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To investigate acoustic waves in an un-magnetized uniform Maxwellian plasma, a set of coupled equations (Fried and Gould, 1961) was considered to obtain numerical solutions of the electrostatic dispersion equation. Watanabe and Taniuti (1977) were the first who pointed out the existence of strongly damped electron-acoustic (EA) waves when cool electron number density is assumed to be higher compared to hot electron number density, otherwise, an EA solitary wave propagate’s in a two-temperature electron Maxwellian plasma.

Gary and Tokar (1985) proposed the necessary conditions such as Tc=Th≪1 and 0 < nc0=nh0 < 0:8 for the propagation of EA waves, where Tc ðThÞ denotes the cool (hot) electron temperature with cool (hot) electron density nc0 ðnh0Þ: Moreover, on ionic-timescale an ion-acoustic (IA) wave can propagate both in unmagnetized and magnetized Maxwellian plasmas under the constraints vti≪vph≪vte: The IA wave was theoretically predicted by Tonks and Langmuir (1929) and later experimentally verified by Revans (1933).

Most of the research works were carried out by considering the single component of the electrons. However, some attempts were involved with the existence of two distinct groups of electrons, affecting the characteristics of IA waves (Yadav et al., 1995) significantly. Kourakis and Shukla (2003) presented the dispersive properties of IA waves with two-temperature electrons to study oblique modulation and envelope excitations. It was found that localized excitations are in good agreement with the satellite observations concerning the atmosphere of magnetosphere.

The electromagnetic radiation is the most important source of energy and momentum for optical investigations. The laser beams can be transformed into the planar and non-planar wavefronts. In non-planar helical wavefronts, the Poynting and wave vectors spiral around the beam axis and each photon carries an additional azimuthal component of the momentum. Light beams showing helical wavefronts can be represented by the Laguerre-Gaussian (LG) beam solutions (Allen et al., 1992). The LG beam functions not only satisfy the basic orthogonality conditions for the beam representation but also carry the spin and orbital angular momenta (Allen et al., 1992) owing to the polarization states and non-uniform phase structures, respectively.

Due to finite orbital angular momentum (OAM) states, many twisted plasma modes and instabilities have been investigated by LG beam solutions instead of plane wave solutions. Potential applications of twisted waves exist in many astrophysical plasma environments (Harwitt, 2003), e.g., in Earth's ionosphere, in magnetoplasmas (Shukla, 2013), in solar corona, as well as in laboratory plasmas. Recently, Khan et al. (2014) discussed the dispersion properties of IA plasma vortice’s with the best agreement of numerical and analytical results. Rehman et al. carried out investigations to find OAM states of twisted EA waves with double Kappa distributions and examined weakly damped regions with strong dependence of OAM states leading to enhanced Landau damping rates.

In this paper, we consider the Vlasov-Poisson coupled set of equations to derive a generalized dielectric constant for electrostatic waves.

**Discussion**

The real wave frequency of TIA waves has strong dependence on hot-to-cool electron number density ratio fð¼ n0h=n0cÞ which can be clearly examined from. The contours represent the variation of normalized TIA wave frequency ω~iað¼ ωia=ωpiÞ against the twisted parameter ηð¼ k=lqφÞ and normalized wave number ~kð¼ kλDÞ for varying (a) f ¼ 0:4 and (b) f ¼ 0 with κh ¼ 4 and κc ¼ 2: A noticeable difference in the magnitudes of TIA wave frequency is shown, the frequency enhances at larger value of density ratio i.e. at f ¼ 0:4: Hence, the hot electrons contribute significantly to enhance the TIA wave frequency. The twisted parameter (η) and density ratio (f) modify the growth rates of TIA waves and their impacts in the limit ~v0 > ~vcr, respectively. The critical value for which TIA waves become unstable can be calculated at κh ¼ 7; κc ¼ 4:0; η ¼ 1:5 and ~k ¼ 0:7; as vcr½¼ vphð1 þ R'=Q'Þ ¼ 926vti. It may be seen that twisted waves show large growth rate and as η increases, we approach towards the planar wave solution. Consequently, the wave shows a reduction in the growth rate. It means that TIA wave have been strong interaction with the resonant plasma species to gain free energy from them. The streaming hot electrons are the ultimate source of the free energy in this region R 15:2Rs. It shows how the density ratio ”f” modifies the growth rate instability

**Summary**

To summarize, we have considered twisted electron-acoustic (TEA) and twisted ion-acoustic (TIA) waves in a collisionless unmagnetized plasma containing superthermal hot and cool electrons with positive ions. Such a plasma situation exist’s in the Saturn's magnetosphere. Within the framework of Vlasov-Poisson theory, was a generalized dielectric constant is derived by expressing the perturbed distribution function and electrostatic potential in terms of Laguerre-Gaussian functions. The dispersion relations and Landau damping rates of the TEA and TIA waves are obtained and analyzed numerically for different plasma parameters such as the dimensionless twisted parameter ηð¼ k=lqφÞ; the density ratios and nonthermal spectral indices κc and κh: It is found that the impact of twisted parameter significantly alters the dispersive and grwoth rate characteristics of the TEA and TIA waves. Since the outer magnetosphere of Saturn is directly influenced by the rotating solar wind, therefore, an attention has been paid to study low density variation ð0:1 0:5Þ cm3 in outer region of magnetosphere ½R ð12 20ÞRs for numerical evaluation of dispersion relations and damping rates.

However, the growth rate instability thresholds for the TEA and TIA waves occur at vcr ¼ 0:6vtc and 926vti; respectively, within the region at distance R15:2Rs: Schippers et al. (2008) have modelled super-thermal hot and cool electrons by using the double Kappa distribution functions and showed a better fit for third (outer) region of Saturn's magnetosphere. They have made their observations (Schippers et al., 2008) by the instruments such as CAPS/ELS (0.6 eV–26 keV) and MIMI/LEMMS (15 keV–10 MeV) on board Cassini. Masters et al. (2010) have already observed the plasma vortex in Saturn's dayside outer magnetosphere and discussed the consistency of observations relating to the inner edge of the boundary layer due to spacecraft encounter with the plasma vortex of highly energetic super thermal electrons. The results are important for understanding the non-thermality effects on the dispersion relations and growth rates of twisted waves in a double κ distributed plasma and may prove useful for particle transport and trapping of plasma particles in Saturn's outer magnetosphere

**Note: This work is partly presented at International Conference on Atomic, Nuclear and Plasma Physics on November 19-20, 2018 at Sydney, Australia.**

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