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Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma
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Advances in Space Research xxx (2013) xxx–xxx www.elsevier.com/locate/asr
Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma Hai-Feng Liu (刘海峰) a,b,⇑, Shi-Qing Wang (王世庆) a,b, Zhi-long Wang (王志龙) b, Fa-Zhan Yang (杨发展) a, Yao-Liu (刘耀) b, Sili-Li (李思丽) b b
a Southwestern Institute of Physics, Chengdu 610041, China The Engineering and Technical College of Chengdu University of Technology, Leshan 614000, China
Received 3 September 2012; received in revised form 3 January 2013; accepted 15 January 2013
Abstract The properties of cylindrical and spherical dust–ion acoustic solitary waves (DIASW) in an unmagnetized dusty plasma comprising of relativistic ions, Boltzmann electrons, and stationary dusty particles are investigated. Under a suitable coordinate transformation, the cylindrical KdV equation can be solved analytically. The change of the DIASW structure due to the effect of geometry, relativistic streaming factor, ion density and electron temperature is studied by numerical calculation of the cylindrical/spherical Kdv equation. It is noted that with ion pressure the effect of relativistic streaming factor to solitary waves structure is different. Without ion pressure, as the relativistic streaming factor decreases, the amplitude of the solitary wave decreases. However, when the ion pressure is taken into account, the amplitude decreases as the relativistic streaming factor increases and is highly sensitive to relativistic streaming factor. Our results may have relevance in the understanding of astrophysical plasmas. Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Solitary waves; Unmagnetized dusty plasma; Cylindrical/spherical Kdv equation; Ion pressure; Relativistic streaming factor; Electron temperature
1. Introduction The linear and nonlinear propagation of electrostatic excitations in dusty plasmas have received considerable attention in the last few decades. Ion acoustic waves in unmagnetized dusty plasmas with phase velocity between the electron and ion thermal velocities were first studied theoretically by Shukla and Silin (1992) and later their existence was confirmed experimentally by Barkan et al. (1995). Dust being a common species in a wide range of ⇑
Corresponding author. Address: Southwestern Institute of Physics, People’s south road in three sections, Chengdu 610041, China. Tel.: +86 15 681 386 018. E-mail addresses: [email protected] (H.-F. Liu (刘海峰)), [email protected] (S.-Q. Wang (王世庆)), [email protected] (Z.-l. Wang (王志龙)), [email protected] (F.-Z. Yang (杨发展)), [email protected] ( Yao-Liu (刘耀)), [email protected] ( Sili-Li (李思丽)).
space and astrophysical plasmas such as the cometary tails and comae, interstellar clouds, Earth’s mesosphere and ionosphere, Saturn’s rings, the gossamer ring of Jupiter, and in laboratory experiments (Barkan et al.,1995; Homann et al.,1997; Mendis and Rosenberg, 1994; Northrop, 1992), the study of dusty plasmas has been an important focus of much recent research. Dusty particles are often of micron to submicron size, with masses in the range of 106 1012 proton masses (Shukla and Mamun, 2002; Verheest, 2000), and are usually found to have negative charge (possibly as large as 104 electron charges), depending on the environment where they occur. On the other hand, smaller dust grains may be found to be positively charged. The presence of the dust modifies the standard ion acoustic mode, giving rise to what is termed the dusty ion acoustic wave (DIAW). One of the important effects of the massive dust grains is that the associated change in the free electron density yields a corresponding
0273-1177/$36.00 Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2013.01.028
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
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H.-F. Liu (刘海峰) et al. / Advances in Space Research xxx (2013) xxx–xxx
change in shielding and hence in the detailed wave behavior. The linear DIAW has an increased phase velocity when the dust is negatively (positively) charged. The DIAW was observed by Nakamura et al. (1999, 2002) in a collisional dominated dusty plasma. Theoretical models for the DIAW is given by in a weakly correlated dusty plasma. The study of DIAW in a strongly coupled dusty plasma is also presented by Shukla and Mamun (2001). Xiao et al. (2008) and Liu et al. (2010) have investigated the finite amplitude nonlinear DIAW in an inhomogeneous dusty plasma. Under the assumption of negligible dust-charge perturbation, a critical point, at which the coefficient of nonlinear term changes from negative to positive, exists. Consequently the nonlinear wave undergoes structural deformation. Roy et al. (2008) have studied DIAW in quantum dusty plasma. Their main findings are that system can sustain both oscillatory and monotonic shock waves depending on quantum parameters (H), there exists a limiting value of H at which shock wave breaks up to solitary wave. Xue (2003a,b) studied spherical and cylindrical DIAW and observed that wave height decreases with an increase in ion temperature and that the cold ion acoustic shock wave has the largest height. Several authors (Melandsø and Shukla, 1995; Popel et al., 2000; Popel et al., 1996; Gupta et al., 2001; Ghosh et al., 2002; Das et al., 1997) show that, when ion viscosity or Landau damping effects are not important in dusty plasmas, the nonadiabatic dusty charge variation provides an alternate physical mechanism causing dissipation, and as a consequence this gives rise to shocks for which both monotonic and oscillatory structures are possible. However, most of these studies are confined to the nonrelativistic plasmas and the unbounded planar geometry, which may not be a realistic situation in laboratory devices and space. When the particles velocities are comparable to the speed of light, relativistic effects can no longer be ignored. Relativistic plasmas occur in space plasma phenomena (Grabbe, 1989), plasma sheet boundary layer of earth’s magnetosphere (Vette, 1970), in the Van Allen radiation belts (Ikezi, 1973) and in laser–plasma interaction (Arons, 1979). Relativistic ions and non-relativistic electrons also occur (Riordan and Zajc, 2006; Harrison et al., 2003; Casarejos, 2011). Many theoretical studies for dust–ion acoustic and ion acoustic waves in nonplanar geometry show (Mamun and Shukla, 2001, 2002, 2009a,b; Mamun, 2008; Xue, 2003a,b; Liu et al., 2010) that the properties of these waves in bounded nonplanar cylindrical/spherical geometry are very different from that in unbounded planar geometry. Recently, Tasnim et al. (2012) investigated the effects of two temperature nonthermally distributed ions and dust kinematic viscosity, which are found to significantly modify the basic features of dust acoustic shock waves. Masud et al. (2012a,b) studied propagation of Gardner solitons in a nonplanar (cylindrical and spherical) geometry by deriving the modified Gardner equation. The basic features (amplitude, width, etc.) of the hump (positive potential) and dip (negative potential) shaped DIA solitons (Gardner solitons, i.e.,
GSs) are found to exist beyond the Korteweg-de Vries (K-dV) limit. These DIA-GSs are qualitatively different from the K-dV and modified K-dV solitons (Masud et al., 2012a,b). In this paper, we investigate the properties of the nonplanar geometry on DIASW in an unmagnetized dusty plasma with relativistic ions, Boltzmann electrons, and stationary dusty particles. By using the standard reductive perturbation method (RPM), a cylindrical/spherical Kdv equation is obtained. Under a suitable coordinate transformation the cylindrical KdV equation turns into the ordinary KdV equation which can be solved analytically. The effects of nonplanar geometry, ion density, relativistic streaming factor, and ion temperature on DIASW structures are studied by numerical calculation of the cylindrical/spherical Kdv equation. 2. Basic set of equations To study cylindrical and spherical acoustic solitary waves in a relativistic dusty plasma, we assume that the DIASW propagate in an axial symmetry cylindrical geometry filled with the unmagnetized collisionless dusty plasma. In equilibrium, the charge neutrality condition is ne0 nio + Zdnd0 = 0, where ne0, nio , and nd0 are the unperturbed electron, ion, and dust number densities, respectively, and Zd is the number of electrons residing on the dust grains. The governing equations of the physical phenomenon under consideration consist of the conservation laws of mass and momentum for relativistic ions and Poisson’s equation. @ni 1 @ðrm ui ni Þ ¼0 þ @r @t rm @ðcui Þ @ðcui Þ @/ r @ni þ ui ¼ 3 @t @r @r ni @r d
1 @ m @/ ðr Þ ¼ expð/Þ dni þ ðd 1Þ rm @r @r
ð1Þ ð2Þ ð3Þ
where m = 0 for a one-dimensional geometry and m = 1(2) for a nonplanar cylindrical (spherical) geometry and r ¼ T i =T e , d = nio/ne0. The variables of time (t), space (r), ion number density (ni), ion fluid velocity (ui), and electrostatic wave potential (u) are normalized to the reciprocal pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pni0 e2 =mi ), Debye raion plasma frequency x1 pi (xpi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dius kD (kD ¼ k B T e =4pni0 e2 ), unperturbed equilibrium plasma density nio, effective ion acoustic velocity Ci pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (C i ¼ k B T e =mi ), and kBTe/e respectively. The relativistic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi factor is given as c ¼ 1 u2i =c2 , where c is the speed of light. The relativistic factor for a weakly relativistic regime can be approximated as c 1 þ u2i =2c2 . 3. Derivation and solution of K-dV equation In order to investigate the DIASW in such a plasma, We employ the RPM to derive the dynamical equation for the nonlinear propagation of the electrostatic waves under
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
H.-F. Liu (刘海峰) et al. / Advances in Space Research xxx (2013) xxx–xxx
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consideration. The RPM is mostly applied to small amplitude nonlinear waves (Taniuti and Wei, 1968). This method rescales both space and time in the governing equations of the system in order to introduce space and time variables, which are appropriate for the description of long-wavelength phenomena. The independent variables are stretched as
significant when s ? 0 and weaker for larger value of |s|. For one-dimensional geometry (m = 0) and for moving frame moving with a speed g0, the stationary propagation of the DIASW governed by Eq. (8) has the following form: rffiffiffiffiffiffi 3g0 g0 2 /¼ ð9Þ sec h ðn g0 sÞ A 4B
n ¼ e1=2 ðr þ v0 tÞ and s ¼ e3=2 t;
To obtain an analytical solution of cylindrical KdV equation, let us first use Hirota’s transformation (Sahu and Roychoudhury, 2003) given by
ð4Þ
where e is a small (0 < e < 1) expansion parameter proportional to the amplitude of the perturbation and v0 represents the normalized phase velocity of the DIASW. The RPM theory general principles are based on multiscale expansion (Jeffrey and Kawahara, 1982), which can be written in the following manner: ni ¼ 1 þ en1 þ e n2 þ ; 2
ð5Þ
ui ¼ u0 þ eu1 þ e2 u2 þ ; / ¼ e/1 þ e2 /2 þ
Substituting Eqs. (4), (5) into Eqs. (1)–(3) and collecting the terms in the different powers of e, we can derive the following expressions from the lowest order in e: ðv0 þ u0 Þ / /1 ; n1 ¼ 1 d d 1=2 d þ 3r v 0 þ u0 ¼ 1 þ 1:5b2
u1 ¼
ð6Þ
where b = u0/c is relativistic streaming factor. To the next order of e, the equations become: @n1 @n2 @ðu1 n1 Þ mu1 þ mu0 n1 @u2 ðv0 þ u0 Þ ¼0 @n @s @n sv0 @n @u1 @u2 ðv0 þ u0 Þð1 þ 1:5b2 Þ ð1 þ 1:5b2 Þ @s @n v þ u @u @/ 0 0 1 2 1 þ 1:5b2 þ 3b2 u1 u0 @n @n @n2 @n1 þ n1 3r @n @n @ 2 /1 /21 dn2 ¼ / þ 2 2 @n2
Now, using the derived expression of n1, u1 and eliminating n2, u2 and u2 from Eq. (7), the cylindrical and spherical KdV equation is written as
where u1 u, A ¼ d2 2ð1þ1:5b2 Þðv0 þu0 Þ
ð8Þ
2þ9r dþ 1þ1:5b2 þ3b2 d 2
v0 þu0 u0
2ð1þ1:5b Þðv0 þu0 Þ
i
ðdþ3rÞ dð1þ1:5b2 Þ
@m Am @m n @m @3m þ þ þB 3 ¼0 @s s @n 2s @n @n
ð10Þ
Again we use a transformation given by s0 ¼ 2s1=2 ;
n0 ¼ ns1=2
Eq. (10) (can be reduced as) reduces to @m @m @3m þ Am þ B ¼0 @s0 @n0 @n03
ð11Þ
Eq. (11) is the usual KdV So the solution of pffiffiffiffi equation. g0 Eq. (11) is given by m ¼ 3gA0 sec h2 4B ðn0 g0 s0 Þ The exact solution of Eq. (8) is given by
rffiffiffiffiffiffiffiffi 1 n 3g0 g0 2 þ /¼ ð12Þ sec h ðn þ 2g0 Þ s 2A A 4Bs
4. Results and discussion
ð7Þ
@/ @/ @3/ m/ þ A/ þB 3 þC ¼0 @s @n s @n h
Under this transformation Eq. (8) transforms to
This solution is valid for s – 0.
¼0 d
m n /¼ þ s 2As
, B¼
and C = 1/2. The second, third and fourth
term on the left-hand side of Eq. (8) is the nonlinearity, the dispersion and the geometry effects, respectively. It is clear from Eq. (8) that the nonplanar geometrical effect is
The properties of cylindrical and spherical DIASW in an unmagnetized dusty plasma consisting of relativistic thermal ions, Boltzmann electrons, and stationary dusty particles are studied. The influence of geometry, relativistic streaming factor, ion density and electron temperature has been investigated. When the geometrical effect is taken into account (m – 0), an exact analytical solution of Eq. (8) is not possible without any transformation. We have numerically solved Eq. (8) and have studied the geometrical effects on the propagation of DIASW. The initial condition that we have used in all our numerical results is the form of the stationary solution of Eq. (9) without the geometry term at s = 70 (at this stage the geometry effect is weaker, so we can take this stage as the initial stage of evolution). Fig. 1 shows the solitary wave structure evolved at s = 4 in different geometry. It is clear that the solitary height in different geometry are different from each other. The spherical solitary wave, with the highest height, is the strongest one. The height of cylindrical solitary wave is larger than that of the one dimensional solitary wave but smaller than that of the spherical solitary wave. This
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
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Fig. 1. Solitary wave structures in different geometry at s = 4.
Fig. 2. Cylindrical Solitary wave structures for different d at s = 4.
result is congruent with the observations made in Refs. [28– 31]. Fig. 2 exhibits variations in solitary structure due to ion density. It is noted that soliton height increases as ion density is increased. This result is correspondent with the observations in Refs. [17,35]. Fig. 3 displays the effect of ion temperature on the structure of soliton. The height of the soliton increases due to decrease in ion temperature. Fig. 4 reveals that for large value of s, the cylindrical and spherical solitary waves are similar to one-dimensional solitary wave in a such relativistic plasma. This is because for large value of |s| the nonplanar geometrical effect is no longer dominant. However, as the value of s decreases, the nonplanar geometrical effect, represented by (m/s)u, will become dominant and the cylindrical, spherical, and one-dimensional solitary waves differ from each other. The variation of soliton height against relativistic streaming factor is plotted in Fig. 5. It is found that without
Fig. 3. Cylindrical Solitary wave structures for different r at s = 4.
Fig. 4. Cylindrical Solitary wave structures for different s.
ion pressure, as the relativistic streaming factor decreases, the amplitude of the solitary wave decreases. However, under consideration of the ion pressure, the amplitude decreases as the relativistic streaming factor increases and is highly sensitive to relativistic streaming factor. 5. Summary In conclusion, we have investigated the nonplanar cylindrical and spherical DIASW in a weakly relativistic dusty plasma which is governed by the modified KdV equation. Under a suitable coordinate transformation the cylindrical KdV equation is reduced into the ordinary KdV equation which can be solved analytically. The nonplanar geometry effect for DIASW is very strong for a small value of s and there are obvious differences between the cylindrical DIASW, spherical DIASW, and one-dimensional DIASW. The height of DIASW decreases with increasing ion tem-
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
H.-F. Liu (刘海峰) et al. / Advances in Space Research xxx (2013) xxx–xxx
Fig. 5. (a) Cylindrical Solitary wave structures with ion pressure for different b at s = 4. (b) Cylindrical Solitary wave structures without ion pressure for different b at s = 4.
perature and increases with the increase of ion density. Through comparing DIASW in a relativistic dusty plasma with ion pressure and without, It is emphasized that the effect of relativistic streaming factor to solitary waves structure is changed. The present investigation may be helpful in better understanding of waves propagation in the pulsar magnetospheres, and the intense laser fields. References Arons, J. Some problems of pulsar physics or I’m madly in love with electricity. Space Sci. Rev. 24, 437–510, 1979. Barkan, A., Merlino, R.L., D’Angelo, N. Laboratory observation of the dust-acoustic wave mode. Phys. Plasmas 2, 3563–3566, 1995. E. Casarejos. Performance of timing RPC detectors for relativistic ions and design of a time-of-flight detector (iToF) for the R3B-FAIR experiment for fission and spallation reactions. In: Advancements in Nuclear Instrumentation Measurement Methods and their Applications (ANIMMA), 2011 2nd International Conference on.
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Das, G.C., Dwivedi, C.B., Talukdar, M., Sarma, J. A new mathematical approach for shock-wave solution in a dusty plasma. Phys. Plasmas 4, 4236–4240, 1997. Ghosh, S., Sarkar, S., Khan, M., Gupta, M.R. Ion acoustic shock waves in a collisional dusty plasma. Phys. Plasmas 9, 378–382, 2002. Grabbe, C. Wave propagation effects of broadband electrostatic noise in the magnetotail. J. Geophys. Res. 94, 17299–17304, 1989. Gupta, M.R., Sarkar, S., Ghosh, S., Debnath, M., Khan, M. Effect of nonadiabaticity of dust charge variation on dust acoustic waves: generation of dust acoustic shock waves. Phys. Rev. E 63, 046406– 046415, 2001. Harrison, M., Ludlam, T., Ozaki, S. RHIC project overview. Nucl. Instrum. Methods Phys. Res. Sect. A 499, 235–244, 2003. Homann, A., Melzer, A., Peters, S., Piel, A. Determination of the dust screening length by laser-excited lattice waves. Phys. Rev. E 56, 7138– 7141, 1997. Ikezi, H. Experiments on ion-acoustic solitary waves. Phys. Fluids 16, 1668–1676, 1973. Jeffrey, A., Kawahara, T. (Eds.). Asymptotic Methods in Nonlinear Wave Theory. Pitman, Boston, 1982. Liu, H.-f., Wang, S.Q., et al. Cylindrical and spherical dust-ion acoustic solitary waves in a relativistic dust plasma. Phys. Scr. 82, 065402– 065407, 2010. Mamun, A.A. Effects of adiabaticity of electrons and ions on dust-ionacoustic solitary waves. Phys. Lett. A 372, 1490–1493, 2008. Mamun, A.A., Shukla, P.K. Spherical and cylindrical dust acoustic solitary waves. Phys. Lett. A 290, 173–175, 2001. Mamun, A.A., Shukla, P.K. Cylindrical and spherical dust ion–acoustic solitary waves. Phys. Plasmas 9, 1468–1471, 2002. Mamun, A.A., Shukla, P.K. Electrostatic solitary and shock structures in dusty plasmas. Phys. Scr., http://dx.doi.org/10.1238/Physica.Topical.098a00107, 2009a. Mamun, A.A., Shukla, P.K. Effects of nonthermal distribution of electrons and polarity of net dust-charge number density on nonplanar dust-ion-acoustic solitary waves. Phys. Rev. E 80, 037401–037405, 2009b. Masud, M.M. et al. Time dependent cylindrical and spherical DIA solitary waves with two populations of thermal electrons in dusty plasma. Astrophys. Space Sci., http://dx.doi.org/10.1007/s10509-012-1244-x, 2012a. Masud, M.M. et al. Dust-ion-acoustic Gardner solitons in a dusty plasma with bi-Maxwellian electrons. Phys. Plasmas 19, 103706–1037011, 2012b. Melandsø, F., Shukla, P.K. Theory of dust-acoustic shocks. Planet. Space Sci. 43, 635–648, 1995. Mendis, D.A., Rosenberg, M. Cosmic dusty plasmas. Annu. Rev. Astron. Astrophys. 32, 419–463, 1994. Nakamura, Y., Bailung, H., Shukla, P.K. Observation of ion-acoustic shocks in a dusty plasma. Phys. Rev. Lett. 83, 1602–1605, 1999. Nakamura, Y. et al. Island dynamics in the large-helical-device plasmas. Phys. Rev. Lett. 88, 055005–055009, 2002. Northrop, T.G. Dusty plasmas. Phys. Scr., http://dx.doi.org/10.1088/ 0031-8949/45/5/011, 1992. Popel, S.I., Yu, M.Y., Tsytovich, V.N. Shock waves in plasmas containing variable-charge impurities. Phys. Plasmas 3, 4313–4316, 1996. Popel, S.I., Gisko, A.A., Golub, A.P., Losseva, T.V., Bingham, R., Shukla, P.K. Shock waves in charge-varying dusty plasmas and the effect of electromagnetic radiation. Phys. Plasmas 7, 2410–2417, 2000. Riordan, M., Zajc, W.A. The first few microseconds. Nat. Sci. Am. 294, 34–41, 2006. Roy, K., Misra, A.P., Chatterjee, P. Ion-acoustic shocks in quantum electron-positron-ion plasmas. Phys. Plasmas 15, 032310, 2008. Sahu, B., Roychoudhury, R. Exact solutions of cylindrical and spherical dust ion acoustic waves. Phys. Plasmas 10, 4162–4166, 2003. Shukla, P.K., Mamun, A.A. Dust-acoustic shocks in a strongly coupled dusty plasma. IEEE Trans. Plasma Sci. 29, 221–225, 2001. Shukla, P.K., Mamun, A.A. (Eds.). Introduction to Dusty Plasma Physics. IOP, Bristol, 2002.
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
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Shukla, P.K., Silin, V.P. Dust ion-acoustic wave. Phys. Scr. 45, 508–513, 1992. Taniuti, T., Wei, C.C. Reductive, Perturbation method in nonlinear wave propagation I. J. Phys. Soc. Jpn. 24, 941–946, 1968. Tasnim, I. et al. Effects of nonthermal ions of distinct temperatures on dust acoustic shock waves in a dusty plasma. Astrophys. Space Sci., http://dx.doi.org/10.1007/s10509-012-1275-3, 2012. Verheest, F. Waves in Dusty Space Plasmas. Kluwer, Dordrecht, 2000. Vette, J.I. Summary of Particle Population in the Magnetosphere. Reidel, Dordrecht, p. 305, 1970.
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Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
Advances in Space Research xxx (2013) xxx–xxx www.elsevier.com/locate/asr
Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma Hai-Feng Liu (刘海峰) a,b,⇑, Shi-Qing Wang (王世庆) a,b, Zhi-long Wang (王志龙) b, Fa-Zhan Yang (杨发展) a, Yao-Liu (刘耀) b, Sili-Li (李思丽) b b
a Southwestern Institute of Physics, Chengdu 610041, China The Engineering and Technical College of Chengdu University of Technology, Leshan 614000, China
Received 3 September 2012; received in revised form 3 January 2013; accepted 15 January 2013
Abstract The properties of cylindrical and spherical dust–ion acoustic solitary waves (DIASW) in an unmagnetized dusty plasma comprising of relativistic ions, Boltzmann electrons, and stationary dusty particles are investigated. Under a suitable coordinate transformation, the cylindrical KdV equation can be solved analytically. The change of the DIASW structure due to the effect of geometry, relativistic streaming factor, ion density and electron temperature is studied by numerical calculation of the cylindrical/spherical Kdv equation. It is noted that with ion pressure the effect of relativistic streaming factor to solitary waves structure is different. Without ion pressure, as the relativistic streaming factor decreases, the amplitude of the solitary wave decreases. However, when the ion pressure is taken into account, the amplitude decreases as the relativistic streaming factor increases and is highly sensitive to relativistic streaming factor. Our results may have relevance in the understanding of astrophysical plasmas. Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Solitary waves; Unmagnetized dusty plasma; Cylindrical/spherical Kdv equation; Ion pressure; Relativistic streaming factor; Electron temperature
1. Introduction The linear and nonlinear propagation of electrostatic excitations in dusty plasmas have received considerable attention in the last few decades. Ion acoustic waves in unmagnetized dusty plasmas with phase velocity between the electron and ion thermal velocities were first studied theoretically by Shukla and Silin (1992) and later their existence was confirmed experimentally by Barkan et al. (1995). Dust being a common species in a wide range of ⇑
Corresponding author. Address: Southwestern Institute of Physics, People’s south road in three sections, Chengdu 610041, China. Tel.: +86 15 681 386 018. E-mail addresses: [email protected] (H.-F. Liu (刘海峰)), [email protected] (S.-Q. Wang (王世庆)), [email protected] (Z.-l. Wang (王志龙)), [email protected] (F.-Z. Yang (杨发展)), [email protected] ( Yao-Liu (刘耀)), [email protected] ( Sili-Li (李思丽)).
space and astrophysical plasmas such as the cometary tails and comae, interstellar clouds, Earth’s mesosphere and ionosphere, Saturn’s rings, the gossamer ring of Jupiter, and in laboratory experiments (Barkan et al.,1995; Homann et al.,1997; Mendis and Rosenberg, 1994; Northrop, 1992), the study of dusty plasmas has been an important focus of much recent research. Dusty particles are often of micron to submicron size, with masses in the range of 106 1012 proton masses (Shukla and Mamun, 2002; Verheest, 2000), and are usually found to have negative charge (possibly as large as 104 electron charges), depending on the environment where they occur. On the other hand, smaller dust grains may be found to be positively charged. The presence of the dust modifies the standard ion acoustic mode, giving rise to what is termed the dusty ion acoustic wave (DIAW). One of the important effects of the massive dust grains is that the associated change in the free electron density yields a corresponding
0273-1177/$36.00 Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2013.01.028
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
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change in shielding and hence in the detailed wave behavior. The linear DIAW has an increased phase velocity when the dust is negatively (positively) charged. The DIAW was observed by Nakamura et al. (1999, 2002) in a collisional dominated dusty plasma. Theoretical models for the DIAW is given by in a weakly correlated dusty plasma. The study of DIAW in a strongly coupled dusty plasma is also presented by Shukla and Mamun (2001). Xiao et al. (2008) and Liu et al. (2010) have investigated the finite amplitude nonlinear DIAW in an inhomogeneous dusty plasma. Under the assumption of negligible dust-charge perturbation, a critical point, at which the coefficient of nonlinear term changes from negative to positive, exists. Consequently the nonlinear wave undergoes structural deformation. Roy et al. (2008) have studied DIAW in quantum dusty plasma. Their main findings are that system can sustain both oscillatory and monotonic shock waves depending on quantum parameters (H), there exists a limiting value of H at which shock wave breaks up to solitary wave. Xue (2003a,b) studied spherical and cylindrical DIAW and observed that wave height decreases with an increase in ion temperature and that the cold ion acoustic shock wave has the largest height. Several authors (Melandsø and Shukla, 1995; Popel et al., 2000; Popel et al., 1996; Gupta et al., 2001; Ghosh et al., 2002; Das et al., 1997) show that, when ion viscosity or Landau damping effects are not important in dusty plasmas, the nonadiabatic dusty charge variation provides an alternate physical mechanism causing dissipation, and as a consequence this gives rise to shocks for which both monotonic and oscillatory structures are possible. However, most of these studies are confined to the nonrelativistic plasmas and the unbounded planar geometry, which may not be a realistic situation in laboratory devices and space. When the particles velocities are comparable to the speed of light, relativistic effects can no longer be ignored. Relativistic plasmas occur in space plasma phenomena (Grabbe, 1989), plasma sheet boundary layer of earth’s magnetosphere (Vette, 1970), in the Van Allen radiation belts (Ikezi, 1973) and in laser–plasma interaction (Arons, 1979). Relativistic ions and non-relativistic electrons also occur (Riordan and Zajc, 2006; Harrison et al., 2003; Casarejos, 2011). Many theoretical studies for dust–ion acoustic and ion acoustic waves in nonplanar geometry show (Mamun and Shukla, 2001, 2002, 2009a,b; Mamun, 2008; Xue, 2003a,b; Liu et al., 2010) that the properties of these waves in bounded nonplanar cylindrical/spherical geometry are very different from that in unbounded planar geometry. Recently, Tasnim et al. (2012) investigated the effects of two temperature nonthermally distributed ions and dust kinematic viscosity, which are found to significantly modify the basic features of dust acoustic shock waves. Masud et al. (2012a,b) studied propagation of Gardner solitons in a nonplanar (cylindrical and spherical) geometry by deriving the modified Gardner equation. The basic features (amplitude, width, etc.) of the hump (positive potential) and dip (negative potential) shaped DIA solitons (Gardner solitons, i.e.,
GSs) are found to exist beyond the Korteweg-de Vries (K-dV) limit. These DIA-GSs are qualitatively different from the K-dV and modified K-dV solitons (Masud et al., 2012a,b). In this paper, we investigate the properties of the nonplanar geometry on DIASW in an unmagnetized dusty plasma with relativistic ions, Boltzmann electrons, and stationary dusty particles. By using the standard reductive perturbation method (RPM), a cylindrical/spherical Kdv equation is obtained. Under a suitable coordinate transformation the cylindrical KdV equation turns into the ordinary KdV equation which can be solved analytically. The effects of nonplanar geometry, ion density, relativistic streaming factor, and ion temperature on DIASW structures are studied by numerical calculation of the cylindrical/spherical Kdv equation. 2. Basic set of equations To study cylindrical and spherical acoustic solitary waves in a relativistic dusty plasma, we assume that the DIASW propagate in an axial symmetry cylindrical geometry filled with the unmagnetized collisionless dusty plasma. In equilibrium, the charge neutrality condition is ne0 nio + Zdnd0 = 0, where ne0, nio , and nd0 are the unperturbed electron, ion, and dust number densities, respectively, and Zd is the number of electrons residing on the dust grains. The governing equations of the physical phenomenon under consideration consist of the conservation laws of mass and momentum for relativistic ions and Poisson’s equation. @ni 1 @ðrm ui ni Þ ¼0 þ @r @t rm @ðcui Þ @ðcui Þ @/ r @ni þ ui ¼ 3 @t @r @r ni @r d
1 @ m @/ ðr Þ ¼ expð/Þ dni þ ðd 1Þ rm @r @r
ð1Þ ð2Þ ð3Þ
where m = 0 for a one-dimensional geometry and m = 1(2) for a nonplanar cylindrical (spherical) geometry and r ¼ T i =T e , d = nio/ne0. The variables of time (t), space (r), ion number density (ni), ion fluid velocity (ui), and electrostatic wave potential (u) are normalized to the reciprocal pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pni0 e2 =mi ), Debye raion plasma frequency x1 pi (xpi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dius kD (kD ¼ k B T e =4pni0 e2 ), unperturbed equilibrium plasma density nio, effective ion acoustic velocity Ci pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (C i ¼ k B T e =mi ), and kBTe/e respectively. The relativistic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi factor is given as c ¼ 1 u2i =c2 , where c is the speed of light. The relativistic factor for a weakly relativistic regime can be approximated as c 1 þ u2i =2c2 . 3. Derivation and solution of K-dV equation In order to investigate the DIASW in such a plasma, We employ the RPM to derive the dynamical equation for the nonlinear propagation of the electrostatic waves under
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
H.-F. Liu (刘海峰) et al. / Advances in Space Research xxx (2013) xxx–xxx
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consideration. The RPM is mostly applied to small amplitude nonlinear waves (Taniuti and Wei, 1968). This method rescales both space and time in the governing equations of the system in order to introduce space and time variables, which are appropriate for the description of long-wavelength phenomena. The independent variables are stretched as
significant when s ? 0 and weaker for larger value of |s|. For one-dimensional geometry (m = 0) and for moving frame moving with a speed g0, the stationary propagation of the DIASW governed by Eq. (8) has the following form: rffiffiffiffiffiffi 3g0 g0 2 /¼ ð9Þ sec h ðn g0 sÞ A 4B
n ¼ e1=2 ðr þ v0 tÞ and s ¼ e3=2 t;
To obtain an analytical solution of cylindrical KdV equation, let us first use Hirota’s transformation (Sahu and Roychoudhury, 2003) given by
ð4Þ
where e is a small (0 < e < 1) expansion parameter proportional to the amplitude of the perturbation and v0 represents the normalized phase velocity of the DIASW. The RPM theory general principles are based on multiscale expansion (Jeffrey and Kawahara, 1982), which can be written in the following manner: ni ¼ 1 þ en1 þ e n2 þ ; 2
ð5Þ
ui ¼ u0 þ eu1 þ e2 u2 þ ; / ¼ e/1 þ e2 /2 þ
Substituting Eqs. (4), (5) into Eqs. (1)–(3) and collecting the terms in the different powers of e, we can derive the following expressions from the lowest order in e: ðv0 þ u0 Þ / /1 ; n1 ¼ 1 d d 1=2 d þ 3r v 0 þ u0 ¼ 1 þ 1:5b2
u1 ¼
ð6Þ
where b = u0/c is relativistic streaming factor. To the next order of e, the equations become: @n1 @n2 @ðu1 n1 Þ mu1 þ mu0 n1 @u2 ðv0 þ u0 Þ ¼0 @n @s @n sv0 @n @u1 @u2 ðv0 þ u0 Þð1 þ 1:5b2 Þ ð1 þ 1:5b2 Þ @s @n v þ u @u @/ 0 0 1 2 1 þ 1:5b2 þ 3b2 u1 u0 @n @n @n2 @n1 þ n1 3r @n @n @ 2 /1 /21 dn2 ¼ / þ 2 2 @n2
Now, using the derived expression of n1, u1 and eliminating n2, u2 and u2 from Eq. (7), the cylindrical and spherical KdV equation is written as
where u1 u, A ¼ d2 2ð1þ1:5b2 Þðv0 þu0 Þ
ð8Þ
2þ9r dþ 1þ1:5b2 þ3b2 d 2
v0 þu0 u0
2ð1þ1:5b Þðv0 þu0 Þ
i
ðdþ3rÞ dð1þ1:5b2 Þ
@m Am @m n @m @3m þ þ þB 3 ¼0 @s s @n 2s @n @n
ð10Þ
Again we use a transformation given by s0 ¼ 2s1=2 ;
n0 ¼ ns1=2
Eq. (10) (can be reduced as) reduces to @m @m @3m þ Am þ B ¼0 @s0 @n0 @n03
ð11Þ
Eq. (11) is the usual KdV So the solution of pffiffiffiffi equation. g0 Eq. (11) is given by m ¼ 3gA0 sec h2 4B ðn0 g0 s0 Þ The exact solution of Eq. (8) is given by
rffiffiffiffiffiffiffiffi 1 n 3g0 g0 2 þ /¼ ð12Þ sec h ðn þ 2g0 Þ s 2A A 4Bs
4. Results and discussion
ð7Þ
@/ @/ @3/ m/ þ A/ þB 3 þC ¼0 @s @n s @n h
Under this transformation Eq. (8) transforms to
This solution is valid for s – 0.
¼0 d
m n /¼ þ s 2As
, B¼
and C = 1/2. The second, third and fourth
term on the left-hand side of Eq. (8) is the nonlinearity, the dispersion and the geometry effects, respectively. It is clear from Eq. (8) that the nonplanar geometrical effect is
The properties of cylindrical and spherical DIASW in an unmagnetized dusty plasma consisting of relativistic thermal ions, Boltzmann electrons, and stationary dusty particles are studied. The influence of geometry, relativistic streaming factor, ion density and electron temperature has been investigated. When the geometrical effect is taken into account (m – 0), an exact analytical solution of Eq. (8) is not possible without any transformation. We have numerically solved Eq. (8) and have studied the geometrical effects on the propagation of DIASW. The initial condition that we have used in all our numerical results is the form of the stationary solution of Eq. (9) without the geometry term at s = 70 (at this stage the geometry effect is weaker, so we can take this stage as the initial stage of evolution). Fig. 1 shows the solitary wave structure evolved at s = 4 in different geometry. It is clear that the solitary height in different geometry are different from each other. The spherical solitary wave, with the highest height, is the strongest one. The height of cylindrical solitary wave is larger than that of the one dimensional solitary wave but smaller than that of the spherical solitary wave. This
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
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Fig. 1. Solitary wave structures in different geometry at s = 4.
Fig. 2. Cylindrical Solitary wave structures for different d at s = 4.
result is congruent with the observations made in Refs. [28– 31]. Fig. 2 exhibits variations in solitary structure due to ion density. It is noted that soliton height increases as ion density is increased. This result is correspondent with the observations in Refs. [17,35]. Fig. 3 displays the effect of ion temperature on the structure of soliton. The height of the soliton increases due to decrease in ion temperature. Fig. 4 reveals that for large value of s, the cylindrical and spherical solitary waves are similar to one-dimensional solitary wave in a such relativistic plasma. This is because for large value of |s| the nonplanar geometrical effect is no longer dominant. However, as the value of s decreases, the nonplanar geometrical effect, represented by (m/s)u, will become dominant and the cylindrical, spherical, and one-dimensional solitary waves differ from each other. The variation of soliton height against relativistic streaming factor is plotted in Fig. 5. It is found that without
Fig. 3. Cylindrical Solitary wave structures for different r at s = 4.
Fig. 4. Cylindrical Solitary wave structures for different s.
ion pressure, as the relativistic streaming factor decreases, the amplitude of the solitary wave decreases. However, under consideration of the ion pressure, the amplitude decreases as the relativistic streaming factor increases and is highly sensitive to relativistic streaming factor. 5. Summary In conclusion, we have investigated the nonplanar cylindrical and spherical DIASW in a weakly relativistic dusty plasma which is governed by the modified KdV equation. Under a suitable coordinate transformation the cylindrical KdV equation is reduced into the ordinary KdV equation which can be solved analytically. The nonplanar geometry effect for DIASW is very strong for a small value of s and there are obvious differences between the cylindrical DIASW, spherical DIASW, and one-dimensional DIASW. The height of DIASW decreases with increasing ion tem-
Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028
H.-F. Liu (刘海峰) et al. / Advances in Space Research xxx (2013) xxx–xxx
Fig. 5. (a) Cylindrical Solitary wave structures with ion pressure for different b at s = 4. (b) Cylindrical Solitary wave structures without ion pressure for different b at s = 4.
perature and increases with the increase of ion density. Through comparing DIASW in a relativistic dusty plasma with ion pressure and without, It is emphasized that the effect of relativistic streaming factor to solitary waves structure is changed. The present investigation may be helpful in better understanding of waves propagation in the pulsar magnetospheres, and the intense laser fields. References Arons, J. Some problems of pulsar physics or I’m madly in love with electricity. Space Sci. Rev. 24, 437–510, 1979. Barkan, A., Merlino, R.L., D’Angelo, N. Laboratory observation of the dust-acoustic wave mode. Phys. Plasmas 2, 3563–3566, 1995. E. Casarejos. Performance of timing RPC detectors for relativistic ions and design of a time-of-flight detector (iToF) for the R3B-FAIR experiment for fission and spallation reactions. In: Advancements in Nuclear Instrumentation Measurement Methods and their Applications (ANIMMA), 2011 2nd International Conference on.
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Please cite this article in press as: Liu (刘海峰), H.-F., et al. Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma. J. Adv. Space Res. (2013), http://dx.doi.org/10.1016/j.asr.2013.01.028