# Influence of suprathermality on the obliquely propagating dust-acoustic solitary waves in a magnetized dusty plasma

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10509-012-1316-y

- Cite this article as:
- Shahmansouri, M. Astrophys Space Sci (2013) 344: 153. doi:10.1007/s10509-012-1316-y

- 9 Citations
- 138 Views

## Abstract

Dust-acoustic (DA) solitary waves are investigated in a magnetized dusty plasma comprising cold dust fluid and kappa-distributed ions and/or electrons. The influence of suprathermal particles, obliqueness, and ion temperature on the DA solitary waves is investigated. We find that only negative DA solitary waves will be excited in this model. Also it is shown that the amplitude of the DA solitary wave decreases with deviation of electrons or ions from Maxwellian distribution via decrease of *κ*_{e} or *κ*_{i}. The effect of the temperature of the ion decreases with the amplitude and steepness of the solitary wave front.

### Keywords

Dust-acoustic wavesSuprathermal electrons/ionsSolitary wavesMagnetized plasmas## 1 Introduction

Dust-acoustic waves, belonging to the most important low frequency electrostatic dust associated waves, were predicted theoretically by Rao et al. (1990). Later, these waves have been observed in laboratory experiments by Barkan et al. (1995). The basic properties of DA waves in a dusty plasma has been studied by a number of authors (Melandso et al. 1993; Rosenberg 1993; D’Angelo 1995; Mamun 1999a, 1999b; Ghosh et al. 2001; Misra and Chowdhury 2006a, 2006b; El-Labany et al. 2002, 2008, 2010; Rahman et al. 2008; Pakzad 2010; Das and Devi 2010; Tribeche and Benzekka 2011; Mayout and Tribeche 2011; Alinejad 2011; Shahmansouri and Tribeche 2012). It should be noted that the harmonic generated nonlinearity leads to a small amplitude for the dust-acoustic solitary waves, which satisfy the Korteweg–de-Vries (KdV) equations (Misra and Chowdhury 2006a, 2006b, 2006c). Mamun et al. have studied DA solitary waves in an electron depleted unmagnetized dusty plasma comprising a cold dust fluid, and Maxwellian (1996a) and non-Maxwellian (1996b, 1998) ions. The effect of trapped electrons and nonisothermal ions on DA solitary waves in a self-gravitating complex plasma has been discussed by Misra and Chowdhury (2006a). The nonlinear propagation of small amplitude dust ion acoustic solitary waves in an ion beam driven plasma has been studied by Adhikary et al. (2010). Furthermore, Misra and Adhikary (2011) have investigated the nonlinear propagation of large amplitude dust ion acoustic solitary waves in an ion-beam plasma. They found that three modes were in existence in the system, and they discussed the necessary conditions for the propagation of these modes as solitary waves.

It is well known that an external magnetic field can modify the propagation properties of electrostatic solitary structures. The effect of an ambient external magnetic field on the electrostatic waves has been studied by a number of authors (Mamun 1998a, 1998b; Alinejad and Mamun 2011; Mamun and Hassan 2000; Zhang and Xue 2005; Mahmood and Akhtar 2008; Anowar and Mamun 2008; El-Labany et al. 2008; Saha and Chatterjee 2009). Obliquely propagating DA solitary waves in a hot magnetized dusty plasma with Maxwell–Boltzmann-distributed ions and electrons have been investigated by Mamun et al. (1998). The effect of charge fluctuations has been considered in the next work of Mamun and Hassan (2000). El-Labany et al. (2004) have studied the DA solitary waves in a hot magnetized dusty plasma through the Zakharov–Kuznetsov (ZK) equation in a homogeneous medium. Dust-acoustic solitary waves in an inhomogeneous magnetized hot dusty plasma have been discussed by Misra and Chowdhury (2006c) taking into account the effect of dust charge fluctuations. They found that the dynamical behavior of DA solitary waves obeys the ZK equation. Oblique propagation of DA solitary waves in the tropical mesospheric plasma in the presence of variable charge and rotation of the plasma have been studied by Mushtaq et al. (2006). Samanta et al. (2007) have studied the oblique propagation of large amplitude DA solitary waves in a magnetized hot dusty plasma consisting of nonthermal ions. Their model supported the coexistence of compressive and rarefactive solitary structures.

The characteristics of linear and nonlinear structures are found to depend significantly on the distribution function of components. Dust-acoustic solitary waves have been studied based on Maxwell–Boltzmann (Mushtaq et al. 2006), cortex-like (Mamun 1998b; El-Labany et al. 2008), nonthermal (Misra and Chowdhury 2006a; Mamun et al. 1996b) and kappa (Shahmansouri and Tribeche 2012)-distributed components. Numerous observations (Vasyliunas 1968; Leubner 1982; Armstrong et al. 1983) and studies (Summers and Thorne 1991; Baluku et al. 2008, 2010; Hellberg et al. 2009; Chatterjee and Ghosh 2011; Sultanu and Kourakis 2012) indicate that the distribution functions of the components can be included in the suprathermal particles. External forces acting on the neutral space plasma or wave-particle interaction may lead to the formation of the suprathermal particles. Suprathermal plasmas, which are characterized by a long tail in the high-energy region, may generally be modeled by a kappa-like distribution. In the limit of large values of the parameter *κ*, the *κ*-distribution reduced to the Maxwellian distribution, and for low values of *κ*, they present a hard spectrum including a strong tail with power-law form at high speed (Vasyliunas 1968; Leubner 1982; Armstrong et al. 1983), since it fits both the thermal as well as the suprathermal parts of the observed velocity spectra (Pierrard and Lemaire 1996; Christon et al. 1988; Maksimovic et al. 1997; Krimigis et al. 1983; Hasegawa et al. 1985). Suprathermal electrons and ions are often present in space and astrophysical plasma environments, such as ionosphere, mesosphere, magnetosphere, lower atmosphere, magneto-sheet, terrestrial plasma-sheet, radiation belts, and auroral zones (Collier 1993; Maksimovic et al. 2000; Antonova et al. 2003; Mori et al. 2004; Pierrard and Lazar 2010). The suprathermal behavior of plasma also was observed in the laboratory plasma, for instance in the case of laser-matter interaction and of plasma turbulence (Magni et al. 2005).

*θ*

^{2}=2

*T*

_{j}(

*κ*−3/2)/

*κm*

_{j}is the effective thermal speed,

*Γ*is the gamma function,

*j*(=

*e*,

*i*) refers to the types of particle, and

*κ*is the spectral index. The spectral index measures the deviation from a Maxwellian distribution, as the smaller values of

*κ*denote the more suprathermal particles in the distribution function tail (and the harder energy spectrum). In the limit of

*κ*→∞, the kappa distribution recovers the Maxwellian distribution.

The suprathermality effect can modify the profile of obliquely propagating electrostatic solitary waves (Sultanu et al. 2010; Nouri Kadijani et al. 2011; Alinejad and Mamun 2011). Sultanu et al. have investigated the influence of electron suprathermality on the oblique propagation of ion acoustic solitary waves in a magnetized plasma. Recently, the combined influence of the suprathermal electrons, the magnetic field, and the direction of propagation on the electrostatic solitary waves has been examined by Alinejad and Mamun (2011).

To complement the previously published papers, here we propose to extend the work of Mamun et al. (1998) to the situations that ions and electrons have a high-energy-tail distribution. As the aim of our study we investigate the basic properties of DA solitary waves in a magnetized dusty plasma, in which the dust grains are fluid, and ions and electrons have a kappa distribution. The reductive perturbation technique is employed to investigate the influence of suprathermality effects, electron population, ion temperature and obliqueness on the DA wave properties.

## 2 Basic equations

*n*

_{e0}+

*Z*

_{d0}

*n*

_{d0}=

*n*

_{i0}, where

*n*

_{e0},

*n*

_{i0}and

*n*

_{d0}are the equilibrium density of electrons, ions, and dust grains, respectively; and

*Z*

_{d0}is the number of charge residing on the dust grain. From the charge neutrality condition we define

*f*=

*n*

_{e0}/

*Z*

_{d0}

*n*

_{d0}=

*n*

_{i0}/

*Z*

_{d0}

*n*

_{d0}−1. We assume this system to be immersed in an external magnetic field \((B_{0}\parallel \hat{z})\). The dynamics of the low phase velocity DA wave is governed by the following normalized equations:

*n*

_{j}→

*n*

_{j}/

*n*

_{j0},

*u*

_{j}→

*u*

_{j}/

*c*

_{d},

*x*→

*x*/

*λ*

_{Dd},

*t*→

*tω*

_{pd},

*ϕ*→

*eϕ*/

*T*

_{e},

*ω*

_{c}→

*ω*

_{c}/

*ω*

_{pd}. Furthermore, \(\omega_{pd} = \sqrt{4\pi n_{d0}e^{2}Z_{d0}^{2}/m_{d}}\) is the dust plasma frequency, \(c_{d} = \sqrt{T_{i}Z_{d0}/m_{d}}\) is the dust-acoustic speed, and

*λ*

_{Dd}=

*c*

_{d}/

*ω*

_{pd}is the dust Debye length. Here,

*j*=

*i*,

*e*,

*d*denote, respectively, ions, electrons, and dust grains, the subscript “0” stands for equilibrium quantities, and the other variables have their usual meaning.

*σ*=

*T*

_{i}/

*T*

_{e}.

*c*

_{1}=

*σ*(2

*κ*

_{e}−1)/(2

*κ*

_{e}−3) and

*b*

_{1}=(2

*κ*

_{i}−1)/(2

*κ*

_{i}−3). It is clear that the linear dispersion relation, which here has been obtained for DA solitary waves, includes the suprathermality effects, external magnetic field, and obliqueness. It should be noted that as the cyclotron frequency tends to zero, in the limit of (

*κ*

_{e},

*κ*

_{i})→∞, Eq. (6) reduces to Eq. (7) of Rao et al. (1990) for an unmagnetized dusty plasma with Maxwell–Boltzmann ions and electrons. Also, in this case Eq. (6) covers Eq. (2) of Pieper and Goree (1996) in the absence of the damping term.

*k*

_{⊥}=0), Eq. (6) reduces to

*k*

_{z}=0) takes this form:

*κ*

_{e},

*κ*

_{i})→∞, Eq. (8) is in accordance with results obtained by Shukla and Rahman (1998). In the long wavelength limit, Eq. (8) reduces to the results of Shukla (1992), which were obtained for a dust-cyclotron wave.

Indeed Eq. (6) shows two distinct modes with different phase velocities in the system, for the plus/minus sign. The phase velocity for the plus sign is larger than for the minus sign, thus the mode with larger phase velocity is called the fast mode, whereas the mode with smaller phase velocity is known as the slow mode. Furthermore, it can be seen that the frequency of the fast mode is larger than the cyclotron frequency (*ω*_{c}≪*ω*), while the frequency of the slow mode is smaller than the cyclotron frequency *ω*≪*ω*_{c}.

*ω*,

*k*) in a magnetized dusty plasma for different several values of

*κ*

_{e}and

*κ*

_{i}. Dispersion relation of DA modes is depicted in Fig. 1, for different values of direction of propagation. When propagation of wave becomes more oblique, the frequency of the fast mode increases while the slow mode experiences a decrease in the frequency. Thus, the separation between the two modes increases with the angle between the wave propagation and the external magnetic field. It is obvious that when a wave propagates perpendicular (or parallel) to the magnetic field, only a single mode exists in the system. Figure 2 shows the effect of the suprathermal character of electrons/ions on the dispersion properties of the DA modes. It is clear that the frequency of both two modes increases with increase of the suprathermality character of system, via a decrease of the spectral index of electrons/ions.

## 3 Derivation of KdV equation

*ε*is a real small parameter measuring the strength of nonlinearity;

*l*

_{x},

*l*

_{y}, and

*l*

_{z}are the directional cosines of the wave vector \(\vec{k}\) along, respectively, the

*x*,

*y*, and

*z*axes; and

*V*

_{0}is the normalized wave phase velocity. The expansion of dependent variables about their equilibrium values are considered:

*ε*, the lowest order of

*ε*leads to

*l*

_{z}=1, then Eq. (12) reduces to Eq. (15) of Baluku et al. (2008). Also for (

*κ*

_{e},

*κ*

_{i})→∞ this relation corresponds to that obtained by Mamun and Hassan (2000), for the case of constant charge.

*ε*leads to the following set of equations:

*c*

_{2}=

*σ*

^{2}(2

*κ*

_{e}−1)(2

*κ*

_{e}+1)/2(2

*κ*

_{e}−3)

^{2},

*b*

_{2}=(2

*κ*

_{i}−1)(2

*κ*

_{i}+1)/2(2

*κ*

_{i}−3)

^{2}. Now, using (12) we can eliminate

*n*

_{d2},

*u*

_{dx2},

*u*

_{dy2},

*u*

_{dz2}, and

*ϕ*

_{2}. The elimination of these second-order variables leads to the KdV equation:

*κ*

_{e},

*κ*

_{i})→∞, Eq. (14) reduces to Eq. (14) of Mamun et al. (1998), and also it recovers Eq. (15) of Mamun and Hassan (2000) in the absence of charge fluctuations. Moreover, if the cyclotron frequency tends to zero in the limit of Maxwellian ions and electrons, Eq. (14) reduces to the KdV equation that has been obtained by Mamun (1999a) for an unmagnetized dusty plasma with Maxwell–Boltzmann distributed ions and electrons.

*α*and

*β*with ion suprathermality character, for different values of the other physical parameters, separately. Figures 3(a)–(c) indicate the behavior of the nonlinear coefficient of the KdV equation,

*α*, with respect

*κ*

_{i}, respectively, for different values of obliqueness, electron population, and ion temperature. It is clear that

*α*has a decreasing behavior with

*κ*

_{i}. This means that deviation of ions from Maxwellian behavior leads to the larger values of the nonlinear coefficient. Figure 3a shows that as propagation goes more oblique the

*α*value maybe decreases. Also it is clear that the solitary structure is highly sensitive to the electrons population, as in the presence of more electrons the value of

*α*maybe increases (Fig. 3b). Figure 3c shows that the nonlinear coefficient decreases with ion temperature. Then, we have plotted the variation of the dispersion coefficient of the KdV equation,

*β*, with respect to

*κ*

_{i}, for different values of obliqueness, electrons population, ion temperature, and strength of the external magnetic field in Figs. 4(a)–(d), respectively. It is evidence that the dispersion coefficient has an increasing behavior with the spectral index of the ions

*κ*

_{i}. These figures show that the dispersion coefficient decreases with electrons population, ion temperature, and magnitude of the external magnetic field, while the obliqueness leads to an increase of the dispersion coefficient.

*ξ*and

*τ*to

*ζ*=

*ξ*−

*U*

_{0}

*τ*′, and

*τ*′=

*τ*. Applying the appropriate boundary conditions, the stationary soliton wave solution of Eq. (14) takes this form (Washimi and Taniuti 1966):

*ϕ*

_{m}=3

*U*

_{0}/

*α*and \(\Delta= \sqrt{4\beta/ U_{0}}\), respectively. It must be noted that the mutual balance between dispersion and nonlinearity leads to the formation of this type of soliton solution. It is clear that an increase in

*U*

_{0}enhances the height of the soliton wave, and decreases its width.

*κ*

_{i}) the DA solitary waves appear with smaller amplitude. Figure 3b shows the influence of the magnetic field on the DA soliton. We see that an increase of the magnetic field strength leads to an increase of the soliton thickness, while its amplitude remains unchanged. Thus, a stronger magnetic field yields steeper solitary structures with similar amplitudes. Figure 5c indicates that the amplitude of DA soliton increases when the wave propagates more obliquely.

## 4 Conclusions

A suprathermal magnetized dusty plasma comprising a cold dust fluid and kappa-distributed ions/electrons, has been considered. The effects of the relevant physical parameters, such as suprathermality, obliqueness, and external magnetic field, on the DA soliton characteristics have been investigated by employing the reductive perturbation method.

When the propagation of the wave is more oblique, the frequency of the fast mode increases, while the slow mode experiences a decrease in frequency. Thus separation between the two modes increases with the angle between wave propagation and magnetic field. The influence of suprathermality on the dispersion properties is in decreasing of the frequency of both modes, fast and slow.

We found that the DA solitary profile is significantly sensitive to the suprathermal character of the dusty plasma. It is shown that as ions tend to thermodynamic equilibrium, DA solitary waves may be produced with larger amplitude. It can be seen that the effect of obliqueness on the DA wave solitary structure is enhanced as regards strength and decrease of steepness; obliqueness leads to broader DA solitary waves. It must be noted that at higher values of *θ*, the DA solitary amplitude reaches large values and our model (which is only valid in the limit of small but finite amplitude) may be unreliable. Also it is clear that in the presence of more electrons the DA soliton may be excited with a smaller amplitude. It is shown that the ion temperature effect appears in a decrease of the DA soliton amplitude. Thus the temperature of the ions has a destructive effect on the formation of solitary waves. Furthermore, we found that the magnitude of the external magnetic field only affects the width of the DA solitons.

The above results should be applicable to the formation of nonlinear DA solitary wave structures in regions in which dust grains are embedded in a kappa-distribution plasma, such as Saturn’s magnetosphere. Furthermore they may also explain the strong spiky waveforms observed in auroral electric field measurements (Ergun et al. 1998) and already predicted by Lotko and Kennel (1983). Particularly, according to the observations of both *κ*-distributed electrons and ions in Saturn’s magnetosphere (Krimigis et al. 1983; Schippers et al. 2008), these results may be applicable to the description of the dust-acoustic solitary waves in Saturn’s magnetosphere.