Shock Waves in a Dusty Plasma with Positive and Negative Dust, where Electrons Are Superthermally Distributed

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1 Bulg. J. Phys. 38 (0) Shock Waves in a Dusty Plasma with Positive and Negative Dust where Electrons Are Superthermally Distributed S.K. Kundu D.K. Ghosh P. Chatterjee B. Das Department of Mathematics Siksha Bhavana Visva Bharati University Santiniketan India Dhaltikuri K.N. High School P.O. Dhaltikuri Dist. Birbhum India Received 9 July 0 Revised 9 December 0 Abstract. Using the reductive perturbation technique Burgers equation is derived for studying the behavior of dust acoustic (DA) shock waves in an unmagnetized dusty plasma whose constituents are kappa-distributed (superthermal) electrons Boltzmann distributed ions and positively and negatively charged dust particles. The solution of Burgers equation is numerically analyzed. The effects of spectral index κ on the properties of DA shock waves is discussed. It is found that the spectral index κ changes the amplitude of the DA shock waves significantly. The present investigation may be helpful in the understanding of the nonlinear propagation of waves in the pulsar magnetosphere. PACS codes: 5.35.Fp 5.35.Sb Introduction Dusty plasma research is one of the most rapidly growing branches of plasma physics. The presence of dusty plasmas in comet tails asteroid zones planetary ring interstellar medium lower part of the earths ionosphere and the magnetosphere makes this subject rapidly important. Dusty plasma plays a vital role to understanding different types of new and interesting aspects in field like low temperature physics radiofrequency plasma discharge plasma crystals etc. Such plasmas are investigated theoretically and experimentally also [-6] Dusty plasma supports mainly two types of acoustic waves: high frequency dust ion acoustic wave (DIAWs) involving mobile ions and static dust grains and a low frequency dust acoustic wave (DAWs) involving mobile dust grains. Both of these modes have been studied theoretically [7-9] and experimentally [0] c 0 Heron Press Ltd. 409

2 Shock Waves in a Dusty Plasma where Electrons Are Superthermally Distributed Usually the Reductive Perturbation Technique (RPT) is applied to study solitary waves. The discoveries of dust acoustic wave (DAW) give a new impact to study the waves in dusty plasma [3]. In most of the studies on dusty plasmas the dust grains are assumed to be negatively charged. However this assumption is questionable in different realistic situations. V.E. Fortov [4] has shown there are three mechanisms by which the dust grains can be positively charged. These three mechanisms are photoemission induced by radiative heating and secondary emission of electrons from the surface of the dust grains. Recently positively charged dust grains are found in space plasma environments and it is seen that these positively charged grains also play significant role. G. Mandal et al. [5] studied dusty plasmas taking into account both positive and negative dust grains. They found negative as well as positive shock potentials (for electron and ions). Armina et al. [6] have considered a four-component dusty plasma containing positively and negatively charged dust grain and Boltzmann distributed electrons and ions. They investigated the possibility for the formation of shock waves and the existence of shock structures. Most of the research papers which discussed DASWs are based on electrons and ions with Maxwell-Boltzmann distributions. In collisionless plasma however particle velocity distributions can often depart from being Maxwellian. For example in naturally occurring plasma in the planetary magnetosphere and in solar wind the particle velocity distribution is observed to have non- Maxwellian (power-law) high-energy tail [7]. Such distribution may be accurately modelled by a Kappa (or Generalized Lorentzian) distribution [8]. A three-dimensional generalized Lorentzian or κ-distribution function takes the form [ ] F κ (v) = A κ l + v (κ+) κθ where A κ is the normalization constant v = vx + v y + v z θ is the effective thermal speed related to the usual thermal speed v th = (k B T/m) / by θ = [(κ 3 / )/κ]vth and κ is the spectral index. The spectral index κ is a measure of the slope of the energy spectrum of the suprathermal particles forming the tail of the velocity distribution function. Kappa (κ) distribution approaches to Maxwellian as κ tends to infinity. However a few observations in space plasmas [90] indicate clearly the presence of superthermal (kappa-distributed) electron and ion structures in a variety of astrophysical plasma environments. Recently A. Shah et al. [] have studied the shock wave structure with kappadistributed electrons and positrons. They derived Korteweg-de Vries (KdV) Burgers equation and solved it analytically. The effect of plasma parameters on the shock strength and steepness are investigated. Recently B. Sahu [] has investigated the cylindrical or spherical dust-ion acoustic shock waves in an adiabatically dusty plasma and found that the nonplanar geometry effects have a very significant role in the formation of shock waves. In this paper we study a four 40

3 S.K. Kundu D.K. Ghosh P. Chatterjee B. Das component unmagnetized dusty plasma comprising of Boltzmann-distributed ions Kappa-distributed electrons and positively and negatively charged dust grains. We have been particularly interested to observe the effect of the superthermal electrons i.e the effect of spectral index κ on the DA shock waves in our four component dusty plasma system. The paper is designed as follows: basic equations associated with the plasma model are given in Section and the Burgers equation is derived using the RPT in Section 3. Solutions results and discussions are highlighted in Section 4. Section 5 is kept for conclusions. Governing Equations We consider one-dimensional unmagnetized collisionless dusty plasma consisting of superthermal electrons Boltzmann-distributed ions and positively and negatively charged dust grains. The nonlinear dynamics of DA waves is governed by n t + (n u ) = 0 () x u t + u u x = z e ϕ m x + η u x () n t + (n u ) = 0 x (3) u t + u u x = z e ϕ m x + η u x (4) n e n i + z n z n = 0 (5) where n (n ) is the negative (positive) dust number density u (u ) is the negative (positive) dust fluid speed z (z ) is the number of electrons (protons) residing on a negative (positive) dust particles m (m ) is the mass of negative (positive) dust particle e is the electron charge ϕ is the wave potential η (η ) is the viscosity coefficient of negative (positive) dust fluid x is the space variable and t is the time variable n e (n i ) is the electron (ion) number density: ( n e = n e0 eϕ/k T ) κ ( B e κ ni = n i0 exp eϕ ). k B T i Here T e (T i ) is the temperature of electrons (ions) and k B is the Boltzmann constant. In Eq. (5) we have assumed the quasi-neutrality condition at equilibrium. Now in terms of normalized variables namely N = n n 0 N = n n 0 U = u c U = u c ψ = eϕ k B T i T = tw pd X = x λ D η = η w pd λ D η = η w pd λ D 4

4 Shock Waves in a Dusty Plasma where Electrons Are Superthermally Distributed where n 0 (n 0 ) is the equilibrium value of n (n ) c = z k B T i m w pd = we can express Eqs. () (5) as where 4πz e n 0 λ m D = z k B T i 4πz e n 0 N T + (N U = 0 (6) X U T + U ]du X = ψ U T + U U ψ = αβ X N = ( + µ e µ i )N µ e ( σψ κ X + η U X (7) N T + N U = 0 X (8) X + η U X (9) ) κ + µ i exp( ψ) (0) α = z z β = m m µ e = n e0 z n 0 µ i = n i0 z n 0 σ = T i T e. 3 Derivation of Burgers Equation Now we derive the Burgers equation from Eqs. (6) (0) by employing the reductive perturbation technique (RPT) and the stretched coordinates ξ = ǫ / (X V 0 T) and τ = ǫ 3 / T [3] where ǫ is a smallness parameter measuring the weakness of the nonlinearity and V 0 is the phase speed of the DA waves normalized by c we now express (6) to (0) in terms of ξ and τ as ǫ 3 / N τ V 0ǫ / N + / (N U ) = 0 ǫ () ǫ 3 / U τ V 0ǫ / U + / U U ǫ = / ψ ǫ + / U η ǫ3 0 () ǫ 3 / N τ V 0ǫ / N + / (N U ) = 0 (3) ǫ ǫ 3 / U τ V 0ǫ / U + / U U ǫ = / αβ ψ ǫ + / U η ǫ3 0 (4) N = ( + µ e µ i )N µ e ( σψ κ where η = ǫ / η 0 and η = ǫ / η 0 are assumed. 4 ) κ + µ i e ψ (5)

5 S.K. Kundu D.K. Ghosh P. Chatterjee B. Das We can expand the variables N U N U and ψ in a power series of ǫ as N = + ǫn () + ǫ N () + (6) U = 0 + ǫu () + ǫ U () + (7) N = + ǫn () + ǫ N () + (8) U = 0 + ǫu () + ǫ U () + (9) ψ = 0 + ǫψ () + ǫ ψ () + (0) Now substituting (6) (0) into () (5) and taking the coefficient of ǫ 3 / from () (4) and ǫ from (5) we get N () = U() V 0 () U () = ψ() V 0 () N () = U() V 0 (3) U () = (αβ) ψ() V 0 (4) N () = ( + µ e µ i )N () κ + σµ e κ Now using () (5) we get N () = ψ() V 0 N () = (αβ) ψ() V 0 ψ () µ i ψ (). (5) (6) (7) V 0 = (κ )[ + αβ( + µ e µ i )] [σµ e (κ + ) + µ i(κ )]. (8) Equation (8) is the linear dispersion relation for the DA shock waves propagating in our dusty plasma system. Similarly substituting (6) (0) into () (5) and equating the coefficient of ǫ 5 / from () (4) and ǫ from (5) one obtains N () τ N () V 0 + U() + (N() U() ) = 0 (9) U () τ U () V 0 + U () U () = ψ() U () + η 0 (30) 43

6 Shock Waves in a Dusty Plasma where Electrons Are Superthermally Distributed U () τ N () τ U () V 0 N () V 0 + U () + U() U () [ N () = ( + µ e µ i )N () µ e σ κ + κ + (N() U() ) = 0 (3) = αβ ψ() ψ () U () + η 0 (3) + σ (κ + )(κ + 3 ) (κ ) (ψ () ) ] + µ i [ ψ () + (ψ() ) ] (33) Now using () (4) and (6) (8) and eliminating N () N() U() U() and ψ () from the above set of equations we finally obtain ψ () τ () ψ() + Aψ = C ψ () (34) where the nonlinear coefficient A and the dissipation coefficient C are given by {( A = κ ) [3α β ( + µ µ i ) 3] V 0 ( [σ µ e κ + ( {V 0 κ )( κ + 3 C = η 0 + η 0 αβ( + µ µ i ) [ + αβ( + µ e µ i )] ) µ i ( κ ) ]} ) [ + αβ( + µ e µ i )]} (35). (36) Equation (34) is known as Burgers equation which can describe the nonlinear propagation of DA shock waves in our four component dusty plasma system. 4 Numerical Results and Discussions The stationary solution of this Burgers equation is obtained by transforming the independent variables ξ and τ to ζ = ξ U 0 τ and τ = τ where U 0 is a constant velocity normalized by C and imposing the appropriate boundary conditions viz. ψ () ψ () 0 0 at ζ. ζ Thus one can express the stationary solution of the Burgers equation as [9] [ ( ζ ψ () = ψ m () tanh (37) )] 44

7 S.K. Kundu D.K. Ghosh P. Chatterjee B. Das where the amplitude ψ () m (normalized by k B T i /ǫ) and the width (normalized by λ D ) are given by ψ m () = U 0 A (38) = C. U 0 (39) It is observed from Eqs. (37) (39) that the amplitude (width) of the shock waves increases (decreases) as U 0 increases (decreases). It is also clear from Eq. (37) that shock potential profile is positive (negative) when A is positive (negative). Figure exhibits the effect of spectral index κ on the propagation of DA shock waves. It is obvious from figure that for the small changes in spectral index κ exhibits different shock wave structures. The shock wave behavior can be seen from the above figure with positive potential. Later we are showing that κ changes the amplitude of the DA shock waves significantly. Figure describes the structure of the positive DA shock wave profiles for different values of µ e. It is seen from the above figure that DA shock wave potential increases as µ e increases. It is seen from Figure 3 that for β = the DA shock structure becomes different. It is clear from the above figure that positive DA shock wave potential decreases as β increases. We also find that for various small values of the parameter α the DA shock structure becomes different (not shown in figure) Figure. Variation of the DA shock wave potential ψ () with spatial coordinate ζ for different values of spectral index κ where α = 0. β = 500 σ = 0.5 µ i = 0.5 µ e = 0.3 η 0 = 0. η 0 = 0. V 0 = 0 κ = 0.8 (solid line) κ =.0 (dotted line) and κ =.5 (dashed line). 45

8 Shock Waves in a Dusty Plasma where Electrons Are Superthermally Distributed Figure. Variation of the DA shock wave potential ψ () with spatial coordinate ζ for different values of µ e where κ =. V 0 = 0 and the other parameters values are the same as in Figure µ e = 0.3 (solid line) µ e = 0.4 (dotted line) and µ e = 0.5 (dashed line) Figure 3. Variation of the DA shock wave potential ψ () with spatial coordinate ζ for different values of β where κ =. and the other parameters values are the same as in Figure β = 400 (solid line) β = 500 (dotted line) and β = 650 (dashed line). In Figure 4 the variation of the amplitude of the DA shock waves is plotted against spectral index κ for different values of the parameter σ (ratio of ion temperature to electron temperature). It is seen from the above figure that when σ increases enhances the amplitude of the DA shock waves structure. In Figure 5 the variation of the amplitude of the DA shock waves is plotted against spectral index κ for several values of parameter µ e. From the above figure it is clear that an increase in µ e decreases the amplitude of the positive 46

9 S.K. Kundu D.K. Ghosh P. Chatterjee B. Das m Figure 4. Variation of the amplitude (ψ m) of the DA shock waves with spectral index κ for several values of σ i.e. σ = 0.5 (solid line) σ = 0.7 (dotted line) and σ =.0 (dashed line) where α = 0. β = 500 µ i = 0. µ e = 0.5 η 0 = 0.5 η 0 = 0.. m Figure 5. Variation of the amplitude (ψ m) of the DA shock waves with spectral index κ for several values of µ e i.e. µ e = 0.7 (solid line) µ e =.3 (dotted line) and µ e =.3 (dashed line) where α = 0. β = 400 σ = 0.5 µ i = 0. η 0 = 0.5 η 0 = 0.. DA shock potential structures. Here the range of spectral index κ lies between to 5 (i.e. < κ < 5). It is also seen from the figure that positive DA shock potential structure rapidly increases when values of the spectral index κ is small (i.e. κ < ) then DA shock wave potential becomes steady in nature. Figure 6 shows the variation of the width ( ) as a function of the parameter µ e for different values of parameter α. It can be clearly seen from the above figure that the width of the DA shock waves decreases as µ e increases and the higher values of α gives shorter width of the shock waves. 47

10 Shock Waves in a Dusty Plasma where Electrons Are Superthermally Distributed e Figure 6. Variation of the width ( ) of the DA shock waves with µ e for several values of α i.e. α = (solid line) α = 0.0 (dotted line) and α = 0.0 (dashed line) where β = 500 µ i = 0.5 η 0 = 0.5 η 0 = 0.. The parameters chosen in our numerical calculations are completely relevant to different regions of space viz. solar wind interstellar medium auroral zone [45]; pulsar s magnetosphere [6] etc. 5 Conclusions We have studied the nonlinear propagation DA shock waves in an unmagnetized dusty plasma comprising of mobile positively and negatively charged dust particles superthermal electrons and Boltzmann distributed ions. The propagation of the small amplitude nonlinear DA shock waves in the unmagnetized dusty plasma system is considered by analyzing the solution of Burgers equation. Burgers equation is derived by using the standard reductive perturbation method. A detailed numerical analysis of the amplitude width of the DA shock waves is performed by varying different important plasma parameters. It is seen that for the small values of viscosity coefficient of both negative (or positive) dust fluid the spectral index κ and the other parameters α β µ e and σ play significant role in the formation of DA shock waves and existence of shock structures in this unmagnetized plasma system. It would be possible that positive DA shock potential may trap negatively charged dust particles which can attract dust particles of opposite polarity to form larger sized dust or to be coagulated into extremely large sized neutral dust in solar wind interstellar medium auroral zone pulsar s magnetosphere or even in laboratory experiments. In conclusion we have shown here how the basic features of the nonlinear DA shock waves are modified by the presence of positive negative dust particles and the superthermal nature of the kappa distributed electrons. The results which 48

11 S.K. Kundu D.K. Ghosh P. Chatterjee B. Das have been obtained from this investigation would be useful in understanding the properties of DA shock waves in laboratory and in space dusty plasmas. The present results may be useful for understanding the existence of nonlinear potential structures that are observed in different regions of space (viz. solar wind interstellar medium auroral zone [45]). The present investigation may be also helpful in the understanding of the nonlinear propagation of waves in the pulsar s magnetosphere [6]. References [] M. Rosenberg (993) Planet. Space Sci [] A.A. Mamun (998) Phys. Plasmas [3] K. Yoshimura S. Watanabe (99) J. Phys. Soc. Jpn [4] M. Horanyi D.A. Mendis (985) J. Geophys [5] M. Horanyi D.A. Mendis (986) Astrophys. J [6] A. Bouchoule A. Plain L.Ph. Blondeau C. Laure (99) J. Appl. Phys [7] Y. Hase S. Watanabe H. Tanaca (985) J. Phys. Soc. Jpn [8] W.S. Duan Y.R. Shi X.R. Hong (004) Phys. Lett. A [9] A.A. Mamun P.K. Shukla (00) Phys. Plasmas [0] J.C. Johnson N. D Angelo R.L. Merlino (990) J. Phys. D [] N. D Angelo (995) J. Phys. D [] N.N. Rao P.K. Shukla M.Y. Yu (990) Planet Space Sci [3] A. Barkan R.L. Merlino N. D Angelo (995) Phys. Plasmas [4] V.E. Fortov A.P. Nefedov O.S. Vaulina (998) J. Exp. Theor. Phys [5] G. Mandal P. Chatterjee (00) Z. Naturforsch. 65a 85. [6] R. Armina A.A. Mamun S.M.K. Alam (008) Astrophys. Space Sci [7] D. Summers R.M. Thome (99) Phys. Fluids B [8] V.M. Vasyliunas (968) J. Geophys. Res [9] W.C. Feldman J.R. Asbridge S.J. Bame M.D. Montgomery (973) J. Geophys. Res [0] J.D. Scudder E.G. Sittler H.S. Bridge (98) J. Geophys. Res [] A. Shah R. Saeed (0) Plasma Phys. Control. Fusion [] B. Sahu (0) Bulg. J. Phys [3] H. Washimi T. Taniuti (966) Phys. Rev. Lett [4] E.I.El-Awady S.A.El-Tantawy W.M. Moslem P.K. Shukla (00) Phys. Lett. A [5] S.H. Chuang L.N. Hau (009) Phys. Plasmas [6] L. Arons (009) Astrophys. Space Sci. Lib