waves. The ranges of different dusty plasma parameters used in the present numerical analysis are very wide (μ = 0.1–0.9, α = 0.1–0.9, and σ = 10−5 –10−3 ), and are relevant to both space [1–3] and laboratory [3–5] dusty plasmas. Thus, the results of the present investigation should help us to explain the basic features of localized electro-acoustic perturbations propagating in space [1–3] and laboratory [3–5] dusty plasmas. References

Fig. 5. Showing how the potential well starts to form in the negative Ψ -axis when M exceeds Mc = 0.411, where μ = 0.9, α = 0.9, σ = 10−3 , M = 0.42 (solid curve), M = 0.44 (dotted curve), and M = 0.46 (dash curve).

negative potential, for μ = 0.9, α = 0.9, σ = 10−3 , M = 0.42 (solid curve), M = 0.44 (dotted curve), and M = 0.46 (dash curve). Figs. 4 and 5 can also provide a visualization of the amplitude (Ψm ) and the width [|Ψm |/|Vm |, where |Vm | is the minimum value of V (Ψ ) in the potential wells formed in the negative Ψ -axis]. Figs. 4 and 5, where the values of M are around its critical value (Mc ), indicate the existence of small amplitude DA solitary waves. However, in the same way, one can

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