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Journal of Fusion Energy

, Volume 31, Issue 6, pp 611–616 | Cite as

Ion Acoustic Solitary Waves and Double-Layers in a Plasma with Nonthermal Electrons and Positrons

Original Research
First Online:

Abstract

The effects of nonthermal electron distribution and ion temperature are incorporated in the investigation of nonlinear ion acoustic waves in a pair-ion plasma. Sagdeev pseudo- potential method which takes into account the full nonlinearity of the plasma equations is used to study solitary wave solutions. It is shown that there is a region in parameter space where both negative and positive potential can coexist. For the fixed value of nonthermal electrons, it is found that the effect of increase in ion temperature is to reduce the range of co-existence of compressive and rarefactive solitons. Particular concentration of nonthermal electrons results in disappearance of rarefactive solitons while the decrease in ion temperature, at this concentration restores the lost rarefactive solitons. Also, the existence of rarefactive double layers solitons is investigated. It is found that the nonthermal electrons and ion-temperature play significant role in determining the region of the existence of double layers.

Keywords

Ion acoustic waves Solitons Nonlinear phenomena Sagdeev potential 

Introduction

The ion-acoustic (IA) waves and their underlying dynamics have been studied for several decades both theoretically and experimentally. Nonlinear theory for these waves was first considered in [1] where their basic features were studied using a mechanical analogy. It has been established that stationary ion-acoustic waves can exist in the form of periodic or solitary waves. The first experimental observation of ion-acoustic solitons was made by Ikezi et al. [2]. Subsequently, and because of quantitative discrepancies between theory and experiment, the nonlinear ion-acoustic wave theory has been developed to include the effects of a finite ion temperature [3, 4] and those due to a trapped electron population [5, 6], and high-order nonlinearity [7]. On another side, a great deal of attention has been devoted to the study of different types of collective processes in electron–positron–ion plasmas [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. It is well known that when positrons are introduced into an electron–ion plasma the response of the latter changes significantly. In contrast to the usual two-component plasma, it has been observed that the nonlinear waves in plasmas with an additional positron component behave differently. Such pair (electron and positron)-ion plasmas are common in supernovae, pulsar environments, cluster explosions, active galactic nuclei, etc.

Numerous observations clearly indicate the presence of energetic electrons as ubiquitous in a variety of astrophysical plasma environments and measurements of their distribution functions revealed them to be highly non thermal. Non thermal distributions are turning out to be a very common and characteristic feature of space plasmas where coherent nonlinear waves and structures are expected to play an important role. Such non thermal populations may be distributed isotropically in velocities or possess a net streaming motion with respect to the background plasma, and their presence has been confirmed by many observations of space plasmas [30, 31, 32, 33]. Observations made by the Viking spacecraft [34] and Freja satellite [35] have found electrostatic solitary structures in the magnetosphere with density depressions. Motivated by these events, Cairns et al. [36] showed that the presence of a non thermal distribution of electrons may change the nature of ion sound solitary structures and allow the existence of rarefactive ion-acoustic solitary structures very like those observed by Freja and Viking. Some recent theoretical work focused on the effects of particle non thermality on different types of linear and nonlinear collective processes [37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. However and to the best of our knowledge, nonlinear ion acoustic solitary waves in a plasma with non thermal electrons and positrons have never been addressed in the plasma literature. Therefore, it seems worthwhile to present a theoretical work to investigate IA solitary waves in a plasma with non thermal electrons and positrons and, in particular, to identify the conditions that favour their existence.

Basic Equations

Let us consider a plasma having warm fluid ions, non thermal electrons, and thermal positrons of density n, n e , and n p , respectively. The dynamics of low frequency ion-acoustic oscillations is governed by the following equations [47]
$$ \frac{\partial n}{\partial t} + \frac{\partial (nu)}{\partial x} = 0 $$
(1)
$$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \frac{\sigma }{n}\frac{\partial P}{\partial x} + \frac{\partial \phi }{\partial x} = 0\,\, $$
(2)
$$ \frac{\partial P}{\partial t} + u\frac{\partial P}{\partial x} + 3P\frac{\partial u}{\partial x} = 0\,\, $$
(3)
$$ \frac{{\partial^{2} \phi }}{{\partial x^{2} }} = n_{e} - n - n_{p} . $$
(4)

Here and in the following, j = i, e, p stands for ions, electrons and positrons, T j are the temperatures, m j the masses, e is the elementary charge, \( \sigma = \frac{{T_{i} }}{{T_{eff} }} \), in which T eff is the effective temperature of the plasma, defined by \( T_{eff} = \frac{{n_{ \circ i} }}{{\frac{{n_{ \circ e} }}{{T_{e} }} + \frac{{n_{ \circ p} }}{{T_{P} }}}} \) [48, 49, 50]. The ion number density n, the ion fluid velocity u, the pressure P, and the electrostatic potential ϕ are normalized, respectively, by n oi (the unperturbed ion density), \( c_{i} = \sqrt {T_{eff} /m} ,\,n_{ \circ i} T_{i} \,{\text{and}}\,\,T_{eff} /e \). The time t and the spatial variable x are normalized by \( \omega_{pi}^{ - 1} = \sqrt {m_{i} /4\pi n_{0} e^{2} } \) and \( \lambda_{D} = \sqrt {T_{eff} /4\pi n_{0} e^{2} } \,\, \), respectively. We assume that P = n 3 for adiabatic process. Note that it is well known that any thermodynamic process which is typically characterized by a relatively faster change of state so that the system undergoing change does not have time to exchange significant amount of heat with its surroundings is called an adiabatic process. In fact, Plasma collective behavior takes the form of different types of waves, characterized by frequency ω and wave-vector k. Depending on the phase velocity compared to the thermal velocity of the media v th , diverse equation of states must be applied leading to various waves. When ω/k ≪ v th , the particles have enough time to thermalize the plasma causing a constant temperature and validating the isothermal equation of state p = nT. In the opposite limit ω/k ≫ v th , the particle movement—and thus the heat flow—is negligible during the characteristic time of the wave (~1/ω) compared to the wavelength of the actual wave. In this case the adiabatic equation of state \( \frac{p}{{n^{\gamma } }} = {\text{constant}} \) is applicable where γ = (d + 2)/d is the specific heat ratio and d stands for the dimensionality of the problem (for our one-dimensional problem, γ = 3). For ion acoustic waves for which the particle mean free paths are small compared to the wavelength, the ion adiabaticity is thus well- justified [51].

The nonthermal electrons are described by the Cairns velocity distribution [36] whose purpose was to show that nonthermal electrons can change the nature of ion-acoustic solitons and may explain some structures observed by the Freja Satellite. This distribution is given by
$$ f_{e} (x,v_{e} ) = \frac{{n_{e0} }}{1 + 3\alpha }\left( {\frac{{m_{e} }}{{2\pi T_{e} }}} \right)^{1/2} \left( {1 + 4\alpha \left( {\frac{{{{m_{e} v_{e}^{2} } \mathord{\left/ {\vphantom {{m_{e} v_{e}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} - e\psi }}{{T_{e} }}} \right)^{2} } \right)\exp \left( { - \frac{{{{m_{e} v_{e}^{2} } \mathord{\left/ {\vphantom {{m_{e} v_{e}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} - e\psi }}{{T_{e} }}} \right) $$
where ψ is the dimensional electrostatic potential. Integrating this Cairns distribution over all velocity space, we obtain the following non thermal electron number density given by
$$ n_{e} = \frac{1}{1 - p}\left( {1 - \beta \phi + \beta \phi^{2} } \right)\exp \left( \phi \right)\, $$
(5)
where p = n p0/n e0, β = 4α/(1 + 3α) and α is a parameter that determines the fraction of energetic nonthermal electrons and characterizes the degree of electron non thermality. On the ion time scale, the positrons are assumed in thermal equilibrium, with the density
$$ n_{p} = \frac{p}{1 - p}\exp ( - \delta \phi ) $$
(6)
where δ = T e /T p .

Note that the ion-acoustic wave arises due to the restoring force provided by the electron thermal motion, while the inertia is due to the ion mass. In practice, non thermality means that a fraction of the electronic component deviates from its Maxwellian thermodynamic equilibrium by some energizing process such as (for example) turbulence or double-layers self-consistent electric fields. Interestingly, the ion-acoustic mode may get modified as there is a change in its restoring force.

To derive the Sagdeev pseudopotential [1] from Eqs. (1)–(6), we assume that all dependent variables depend on a single independent variable ξ = x − Mt, where M is the soliton velocity normalized by c i . Equations (1) and (2) in the stationary frame can be integrated to give
$$ n = \frac{\sqrt 2 M}{{\sqrt {M^{2} - 2\phi + 3\sigma + \sqrt {(M^{2} - 2\phi + 3\sigma )^{2} - 12\sigma M^{2} } } }} $$
(7)
where we have used boundary conditions for localized disturbance, viz, u → 0, n → 1, ϕ → 0, P → 1, as ξ → ∞. Substituting n from (7) into (4) and following Sagdeev’s pseudopotential method along with appropriate boundary conditions, we obtain the quadrature
$$ \frac{1}{2}\left[ {\frac{d\phi }{d\chi }} \right]^{2} + V(\phi ) = 0 $$
(8)
where
$$ \begin{aligned} V(\phi ) & = \left(\frac{1}{1 - p}\right)\left\{ {\left[ {(1 - p)(M^{2} + \sigma ) + 1 + 3\beta + \frac{p}{\delta }} \right] - \left[ {1 + 3\beta + \beta \phi^{2} - 3\beta \phi } \right]e^{\phi } - \frac{p}{\delta }e^{ - \delta \phi } } \right\} \\ & - \frac{\sqrt 2 }{2}M \times \left\{ {\left[ {M^{2} - 2\phi + 3\sigma + \sqrt {(M^{2} - 2\phi + 3\sigma )^{2} - 12M^{2} \sigma } } \right]^{\frac{1}{2}} + 4M^{2} \sigma \left[ {M^{2} - 2\phi + 3\sigma + \sqrt {(M^{2} - 2\phi + 3\sigma )^{2} - 12\sigma M^{2} } } \right]^{{\frac{ - 3}{2}}} } \right\} \\ \end{aligned} $$
(9)
In the absence of positrons (p = 0) and for cold ions (σ = 0), (9) reduces to the one already derived in Ref. [52]. Furthermore, for α = 0 and σ = 0, (9) reduces to the one already derived in Ref. [19]. Equation (9) may be regarded as an energy integral of a classical particle which oscillates in a potential V(ϕ), with a unit mass and a velocity dϕ/dξ at position ϕ. In deriving Eq. (9), we have imposed the boundary conditions ϕ → 0, dϕ/dξ → 0, at ξ → ∞. For solitary wave solutions, the following conditions must be satisfied
$$ \left. {V(\phi )} \right|_{\phi = 0} = 0,\,\left( {\frac{\partial V}{\partial \phi }} \right)_{\phi = 0} = 0 $$
(10)
and
$$ \left( {\frac{{\partial^{2} V}}{{\partial \phi^{2} }}} \right)_{\phi = 0} < 0 $$
(11)
and V(ϕ) < 0 when ϕ lies between zero and ϕ m (ϕ m is the amplitude of the solitons.), i.e., either for 0 < ϕ < ϕ m (compressive solitons) or ϕ m  < ϕ < 0 (rarefactive solitons). For double-layers, the following conditions must be satisfied
$$ \left. {V(\phi )} \right|_{\phi = 0} = 0,\,\left( {\frac{\partial V}{\partial \phi }} \right)_{\phi = 0} = 0\,{\text{and}}\,\left( {\frac{{\partial^{2} V}}{{\partial \phi^{2} }}} \right)_{\phi = 0} < 0 $$
(12)
$$ V(\phi_{m} ) = 0,\,\left( {\frac{\partial V}{\partial \phi }} \right)_{{\phi_{m} }} = 0\,{\text{and}}\,\left( {\frac{{\partial^{2} V}}{{\partial \phi^{2} }}} \right)_{{\phi_{m} }} < 0 $$
(13)
where ϕ m stands for the amplitude of the double-layer.
Let us return to the reality of n, as given in (7). This is real as long as the arguments under the square roots are positive. Introducing an abbreviated notation N = M 2 – 2ϕ + 3σ, one needs N 2 > 12 M 2 and \( N + \sqrt {N^{2} - 12M^{2} \sigma } \), but the latter only works for N > 0. Rewrite
$$ \sqrt {N^{2} - 12M^{2} \sigma } = (N - 2M\sqrt {3\sigma } )(N + 2M\sqrt {3\sigma } ) $$
(14)
to see that the first factor under the square root sign is positive, and so should the second factor be, leading to
$$ N - 2M\sqrt {3\sigma } = (M - \sqrt {3\sigma } )^{2} - 2\phi > 0. $$
(15)

There is on the positive potential side an upper limit, \( \phi_{c} = \frac{1}{2}(M - \sqrt {3\sigma } )^{2} \), which changes only with σ for given M.

Numerical Investigation and Discussion

Let us now investigate the properties of IA solitary waves. Keeping the non thermal parameter at a constant value α = 0.2, Fig. 1 displays the profile of the Sagdeev potential V(ϕ) with respect to ϕ for four different values of the temperature ratio σ. It can be seen that compressive and refractive solitons coexist for σ = 0, 0.001. Once the value of σ increases, the rarefactive solitons immediately disappear. However, compressive solitons persist till a critical value of σ beyond which neither compressive nor rarefactive solitons exist.
Fig. 1

Sagdeev potential V(ϕ) with respect to ϕ for different values of σ. The parameters are indicated on the figure

Next, we consider the effect of non thermal electrons on the nonlinear IA solitary potential. Figure 2 depicts the variation of V(ϕ) for different values of the electron non thermal parameter α. It is obvious from the graphs in Fig. 2 that a decrease in the non thermal parameter drastically affects the compressive IA solitons. It is also observed that the compressive solitons which had disappeared in Fig. 1 for σ = 0.03, have reappeared upon decreasing the value of α.
Fig. 2

Sagdeev potential V(ϕ) with respect to ϕ for different values of α. The parameters are indicated on the figure

Let now investigate the effect of the Mach number M on the IA solitary wave. Figure 3 shows the formation of compressive as well as rarefactive IA solitons. The following partameters α = 0.2, σ = 0.03, p = 0.1, δ = 0.1 and M = 1.35, 1.37, 1.38, 1.39, have been chosen. It is clear that beyond the critical value M c  = 1.35 neither compressive, nor rarefactive localized potential structures exist.
Fig. 3

Sagdeev potential V(ϕ) with respect to ϕ for different values of M. The parameters are indicated on the figure

Thus on using Sagdeev pseudopotential approach it is possible to investigate the nonlinear wave structures over a wider range of parameter space. Clearly the functional dependence of V(ϕ) is very sensitive to the variation in parametersα, σ and M. Now let us examine, numerically, the coexistence of rarefactive and compressive IA solitons. As is evident from Table 1, as the non thermal parameter α increases, higher Mach numbers are involved for the coexistence of rarefactive (R) and compressive (C) ion-acoustic solitary potentials.
Table 1

Effect of M and α on the coexistence of rarefactive (R) and compressive (C) IA solitons, with σ = 0.0001, p = 0.1, and δ = 0.1

M\α

0.15

0.2

0.25

0.3

1.23

–

–

–

–

1.25

C

–

–

–

1.32

R,C

–

–

–

1.33

R,C

C

–

–

1.36

R,C

R,C

–

–

1.42

R,C

R,C

C

–

1.54

R,C

R,C

R,C

C

1.6

R,C

R,C

R,C

R,C

Figure 4 for which δ = p = σ = 0.1 indicates that under certain conditions (α = 0.5, M = 3.63), the ion-acoustic wave develops into a spatially localized double-layer (DL) structure. DLs occur naturally in a variety of space plasma environments (aurorae, solar wind, extra-galactic jets, etc.). The potential jump that they can sustain over a narrow region can energize and accelerate charged particles.
Fig. 4

Sagdeev potential V(ϕ) with respect to ϕ, with δ = p = σ = 0.1. Solid line: α = 0.15, M = 1.506. Dashed line: α = 0.25, M = 1.5825. Dash-dotted line: α = 0.5, M = 3.63

Occurrence of IA-DLs is very sensitive to the electron non thermality (Fig. 5) and the addition of a small concentration of non thermal electrons can even prevent their formation, giving rise to localized solitary potentials.
Fig. 5

Sagdeev potential V(ϕ) with respect to ϕ for different values of α

Similarly, an increase of the ion temperature may destroy these IA-DLs favoring therefore the emergence of IA solitary waves (Fig. 6).
Fig. 6

Sagdeev potential V(ϕ) with respect to ϕ for different values of σ

Figures 4, 5, 6 indicate that the plasma model can just support DL with negative potential. Finally, the small-amplitude expansion of V(ϕ) is given to obtain an analytical solution of double layers. Expanding in ϕ ≪ 1 and neglecting terms of 0(ϕ5), then V(ϕ) can be written
$$ V(\phi ) = - A_{1} \left(\frac{{\phi^{2} }}{2}\right) + A_{2} \left(\frac{{\phi^{3} }}{6}\right) - A_{3} \left(\frac{{\phi^{4} }}{24}\right) $$
(16)
where
$$ A_{1} = p\delta - 1 - 7\beta + 24\sqrt 2 M^{3} (M^{2} + 3\sigma ) $$
(17)
$$ A_{2} = (p\delta^{2} - 1) - 16\sqrt 2 \sigma M^{3} $$
(18)
$$ A_{3} = 3(1 + 3\beta + p\delta^{3} ) + 8\sqrt 2 \sigma M^{3} . $$
(19)
Performing the last step in deriving double layer solution, one gets [53]
$$ \phi (\xi ) = \frac{{\phi_{m} }}{2}\left\{ 1 - \tanh \left[\sqrt {\frac{{A_{3} }}{48}} \phi_{m} \xi \right]\right\} $$
(20)
where \( \phi_{m} = \frac{{2A_{2} }}{{A_{3} }} \) is the amplitude of the double layer and the thickness of the double layer is given by
$$ W_{DL} = \frac{2}{{\left| {\phi_{m} } \right|}}\left( {\frac{48}{{A_{3} }}} \right)^{1/2} . $$
(21)

It is obvious that in the small amplitude limit, A 2 is always negative and A 3 is always positive. So, there exists only DL with negative potential. This result also is in conformity with the obtained results in Figs. 4, 5, 6 for long amplitude IA-DLs. It is obvious that the domain of allowable double layers depend drastically on the plasma parameters and, in particular, on the nonthermal electron distribution and ion temperature effects. In view of Eqs. (16–20), it becomes evident that our plasma model can admit both large and small amplitude ion acoustic double layers with only negative potential.

Conclusion

To conclude, localized ion-acoustic waves have been addressed in a plasma having warm fluid ions, non thermal electrons, and thermal positrons. Our results reveal that in such a plasma, ion-acoustic solitons as well as double-layers may exist. Their spatial patterns depend sensitively on the degree of electron non thermality and ion temperature. In particular, as the non thermal parameter α increases, higher Mach numbers are involved for the coexistence of rarefactive and compressive ion-acoustic solitary potentials. Under certain conditions, the ion-acoustic wave develops into a spatially localized double-layer (DL) structure. The latter occur naturally in a variety of space plasma environments (aurorae, solar wind, extra-galactic jets, etc.). The potential jump that they can sustain over a narrow region can energize and accelerate charged particles. Occurrence of the IA-DLs is very sensitive to the electron non thermality and the addition of a small concentration of non thermal electrons can even prevent their formation, giving rise to localized solitary potentials. Similarly, an increase of the ion temperature may destroy these IA-DLs favoring therefore the emergence of IA solitary waves. The ranges of different plasma parameters used in this investigation are very wide. Thus the results of the present investigation may help to explain the basic features of nonlinear localized ion-acoustic waves in polar cusp region of pulsars and around active galactic nuclei where e–p–i plasma can exist. These results can also help to understand nonlinear structures in plasmas containing nonthermal electrons, as in certain heliospheric environments.

References

  1. 1.
    R.Z. Sagdeev, in Reviews of plasma physics, vol. 4, ed. by M.A. Leontovich (Consultants Bureau, New York, 1966), p. 23Google Scholar
  2. 2.
    H. Ikezi, R. Taylor, D. Baker, Phys. Rev. Lett. 25, 11 (1970)ADSCrossRefGoogle Scholar
  3. 3.
    F.D. Tappert, Phys. Fluids 15, 2446 (1972)ADSCrossRefGoogle Scholar
  4. 4.
    S. Tagare, Plasma Phys. 15, 1247 (1973)ADSCrossRefGoogle Scholar
  5. 5.
    H. Schamel, Plasma Phys. 14, 205 (1972)CrossRefGoogle Scholar
  6. 6.
    H. Schamel, J. Plasma Phys. 9, 377 (1973)ADSCrossRefGoogle Scholar
  7. 7.
    S. Watanabe, J. Phys. Soc. Jpn. 44, 611 (1978)ADSCrossRefGoogle Scholar
  8. 8.
    A.D. Verga, C. Ferro Fontán, Phys. Lett. A 101, 494 (1984)ADSCrossRefGoogle Scholar
  9. 9.
    P.K. Shukla, N.N. Rao, M.Y. Yu, N.L. Tsintsadze, Phys. Rep. 135, 1 (1986)ADSCrossRefGoogle Scholar
  10. 10.
    M.Y. Yu, P.K. Shukla, L. Stenflo, Astrophys. J. 309, L63 (1986)ADSCrossRefGoogle Scholar
  11. 11.
    F.B. Rizzato, J. Plasma Phys. 40, 289 (1988)ADSCrossRefGoogle Scholar
  12. 12.
    T. Tajima, T. Taniuti, Phys. Rev. A 42, 3587 (1990)ADSCrossRefGoogle Scholar
  13. 13.
    F.C. Michel, Theory of neutron star magnetospheres (Chicago University Press, Chicago, 1991)Google Scholar
  14. 14.
    P.K. Shukla, N.L. Tsintsadze, L.N. Tsintsadze, Phys. Fluids B5, 811 (1993)Google Scholar
  15. 15.
    P.K. Shukla, L. Stenflo, Astrophys. Space Sci. 209, 323 (1993)ADSMATHCrossRefGoogle Scholar
  16. 16.
    L.N. Tsintsadze, P.K. Shukla, Plasma Phys. Rep. 19, 134 (1993)ADSGoogle Scholar
  17. 17.
    V.I. Berezhiani, M.Y. El- Ashry, U.A. Mofiz, Phys. Rev. E 50, 448 (1994)ADSCrossRefGoogle Scholar
  18. 18.
    L.N. Tsintsadze, N.L. Tsintsadze, P.K. Shukla, L. Stenflo, Astrophys. Space Sci. 222, 259 (1994)ADSMATHCrossRefGoogle Scholar
  19. 19.
    S.I. Popel, S.V. Vladimirov, P.K. Shukla, Phys. Plasmas 2, 716 (1995)ADSCrossRefGoogle Scholar
  20. 20.
    J. Srinivas, S.I. Popel, P.K. Shukla, J. Plasma Phys. 55, 209 (1996)ADSCrossRefGoogle Scholar
  21. 21.
    G.S. Lakhina, F. Verheest, Astrophys. Space Sci. 253, 97 (1997)ADSMATHCrossRefGoogle Scholar
  22. 22.
    A.M. Mirza, Phys. Plasmas 7, 3588 (2000)ADSCrossRefGoogle Scholar
  23. 23.
    A.A. da Costa, D.A. Diver, G.A. Stewart, Astron. Astrophys. 366, 129 (2001)ADSCrossRefGoogle Scholar
  24. 24.
    Q. Haque, H. Saleem, J. Vranjes, Phys. Plasmas 9, 474 (2002)ADSCrossRefGoogle Scholar
  25. 25.
    H. Saleem, Q. Haque, J. Vranjes, Phys. Rev. E 67, 057402 (2003)ADSCrossRefGoogle Scholar
  26. 26.
    R.S. Tiwari, A. Kaushik, M.K. Mishra, Phys. Lett. A 365, 335 (2007)ADSCrossRefGoogle Scholar
  27. 27.
    R.S. Tiwari, Phys. Lett. A 372, 3461 (2008)ADSMATHCrossRefGoogle Scholar
  28. 28.
    S. Ghosh, R. Bharuthram, Astrophys. Space Sci. 314, 121 (2008)ADSMATHCrossRefGoogle Scholar
  29. 29.
    H.R. Pakzad, Phys. Lett. A 373, 847 (2009)ADSMATHCrossRefGoogle Scholar
  30. 30.
    M.V. Goldman, M.M. Oppenheim, D.L. Newman, Nonlinear Process. Geophys. 6, 221 (1999)ADSCrossRefGoogle Scholar
  31. 31.
    W.C. Feldman, S.J. Anderson, S.J. Bame, S.P. Gary, J.T. Gosling, D.J. McComas, M.F. Thomsen, G. Paschmann, M.M. Hoppe, J. Geophys. Res. 88, 96 (1983). doi: 10.1029/JA088iA01p00096 ADSCrossRefGoogle Scholar
  32. 32.
    R. Lundlin, A. Zakharov, R. Pellinen, H. Borg, B. Hultqvist, N. Pissarenko, E.M. Dubinin, S.W. Barabash, I. Liede, H. Koskinen, Nat. Lond. 341, 609 (1989)ADSCrossRefGoogle Scholar
  33. 33.
    Y. Futaana, S. Machida, Y. Saito, A. Matsuoka, H. Hayakawa, J. Geophys. Res. 108, 151 (2003). doi: 10.1029/2002JA009366 CrossRefGoogle Scholar
  34. 34.
    R. Boström, IEEE Trans. Plasma Sci. 20, 756 (1992)ADSCrossRefGoogle Scholar
  35. 35.
    P.O. Dovner, A.I. Eriksson, R. Boström, B. Holback, Geophys. Res. Lett. 21, 1827 (1994)ADSCrossRefGoogle Scholar
  36. 36.
    R.A. Cairns, A.A. Mamun, R. Bingham, R. Boström, R.O. Dendy, C.M.C. Nairn, P.K. Shukla, Geophys. Res. Lett. 22, 2709 (1995)ADSCrossRefGoogle Scholar
  37. 37.
    S. Ghosh, R. Bharuthram, M. Khan, M.R. Gupta, Phys. Plasmas 11, 3602 (2004)ADSCrossRefGoogle Scholar
  38. 38.
    L.P. Zhang, J.K. Xue, Phys. Plasmas 12, 042304 (2005)ADSCrossRefGoogle Scholar
  39. 39.
    J.K. Xue, L.P. Zhang, Phys. Plasmas 13, 022104 (2006)ADSCrossRefGoogle Scholar
  40. 40.
    W.F. El-Taibany, I. Kourakis, Phys. Plasmas 13, 062302 (2006)ADSCrossRefGoogle Scholar
  41. 41.
    W.F. El-Taibany, M. Wadati, R. Sabry, Phys. Plasmas 14, 032304 (2007)ADSCrossRefGoogle Scholar
  42. 42.
    M. Tribeche, R. Amour, Phys. Plasmas 14, 103707 (2007)ADSCrossRefGoogle Scholar
  43. 43.
    M. Tribeche, G. Boumezoued, Phys. Plasmas 15, 053702 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    H. Hakimi Pajouh, H. Abbasi, Phys. Plasmas 15, 103705 (2008)ADSCrossRefGoogle Scholar
  45. 45.
    H. Alinejad, Astrophys. Space Sci. 325, 209 (2010)ADSMATHCrossRefGoogle Scholar
  46. 46.
    H. Alinejad, Astrophys. Space Sci. 327, 131 (2010)ADSMATHCrossRefGoogle Scholar
  47. 47.
    A.A. Mamun, Phys Rev. E 55, 1852 (1997)ADSCrossRefGoogle Scholar
  48. 48.
    H.K. Malik, Phys. Plasmas 15, 072105 (2008)ADSCrossRefGoogle Scholar
  49. 49.
    H.K. Malik, K. Singh, IEEE Trans. Plasma Sci. 33, 1995 (2005)ADSCrossRefGoogle Scholar
  50. 50.
    Y. Nishida, Phys. Fluids 27, 2176 (1984)ADSCrossRefGoogle Scholar
  51. 51.
    F.F. Chen, Introduction to plasma physics (Plenum, New York, 1974)Google Scholar
  52. 52.
    A.A. Mamun, Eur. Phys. J. D. 11, 143 (2000)ADSGoogle Scholar
  53. 53.
    R. Sabry, Phys. Plasmas 16, 072307 (2009)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Physics, Bojnourd BranchIslamic Azad UniversityBojnourdIran
  2. 2.Theoretical Physics Laboratory, Plasma Physics Group, Faculty of Sciences-PhysicsUniversity of Bab-Ezzouar, USTHBAlgiersAlgeria

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