Nonlinear localized dust acoustic waves in a charge varying dusty plasma with trapped ions

Physics Letters A 372 (2008) 5181–5188

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Nonlinear localized dust acoustic waves in a charge varying dusty plasma with trapped ions Smain Younsi, Mouloud Tribeche ∗ Plasma Physics Group, Theoretical Physics Laboratory, Faculty of Sciences-Physics, U.S.T.H.B, Bab-Ezzouar, B.P. 32, El Alia, Algiers 16111, Algeria

a r t i c l e

i n f o

Article history: Received 17 February 2008 Received in revised form 18 May 2008 Accepted 8 June 2008 Available online 20 June 2008 Communicated by F. Porcelli

a b s t r a c t The problem of nonlinear variable charge dust acoustic waves in a dusty plasma with trapped ions is revisited. The correct non-isothermal ion charging current is presented for the first time based on the orbit motion limited (OML) approach. The variable dust charge is then expressed in terms of the Lambert function and we take advantage of this new transcendental function to investigate nonlinear localized dust acoustic waves in a charge varying dusty plasma with trapped ions more rigorously. © 2008 Elsevier B.V. All rights reserved.

PACS: 52.27.Lw 52.35.Fp 52.35.Sb 52.35.Tc 52.35.Mw

1. Introduction Recently, there has been a great deal of interest in understanding different types of collective processes in dusty plasmas, because of their relevance and vital role in laboratory, space, and astrophysical plasma environments [1–5]. A dusty plasma is a normal electron– ion plasma with an additional highly charged component of small micron or sub-micron sized extremely massive charged particulates (dust grains). Unique and novel features of dusty plasmas when compared with the usual electron–ion plasmas are the existence of a new, ultra-low frequency regime for wave propagation and the highly charging of the grains which can fluctuate due to the collection of plasma currents onto the dust surface. It has been shown both theoretically and experimentally that the presence of these extremely massive and highly charged dust grains in a plasma can either modify the behavior of the usual waves and instabilities or introduce new eigenmodes [6–11]. The most well studied of such modes is the so-called “Dust Acoustic Wave” (DAW) which arises due to the restoring force provided by the plasma thermal pressure (electrons and ions) while the inertia is due to the dust mass. Some recent theoretical work focused on the effects of particle trapping on different types of linear and nonlinear collective processes in charge varying dusty plasmas [12–26]. It has been demonstrated that the presence of such non-isothermal particles can significantly modify the wave propagation characteristics in collisionless charge varying dusty plasmas. However, for the sake of mathematical simplicity and to keep their analysis more tractable analytically, Refs. [12–26] made use of the charging current expressions derived for thermal electrons and ions overlooking the fact that the ions or electrons depart from their thermodynamic equilibrium. The effects of electron or ion non-isothermality on the dust charging enter then through the particle densities and the electrostatic potential appearing in these predetermined currents. Therefore, it seems worthwhile to revisit the nonlinear dust acoustic waves in charge varying dusty plasma with trapped ions and rederive the appropriate trapped ion charging current based on the orbit-limited motion theory. The variable dust charge is then expressed in terms of the Lambert function and we take advantage of this new transcendental function that already enabled us to interpret different waves in charge varying dusty plasmas [27,28]. 2. Theoretical model and basic equations Let us consider a collisionless, unmagnetized three component dusty plasma having electrons, positive ions, and dust grains of density ne , ni , and nd , respectively. Although the size (and thus the charge) of the dust grains varies from one grain to another, we assume for

*

Corresponding author. E-mail address: [email protected] (M. Tribeche).

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simplicity that all the grains have the same negative charge, qd = − Z d e, where Z d in the number of charges residing on the dust grain. On the dust time scale, the electrons are assumed to be in thermal equilibrium, with the density given by



ne (φ) = ne0 exp

eφ



Te

(1)

.

To model the hot non-isothermal ion distribution, we refer to a possible three-dimensional equilibrium state ion velocity distribution function which solves the collisionless Vlasov equation with a population of trapped particles. Accordingly, we employ a vortex-like ion distribution function of Schamel [29] which solves the ion Vlasov equation. Thus, we have F i ( v i ) = F i f ( v i ) + F it ( v i ),

(2)

with



2π T i f

 F it ( v i ) = ni0

 3/ 2

mi

F i f ( v i ) = ni0

mi

 exp −

3/2

2π T i f

mi v 2i /2 + e φ Tif

 for v i >

 −2e φ/mi ,

    mi v 2i /2 + e φ exp −β for 0  v i  −2e φ/mi ,

(3)

Tif

where φ is the electrostatic potential, q j =e,i = ∓e are the charges, T j the temperatures, and m j the masses. The subscript “0” stands for equilibrium values. The subscript f (t ) represents the free (trapped) ion contribution. Here β is a parameter determining the number of trapped ions and its magnitude is defined as the ratio of the free hot ion temperature T i f to the hot trapped ion temperature T it , i.e., |β| = T i f / T it . The introduction of the parameter allows for a different temperature in the Maxwellian distribution for the trapped ions, β = 0 (β = 1) corresponding to a flat-topped (Maxwellian) ion distribution. Note that a hole in the trapped region, corresponding to an underpopulation of trapped ions, is represented by negative β ’s (a vortex-like excavated trapped ion distribution). The dynamics of low phase velocity dust-acoustic oscillations is then governed by the following normalized equations

∂( N d V d ) ∂ Nd + = 0, ∂T ∂X ∂ Vd ∂ Vd ∂Ψ + Vd = −Q d , ∂T ∂X ∂X ∂ 2Ψ Qd = N e − f N i + ( f − 1) Nd , Q d0 ∂ X2

(4) (5) (6)

N d is the dust particle density normalized by  nd0 , N i (e) is the ion (electron) density normalized by ni (e)0 , V d is the dust fluid velocity normalized by the dust-acoustic speed C d = Z d0 T i f /md , Ψ is the electrostatic wave potential normalized by T i f /e, and Q d is the dust charge normalized by rT e /e. The time and space variables are in the units of

1 ω− = (md /4π ne0 Z d0 e 2 )1/2 , the dust plasma period, and pd

the Debye length λ Dm = ( T i f /4π ne0 e ) . Charge neutrality at equilibrium requires f = ni0 /ne0 = 1 + Z 0 nd0 /ne0 . To study the timeindependent arbitrary amplitude dust acoustic waves, we suppose that all dependent variables in Eqs. (4)–(6) depend only on a single variable ξ = X − M T (where, again, ξ is normalized by λ Dm and M = wave speed/C d ), and write our basic set of equations as 2 1/ 2

∂ Nd ∂( V d N d ) + = 0, ∂ξ ∂ξ ∂ Vd ∂ Vd ∂Ψ −M + Vd = −Q d , ∂ξ ∂ξ ∂ξ −M

(7) (8)

∂ 2Ψ Qd = N e − f N i + ( f − 1) Nd . Q d0 ∂ξ 2

(9)

Now, under the appropriate boundary conditions, viz., Ψ → 0, V d → 0, and N d → 1 at ξ → ±∞, Eqs. (7)–(8) can be integrated to give Nd = 

1 1 − 2χ / M 2

(10)

,

where

Ψ

χ=

Q d dΨ .

(11)

0

In the standard orbit-limited probe model for the dust grain [30–32], the latter is charged by the plasma currents at the grain surface. The charging current originates from electrons and ions hitting the grain surface. Accordingly, the variable dust charge qd = −e Z d is determined self-consistently by vd

dqd dx

= Ie + Ii ,

(12)

where I e and I i are the average microscopic electron and ion currents entering the dust grains. The grain current from the thermal electrons is

 I e = −π rd2 e

8T e

π me

 1/ 2 ne (φ) exp(eqd /rT e ),

(13)

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where rd is the grain radius and me the electron mass. For non-isothermal distributions such as Eq. (3), one should first rederive the ion dust charging current by using the orbit-limited motion theory. The latter is obtained by averaging the effective collision cross section σi ( v i , qd ) = π rd2 (1 − 2eqd /mi C v 2i ) for charged ions impacting the dust grains over the ion distribution, and C  rd is the effective grain capacitance. Thus, we have



  

√

− 2e φ/mi

3

Ii = e

F i σi v i d v i = 4π e



∞ v 3i i F it

σ

v 3i i F i f

dv i + √

0

σ

dv i .

(14)

−2e φ/mi

After performing the integrals in Eq. (14), one can obtain the following ion charging current

 I i = eni0 π rd2 where

8T i f

 1/ 2

exp(−βΨ )

π mi

1



β

1

β

Qd

−



σ

 +

  Qd 1 +Ψ − −1 , −1 β σ β 1

(15)

σ = T i f / T e and β > 0 has been assumed. The latter reduces to the well-known thermal ion current  I i = eni0 π rd2

8T i f

 1/ 2

 exp(−Ψ ) 1 −

π mi

Qd

 (16)

σ

for Boltzmann distributed ions (β = 1). Eq. (12) is the additional dynamical equation that is coupled self-consistently to the plasma equations through the plasma currents. We note that the characteristic time for dust motion is of the order of tens of milliseconds for micrometer sized grains, while the dust charging time is typically of the order of 10−8 s. Within the time of charging, the displacement of the grain is thus negligible compared to the spatial scale of the problem. It follows that the charging process can be treated as a local phenomenon, and the convective term on the left-hand side of the charging equation (12) can be neglected. It follows that we have Ie + Ii  0

(17)

and the grains are always at the floating potential. Thus, Eq. (17) leads to

√ Q d = −W

σ /μ β

f g (Ψ )

 exp

σ {exp(−βΨ )(βΨ + 1) + β 2 − 1} β g (Ψ )

 +

h(Ψ ) β g (Ψ )

(18)

where g (Ψ ) = exp(−βΨ ) + β − 1,



h(Ψ ) = σ exp(−βΨ ) + (β − 1)(−βΨ + β + 1)

(19)

and μ = me /mi . The Lambert function [33] also called the omega function, is the inverse function of v (W) = We W and several well-known problems in electrostatic and in quantum mechanics can be solved with greater facility using it. Charge neutrality at equilibrium (Ψ = 0) requires the following constraint

√ f =

σ /μ exp( Q d0 ) . σ − Q d0

(20)

In the following numerical simulation, the value of f is deduced from the latter relation while the remaining other parameters are given first. It may be useful to note that in deriving (18), Eq. (17) has been first cast in the form θ1 (Ψ ) + [ Q d + θ2 (Ψ )] exp(−Q d ) = 0, the solution of which is given by Q d = −W[θ1 (Ψ ) exp(−θ2 (Ψ ))] − θ2 (Ψ ). Now, multiplying both sides of Eq. (9) by dΨ/dξ , and integrating once, we obtain the quadrature 1 2



dΨ

2

dξ

+ V (Ψ ) = 0

(21)

where the Sagdeev [34] or pseudo-potential, for our purpose, reads as V (Ψ ) =

1 − exp(σ Ψ )



σ

+

( f − 1) Q d0

+ f 1 − exp(−Ψ ) erf c

 M2

√

1−

2χ M2



−Ψ −

β

 1/ 2

 −1 

  1  2 √ 1 exp(−βΨ ) erf −βΨ − √ −Ψ 1 − 3/ 2

π

β

 .

(22)

Eq. (21) is integrated numerically using a scheme suitable for stiff problems and assuming a hydrogen plasma. One can numerically solve this equation either as a boundary-value problem or as an initial-value problem. For simplicity, we have solved it as an initial value problem. To start the numerical integration, the initial value Ψ ( X = 0) = 0 and a small edge electric field E 0 = −( ddΨX )( X = 0) = −10−4 are assumed. The electrostatic potential Ψ is given in Fig. 1, which exhibits a spatially localized (soliton-like) structure, as is evident from the well structure of the Sagdeev potential in Fig. 2. Each pick value of Ψ corresponds to a left zero of V (Ψ ) in Fig. 2. The following parameters Q d0 = −1.5, σ = 0.1, and M = 1.1 have been chosen. For the sake of comparison, we have plotted Ψ for various values of the trapping parameter β = 0.1 (dashed line), 0.5 (solid line), and 1 (dotted line). The results reveal that the spatial patterns of the variable charge solitary wave are significantly modified by the trapping effects. With an decrease of β , the pulse amplitude increases while its width is narrowed, i.e., trapping makes the solitary structure more spiky. The dust charge Q d (Fig. 3) adopts the same localized profile and remains negative. Fig. 3 indicates that as β increases, the dust grain charge Q d becomes less negative. The dust grains (Fig. 4) are found to be highly localized. This localization (accumulation) caused by a balance of the electrostatic forces acting on the dust grains is

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Fig. 1. Soliton-like solution for the electrostatic potential Ψ for different values of the trapping parameter β = 0.1 (dashed line), 0.5 (solid line), 1 (dotted line), with Q d0 = −1.5, M = 1.1, and σ = 0.1. The value of f is 1.88.

Fig. 2. Plot of the Sagdeev potential associated with the nonlinear localized structure of Fig. 1.

Fig. 3. Spatial profile of dust grain charge Q d for different values of the trapping parameter β . The values of the parameters are those used for Fig. 1.

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Fig. 4. Spatial profile of the dust grain density N d for different values of the trapping parameter β . The values of the parameters are those used for Fig. 1.

Fig. 5. Subsonic soliton-like solution for the electrostatic potential Ψ for different values of the Mach number M = 0.8 (dashed line), 0.82 (solid line) and 0.84 (dotted line). The values of the parameters are those used for Fig. 1 with β = 0.5.

more effective for higher values of β . Rarefactive variable charge dust acoustic solitons involving cuspid density humps can therefore exist. Next, keeping β at a constant value β = 0.5, the effect of ion trapping on the allowable Mach numbers for the existence of soliton-like solutions is investigated. Interestingly, one finds that due to ion non-isothermality, our present dusty plasma model can support subsonic dust acoustic solitary waves (Fig. 5). Let now consider a situation in which I e + I i = 0. Let us then look at a simple case, in which the dust component is a cold beam of particles, each particle having the same speed at a given position. Thus, we choose [35] f d (x, v d ) = nd0

v d0 v˜ d

δ( v d − v˜ d ),

(23)

where

 v˜ d = v d0 1 −

2 2 md v d0

1/2

φ qd d φ

.

(24)

0

Integrating the dust distribution function f d over all velocity space, we find Nd =

1

(1 − γ χ )1/2

(25)

where

Ψ

χ=

Q d dΨ 0

(26)

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Fig. 6. Soliton-like solution for the electrostatic potential Ψ with Q d0 = −1.8, T e = 1.5 eV, σ = 0.2, ni0 = 8 × 1012 cm−3 , md /mi = 1012 , v d0 = 40 cm s−1 , r = 6 μm, and β = 0.9. The values of k and f are respectively 3.56 × 103 and 1.58. Solid line: the charge on the dust grain is constant. Dashed line: the charge on the dust grain is variable.

Fig. 7. Soliton-like solution for the electrostatic potential Ψ (in the variable dust charge case) with Q d0 = −1.4, T e = 1.5 eV, σ = 0.2, ni0 = 8 × 1012 cm−3 , md /mi = 1012 , p v d0 = 40 cm s−1 , r = 6 μm, and β = 0.1. The values of k and f are respectively 2.60 × 103 and 2.95. Solid line: I i is used. Dashed line: I i is used.

and

γ=

2r σ T e2 2 2 md v d0 e

(27)

.

The Poisson and charging equations become, respectively, d2 Ψ dX2

= N e − f N i + ( f − 1)

Qd Q d0

Nd ,

(28)

  1/ 2       1 1 1 Qd 1 Qd 1 dQd + , = kN d − exp(σ Ψ ) exp( Q d ) + f exp(−βΨ ) − −1 − −1 Ψ + dX σμ β β σ β σ β where

 k=

(29)

2ne0 e 2 r 2 σ 2

(30)

2 mi v d0

and X = x/λ Dm = x/( T i f /4π ne0 e 2 )1/2 . Eqs. (26), (28), and (29) are again integrated numerically. Fig. 6 shows that when the charge of the dust grain is variable, the depth of the localized potential structure increases while its width is narrowed. Let us now explain what is the main significance between our derived current I i [Eq. (15)] and what has been discussed before in p Refs. [12–26] (where the following predetermined current I i = π rd2 e (8T i f /π mi )1/2 ni (φ)(1 − eqd /rT i f ) has been used). To this end, we plot p

in Fig. 7 the soliton-like solution for the electrostatic potential Ψ for the two different currents I i (solid line) and I i (dashed line). Our

S. Younsi, M. Tribeche / Physics Letters A 372 (2008) 5181–5188

5187

Fig. 8. Shock-like solution for the electrostatic potential Ψ in the case of variable charge dust grain for different values of the trapping parameter β . The values of the remaining parameters are those used for Fig. 8 with ni0 = 104 cm−3 . The values of k and f are respectively 0.092 and 2.95. p

results reveal that the use of I i (instead of I i ) leads to soliton-like solution which shows larger amplitude. This additional enlargement is more significant for lower values of the trapping parameter β (i.e., as the ions evolve far away their thermodynamic equilibrium and deviate from isothermality). Fig. 8 shows that under certain conditions the effect of dust charge variation can be quite important: the dust charge variation provides an alternate physical mechanism causing dissipation and as a consequence causes the wave amplitude to decay and transfer to the socalled noise tail. A similar effect has been reported in a recent paper dealing with the effects of non-adiabaticity of dust charge variation on the generation of dust acoustic shock waves [36]. This is a collisionless shock wave in the sense that no viscous or damping effects resulting from collisions between dust and plasma particles are involved [37]. The influence of β on the nature of this shock structure (monotonic or oscillatory behavior) and its front height is clearly displayed in Fig. 8. We note the effect of separation of charges which is manifested by the appearance of some oscillations in the shock wave profile (dispersion-dominant case). This effect decreases when the ions deviate from isothermality (anomalous dissipation-dominant case). We have then carried a numerical investigation over a wide range of plasma parameters and noticed that the anomalous damping is intimately related to the constant k [see (30)]: larger values of k favor the development of coherent structures, whereas smaller ones are required for the existence of dissipative structures. To conclude, we have revisited the problem of nonlinear dust acoustic waves in charge varying dusty plasmas with trapped ions. The appropriate trapped ion current has been derived based on the orbit-limited motion theory. Considering the wide relevance of nonlinear oscillations, the results contained in the present article should help to understand the salient features of coherent nonlinear structures in non-equilibrium plasmas which contain particle distributions that are far away from the Maxwellian. Such non-isothermal particle distributions and associated electrostatic structures are frequently found in space and laboratory experiments. We hope they should also enhance our knowledge on nonlinear dust phase-space vortices (holes) in charge varying dusty plasmas and dusty plasma sheaths where voids are formed. Acknowledgements This work was supported in part by the Ministère de l’Enseignement Supérieur et de la Recherche Scientifique Contract No. D00220060041. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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