Dust-ion-acoustic shock waves due to dust charge fluctuation

Physics Letters A 373 (2009) 1287–1289

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Dust-ion-acoustic shock waves due to dust charge fluctuation S.S. Duha ∗ , A.A. Mamun Department of Physics, Jahangirnagar University, Savar, Dhaka, 1342, Bangladesh

a r t i c l e

i n f o

Article history: Received 2 September 2008 Received in revised form 30 October 2008 Accepted 31 January 2009 Available online 5 February 2009 Communicated by F. Porcelli

a b s t r a c t A dusty plasma system containing Boltzmann electrons, mobile ions and charge fluctuating stationary dust has been considered. The nonlinear propagation of the dust-ion-acoustic waves in such a dusty plasma has been investigated by employing the reductive perturbation method. It has been shown that the dust charge fluctuation is a source of dissipation and is responsible for the formation of the dust-ionacoustic shock waves. The basic features of such dust-ion-acoustic shock waves have been identified. The implications of our results in space and laboratory dusty plasmas are discussed. © 2009 Elsevier B.V. All rights reserved.

It has been first theoretically [1] shown that, due to the conservation of equilibrium charge density ni0 e − ne0 e = zd0 nd0 e and the strong inequality ne0  ni0 [where ns0 is the equilibrium particle number density of the species s with s = e, i, d for electrons, ions, dust particles, zd0 is the equilibrium number of electrons residing onto the dust grain surface, and e is the magnitude of the electronic charge], a dusty plasma (with negatively charged static dust grains) supports low frequency dust-ion-acoustic (DIA) waves with phase speed much smaller (larger) than electron (ion) thermal speed. The dispersion relation (a relation between the wave frequency ω and the wave number k) of the linear DIA waves is [1] ω2 = (ni0 /ne0 )k2 C i2 /(1 + k2 λ2De ), where C i = (k B T e /mi )1/2 is the ion-acoustic speed (with T e being the electron temperature and mi being the ion mass) and λDe = (k B T e /4π ne0 e 2 )1/2 is the electron Debye radius. When we consider a long wavelength limit (viz. kλDe  1), the dispersion relation for the DIA waves becomes ω = (ni0 /ne0 )1/2 kC i . This form of spectrum is similar to the usual ion-acoustic wave spectrum [2–4] for a plasma with ni0 = ne0 and T i  T e (where T i is the ion temperature). However, in dusty plasmas we usually have ni0  ne0 and T i  T e . Therefore, a dusty plasma cannot support the usual ion-acoustic waves, but can support the DIA waves of Shukla and Silin [1]. The phase speed (ω/k) of the DIA waves is larger than C i because of ni0  ne0 for negatively charged dust grains. The increase in the phase velocity is attributed to the electron density depletion in the background plasma, so that the electron Debye radius becomes larger. As a result there appears to be a stronger space charge electric field which is responsible for the enhanced phase velocity of the DIA waves. The DIA waves have been observed in laboratory experiments [5,6]. The linear properties of the DIA waves in dusty

*

Corresponding author. E-mail address: [email protected] (S.S. Duha).

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plasmas are now well understood from both theoretical [1] and experimental [5–8] points of view. The nonlinear waves associated with the DIA waves, particularly the DIA solitary [9] and shock waves [10] have also received a great deal of interest in understanding the basic properties of localized electrostatic perturbations in space and laboratory dusty plasmas [5–10,12,13]. The DIA solitary and shock waves have been investigated theoretically by several authors [9–15]. However, all these investigations [9–15] are limited to constant dust grain charge which may not be a realistic situation in space and laboratory devices, since in space as well in laboratory devices the charge on a dust grain is not constant but varies with space and time [8]. Ghosh et al. [16–18] have studied nonlinear propagation of the DIA waves, particularly solitary [16] and shock waves [17] in a dusty plasma by taking into account the dust charge fluctuation. However, the expression for the ion current used by Ghosh et al. [16,17] is valid for Maxwellian distribution of ions, and is, therefore, inconsistent with their dusty plasma model for the study of the DIA waves in which the electron thermal pressure gives rise to the restoring force and the ion mass provides the inertia. Therefore, in the present Letter, we consider an unmagnetized dusty plasma model consisting of the ion fluid, Boltzmann electrons, and charge fluctuating stationary dust, and properly analyze the nonlinear propagation of the DIA waves. We have shown how the effects of the dust grain charge fluctuation significantly modify the nonlinear propagation characteristics of the DIA waves in order to produce monotonic shock waves. It has been shown that the dust charge fluctuation is a source of dissipation and is responsible for the formation of the dust-ion-acoustic shock waves. We consider a one-dimensional, collisionless dusty plasma consisting of electrons, mobile ions, and charge fluctuating stationary dust. We assume that (i) the electrons follow the Boltzmann distribution, (ii) the ions are mobile, and (iii) charge fluctuating neg-

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S.S. Duha, A.A. Mamun / Physics Letters A 373 (2009) 1287–1289

atively charged stationary dust. We consider the propagation of a low phase speed (in comparison with the electron thermal speed) electrostatic perturbation mode whose nonlinear dynamics is described by

where is a smallness parameter (0 < < 1) measuring the weakness of the dispersion, and V 0 is the Mach number (the phase speed normalized by C i ), and expand N i , U i , Φ , and Z˜ d about their equilibrium values in power series of , viz.

∂ ∂ ni + (ni u i ) = 0, ∂t ∂x ∂ ui ∂ ui e ∂φ k B T i ∂ ni + ui =− − , ∂t ∂x mi ∂ x ni mi ∂ x eφ   ∂ 2φ = 4π e ne0 e k B T e − ni + zd nd0 , ∂ x2

Ni = 1 + Ni

(1) (2) (3)

where ni is the ion number density, u i is the ion fluid speed, zd is the number of electrons residing onto the dust grain surface, and φ is the electrostatic wave potential. We consider a simple situation where the dust grain is charged only by the collections of primary plasma particles (electrons and ions). The dust grain immersed in such a plasma will be charged up accordingly to a dynamical equation.

∂ qd = Ie + Ii , ∂t

(4)

where I e (I i ) is the electron (ion) current flowing on the dust grain surface, the electron current I e is mainly due to the electrons, and the ion current I i is due to the mobile ions speed u i . Therefore, I e and I e0 are given by [8]

 I e = −4π rd2 ne e

kB Te 2π me

 I e0 = −4π

rd2 ne0 e

 12

kB Te 2π me

 exp −

 12



e 2 zd



rd k B T e

exp −

e 2 zd0 rd k B T e

(5)

,  ,

(6)

where rd is the radius of the dust which are assumed to be spherical. We assume that the ion fluctuation current ( I i ) is very small in comparison with the electron current ( I e ), i.e., I i  I e . We also assume that I i  I i0 , and at equilibrium I i0  − I e0 . Therefore, Eq. (4) can be written as

∂ z˜ d  − ˜I e /e , ∂t

(7)

where ˜I e is the perturbed part of the electron current ( I e ), and qd = −e ( zd0 + z˜ d ), in which z˜ d is the perturbed part of the zd . Now, introducing normalized variables N i = ni /ni0 , U i = u i /C i , Φ = e φ/k B T e , Z˜ d = z˜ d / zd0 , X = x/λDm (where λDm = 1 (k B T e /4π ni0 e 2 )1/2 ), and T = t ω pi (where ω− = (mi /4π ni0 e 2 )1/2 ), pi one can reduce (1)–(7) to

∂ ∂ Ni + ( N i U i ) = 0, ∂T ∂X ∂Ui ∂Ui σi ∂ N i ∂Φ + Ui =− − , ∂T ∂X ∂X Ni ∂ X ∂ 2Φ = μ exp(Φ) − N i + (1 − μ)(1 + Z˜ d ), ∂ X2   ∂ Z˜ d = ν exp(Φ − α Z˜ d ) − 1 , ∂T

(8) (9)

(1)

(2)

+ Ui

+ ···,

(15)

(1)

2

(2)

+ ···,

(16)

Z˜ d = Z d + Z d + · · · .

(17)

Ui = Ui

Φ = Φ

+ Φ

(1)

2 (2)

Now, substituting (12)–(17) into (8)–(11), we develop equations in various powers of . To the lowest order of we have (1)

Ui

(1)

Ni

= =

V 0 Φ (1)

,

(18)

Φ (1) , V 02 − σi

(19)

V 02 − σi

V 02 − σi = (1)

Zd

=

1

Φ (1)

α

(20)

,

μ+ f

(21)

,

where f = (1 − μ)/α . Eq. (20) represents the linear dispersion relation for the ion-acoustic (IA) shock waves significantly modified by the presence of the charge fluctuating dust. This implies that for no dust ( f = 0) the phase speed of the IA waves is equal to C i . However, in a plasma with negatively charged dust f  3.23 × 109 for the laboratory plasma [6,12,22] conditions (nd0  109 m−3 , rd = 5 μm, ni0  2 × 1013 m−3 , T e  2.4 × 104 K) and f  323 for the space plasma [8] conditions (nd0  10−5 m−3 , rd = 0.5 μm, ni0  5 × 106 m−3 , T e  6 × 105 K). Therefore, it is obvious from (20) that the presence of the charge fluctuating dust significantly reduces the phase speed of the DIA waves, and introduces a new low phase speed (in comparison with the dust-ion-acoustic speed) electrostatic mode associated with the dust charge fluctuation since f  1, for both the space [8] and laboratory [6,12,22] dusty plasma conditions.  phase speed of this dust-ion-acoustic mode  The is roughly C i / f = ni0 e 2 /nd0 rd mi . Therefore, for f  1, which is valid for both the space and laboratory dusty plasma conditions, the phase speed of this dust-ion-acoustic mode is directly √ proportional to e 2 /rd mi and is inversely proportional to nd0 /ni0 . To the next higher order of , one obtains (1)

∂ Ni ∂τ

(2)

− V0

∂ Ni ∂ξ

− V0

∂Ui ∂ξ

(1)

∂Ui ∂τ

(1)

μ −

(2)

+

∂Ui ∂ξ

(2)

= −N i

(10)

where σi = T i / T e , μ = ne0 /ni0 , α = e 2 zd0 /rd k B T e , ν = e −α (2μ × μm μd ne0 /α zd0 )1/2 , where μm = mi /me and μd = rd3 . To derive a dynamical equation for the nonlinear propagation of the electrostatic waves in a dusty plasma under consideration, we use (8)–(11) and employ the reductive perturbation technique [19]. We introduce the stretched coordinates [20,21]

τ = T,

(14)

2

(11)

2

+ ···,

(1)

f

ξ = ( X − V 0 T ),

(2)

+ 2 Ni

Φ (2) +

∂Φ (1)

∂ξ 1 μ 2 f

+

(1) (1) ∂ U i

+ Ui −

Φ (1)

∂  (1) (1)  N Ui = 0, ∂ξ i

∂ξ

∂Φ (2) ∂ξ 2

−

1 f

(22)

(1) (1) ∂ U i

− V 0 Ni

∂ξ

(2)

− σi (2)

Ni

∂ Ni , ∂ξ

(23)

(2)

+ α Z d = 0,

(24)

(1)

(12) (13)

V 0 ∂ Zd

ν

∂ξ

(2)

= Φ (2) − α Z d .

(25) (2)

(2)

(2)

Now, using (18)–(25), one can eliminate N i , U i , Z d , and Φ (2) and can finally obtain

∂Φ (1) ∂Φ (1) ∂ 2 Φ (1) + A Φ (1) , =B ∂τ ∂ξ ∂ξ 2

(26)

where the nonlinear coefficient A and the dissipation coefficient B are given by

S.S. Duha, A.A. Mamun / Physics Letters A 373 (2009) 1287–1289

(3V 02 − σi )(μ + f )3 − μ , 2V 0 (μ + f )2

A= B=

f 2να (μ + f )2

(27) (28)

.

Eq. (26) is the well-known Burgers equation describing the nonlinear propagation of the IA shock waves in the dusty plasma under consideration. It is obvious from (26) and (28) that the dissipative term, i.e., the right-hand side of (26) is due to the presence of the charge fluctuating dust. We are now interested in looking for the stationary shock wave solution of (26) by introducing ζ = ξ − U 0 τ  and τ  = τ , where U 0 is the shock wave speed (in the reference frame) normalized by C i , ζ is normalized by λDm , and τ  is normalized by ω pi −1 . This leads us to write (26), under the steady state condition ∂/∂ τ  = 0, as

−U 0

∂Φ (1) ∂Φ (1) ∂ 2 Φ (1) . + A Φ (1) =B ∂ζ ∂ζ ∂ζ 2

(29)

It can be easily shown that [23,24] (29) describes the shock waves whose speed U 0 (in the reference frame) is related to the extreme values Φ (1) (−∞) and Φ (1) (∞) by Φ (1) (−∞) − Φ (1) (∞) = 2U 0 / A. Thus, under the condition that Φ (1) is bounded at ζ = ±∞, the shock wave solution of (29) is [23,24]

  Φ (1) = Φ (m) 1 − tanh(ζ /Δ) ,

(30)

where Φ (m) = U 0 / A and Δ = 2B /U 0 are, respectively, the height and thickness of the shock waves moving with the speed U 0 . It is obvious that formation of such shock waves is due to the presence of the charge fluctuating static dust. It has been already shown that in a plasma with negatively charged static dust V 02  1/ f = αni0 /zd0nd0 , where f  3.23 × 109 and α  1.54 × 10−10 for the laboratory plasma [6,12,22] conditions (nd0  109 m−3 , rd = 5 μm, ni0  2 × 1013 m−3 , T e  2.4 × 104 K and zd0 = 104 ), whereas f = 323 and α  61.8 × 10−12 for the space plasma [8] conditions (nd0  10−5 m−3 , rd = 0.5 μm, ni0  5 × 106 m−3 , T e  6 × 105 K, and zd0 = 104 ). Since for both the space and laboratory plasma α  1 and f  1, one can express Φ m and Δ as



Φ

(m)

Δ

2U 0



3

2

ni0

k B T e nd0 rd

  e2 √

1 U0

e2

n2i0

2 k B T e rd4 nd0

,

.

(31)

(32)

It is obvious from (31) that the height (normalized by k B T e /e) of the potential structures in the form of the shock waves is directly proportional to the shock speed U 0 and to the square root of the ion number density ni0 , and it is inversely proportional to the square root of the dust radius rd , the dust number density nd0 , and the electron temperature T e . On the other hand, (32) implies that the thickness (normalized by λDm ) of these shock structures is directly proportional to the ion number density ni0 , and it is inversely proportional to the shock speed U 0 , to the square of the dust radius, to the dust number density and to the square root of the electron temperature T e . We note that the DIA shock structures

1289

studied by Popel et al. [25] do not follow our dusty plasma model and mathematical procedures. However, some of the basic features of the shock structures predicted here (by us) are very similar to those studied by Popel et al. [25]. To summarize, the nonlinear propagation of the low phase speed dust-ion-acoustic waves in a dusty plasma containing electrons, mobile ions, and charge fluctuating static dust has been theoretically investigated by the reductive perturbation method. It has been found that the dust charge fluctuation is the source of the dissipation, and is responsible for the formation of the dust-ionacoustic shock waves in a dusty plasma. It has also been shown that the basic features (Mach number, height, and thickness) of such dust-ion-acoustic shock structures are completely different from those of the ion-acoustic shock structures. To conclude, we propose to design a new laboratory experiment with a double plasma (DP) device modified by the dust dispersing set up [6] or modified by the rotating dust dispenser [22], which will be able to detect the dust-ion-acoustic shock structures, and to identify their basic features predicted in this theoretical investigation. Acknowledgements One of the authors (S.S. Duha) gratefully acknowledges the financial support of University Grant Commission (Bangladesh) and also to the Government of the Peoples Republic of Bangladesh for granting the deputation during this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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