The characteristics of nonlinear and supernonlinear Alfvén waves propagating in a multicomponent plasma composed of a double spectral electron distribution and positive and negative ions were investigated. The Sagdeev technique was employed, and an energy equation was derived. Our findings show that the proposed system reveals the existence of a double-layer solution, periodic, supersoliton, and superperiodic waves. The phase portrait and potential analysis related to these waves were investigated to study the main features of existing waves. It was also found that decreasing the electron temperature helps the superperiodic structure to be excited in our plasma model. Our results help interpret the nonlinear and supernonlinear features of the recorded Alfvén waves propagating in the ionosphere D-region.
A A El-Tantawy, W F El-Taibany, S K El-Labany, A M Abdelghany. Formation of Alfvénic supernonlinear waves in a plasma containing double spectral distributed electrons[J]. Communications in Theoretical Physics, 2024, 76(12): 125501. DOI: 10.1088/1572-9494/ad6de4
1. Introduction
Alfvén waves are low-frequency electromagnetic waves that propagate in the oblique direction of the background magnetic field [1, 2]. These waves are dispersive owing to the finite ion gyroradius effect, and they preserve nonzero electric field fluctuation in the direction of the magnetic field. These waves are classified into two modes: inertial and kinetic Alfvén waves. The inertial dispersive Alfvén wave can propagate in a plasma medium with $\beta \ll \tfrac{{m}_{e}}{{m}_{i}}$, where β denotes the ratio of the kinetic to magnetic pressures [3], and me and mi are the masses of electrons and ions, respectively. In this mode, electron inertia is considered in the direction of the magnetic field, and the electron thermal speed is much lower than the Alfvén velocity. This is known as a slow shear Alfvén wave. However, the kinetic Alfvén wave can propagate in a low-β plasma, but with $\beta \gg \tfrac{{m}_{e}}{{m}_{i}}$, in this type, the electron thermal speed is much higher than the Alfvén velocity. Accordingly, the electrons are considered to be hot and inertialess. The latter case is known as a fast shear Alfvén wave and is also the proposed case in our study. Dispersive Alfvén waves help in interpreting the phenomena of energy transport processes in several plasma media, such as magnetopause [4, 5], and also aid in understanding the mechanism of the acceleration process of the particles existing in the solar wind [6]. According to Freja satellite data, strong and spiky electromagnetic waves prevailed in the auroral region; hence, an Alfvénic solitary structure was created [7]. The study of the nonlinear features of Alfvén waves has been the focus of many researchers [8–11]. Hasegawa and Mima [1] pioneered the investigation of the solitary Alfvénic structure that propagates in low-β plasma using the Sagdeev approach. Shukla et al [12] continued to study solitary Alfvén waves, and they displayed super-Alfvénic solitons that differed from the previously deduced solitons [1]. Nauman and Mushtaq [11] studied the linear and nonlinear features of kinetic Alfvén waves considering the presence of sub-Alfvénic solitary structures. Shahida et al [10] derived the Korteweg–de Vries equation by using the Poincaré–Lighthill–Kuo method to calculate the phase shift collision that results from the interaction of two opposite Alfvénic solitons. Hafeez et al [13] studied the linear and nonlinear features of the coupled Alfvén and ion acoustic waves that propagate in a magnetized electron–ion plasma, where the cnoidal and soliton wave structures were obtained. They concluded that a low-compressive-amplitude structure could exist in a magnetized plasma.
Various types of nonlinear structures exist in diverse plasma media, such as solitons and periodic and shock waves. Such nonlinear waves play a significant role in both theoretical and experimental investigations [14]. Recently, Dubinov and Kolotkov [15] introduced a new interesting structure called supernonlinear waves in five-species plasma. Later, they managed to prove the existence of supernonlinear waves in four-species plasma [16]. Supernonlinear waves are known for their very high amplitudes and very long wavelengths [17, 18]. They are characterized by the nontrivial topology of their phase portraits, which means that their trajectories in phase space can have multiple stable equilibrium points and separatrix layers [15, 16], giving a more complex structure to their phase portraits. Therefore, a supernonlinear structure can be distinguished from other types of nonlinear structures in plasma, such as solitons, by its complex phase portrait. Accordingly, there is a wide range of different wave behaviors that provide additional information. It is known that the topological characteristics of supernonlinear waves vary with changes in the physical oscillating system. One of such supernonlinear waves is the supersoliton structure. Supersolitons or supersolitary waves are novel nonlinear coherent structures that represent a group of solitary supernonlinear waves with the unique characteristics of solitary structures and extreme properties of supernonlinear waves. Such a structure can be found beyond the existence range of a double-layer (DL) structure, and when we approach the DL domain, a supersoliton structure appears. Supersolitons can be distinguished by their unusual Sagdeev pseudopotential because they have three local maxima and a distorted electric field [18]. The supersoliton electric field has a distinct bipolar structure (wiggle shape). Sometimes, it is deceitful to distinguish the actual existence of this structure from the regular soliton. Accordingly, to ensure the existence of a supersoliton, its electric field should be obtained because it represents a signature for its existence [19]. This electric field has been recorded in space plasma [20, 21]. Another supernonlinear structure is the superperiodic structure, which has a Sagdeev potential curve with at least two minima separated by a maximum. These minima and maximum correspond to the center and saddle points, respectively. In addition, superperiodic waves can be distinguished in phase portraits by closed trajectories that indicate the presence of invariant energy in the corresponding dynamic system [18]. Ragab and El-Taibany [22] demonstrated the existence of superperiodic and supersolitons through phase portrait analysis in three-component dusty plasma. They also proved the existence of rarefactive and compressive soliton structures. Verheest [23] proposed a three-component dusty plasma model composed of negative dust and two hot ion species. He [23] employed the Sagdeev potential to obtain a supersoliton structure and found that it exists between two domains. Abulwafa et al [19] investigated the existence of supernonlinear waves in six-component plasma. They also employed the Sagdeev approach to obtain DL and supersoliton structures in addition to the soliton structure. Moreover, they found that the potential curves are sensitive to the mass ratio because by decreasing the value of the mass ratio, the potential curve corresponding to the supersoliton wave appears. They applied their findings to cometary plasma [19].
Plasma species significantly affect the properties of existing nonlinear waves, such as the presence of negative ions, as postulated in this study. Such plasmas have attracted the attention of many researchers [24, 25]. Plasma containing negative ions is found near Earth [26] as well as in laboratories [27]. In addition, it has significant technical applications, such as plasma processing reactors [28], neutral beam sources [29], and semiconductor materials [30]. Atteya et al [25] proposed a model of hot negative and positive ions with superthermal electrons. They [25] studied the influence of the number density of negative ions on the structure of soliton waves. They found that the amplitude and width of the soliton structure are enhanced by increasing the number density of negative ions [25].
Most of the studies on Alfvén waves were conducted by postulating that electrons are governed by the Maxwell distribution because the electrons are assumed to be in thermal equilibrium. However, observations proved that space plasmas, such as solar wind [31] and ionosphere region [32], possess high-energy tails or flat shoulders in the distribution profile [33, 34]; therefore, the plasma particles deviate from the Maxwell distribution, and the non-Maxwellian distributions are more favorable to use in such regions [31, 32]. Such non-Maxwellian distributions introduce a realistic fit to the actual observed distributions. This means that there is a strong need to restudy the dispersion relations and stability and instability properties of the space regions, which were discussed previously, to obtain a much more accurate physical interpretation for many physical phenomena that occur in space regions [35]. One of these non-Maxwellian distributions is the kappa (κ) distribution [34, 36], where the spectral index κ represents a measuring tool for high-energy particles that reside on the tail of the distribution function. Many studies have used kappa distribution [37–39]. Masood et al [37] employed both kappa and Cairns distributions to study Alfvénic solitary waves, and they obtained compressive and rarefactive solitary structures. On the other hand, for the Maxwellian limit (κ → ∞ ), they obtained only a compressive soliton structure. However, many space events cannot be interpreted using the previously mentioned distributions [33]. The (r,q) distribution is sometimes called double spectral distribution (DSD) [40, 41] because it has two dimensionless spectral indices: r, which describes the flat shoulders, and q, which describes the high-energy tails of the distribution function. The DSD is the most generalized distribution because it can describe both low- and high-energy regions in the concerned domain [42]. El-Labany et al [42] adopted the derivative expansion perturbation technique to derive a nonlinear Schrödinger equation, and the modulation instability of dust acoustic waves in a plasma composed of Maxwellian ions, electrons with DSD, and positive and negative dust grains was investigated. They found that the modulation instability growth rate is significantly affected by varying the values of the spectral indices r and q as it decreases by increasing both r and q.
In this paper, we introduce the results of a study of Alfvénic supernonlinear waves that propagate in a plasma containing a double spectral electron distribution (DSED). In addition, the two parameters ${\beta }_{+}={T}_{{\rm{e}}}/({V}_{{\rm{A}}}^{2}{m}_{+})$ and ${\beta }_{-}\,={T}_{{\rm{e}}}/({V}_{{\rm{A}}}^{2}{m}_{-}$) are introduced because of the omnipresence of positive and negative ions in our model, where Te, ${V}_{{\rm{A}}}={B}_{0}/\sqrt{4\pi {n}_{{0}_{+}}{m}_{+}}$, m+, m−, and ${n}_{{0}_{+}}$ are the electron temperature, Alfvén velocity, mass of positive ions, mass of negative ions, and unperturbed number density of positive ions, respectively. The remainder of this paper is organized as follows. Section 2 presents the basic equations. Section 3 presents a Sagdeev pseudopotential analysis. Section 4 discusses the existing nonlinear waves. Finally, section 5 concludes the study.
2. Basic equations
To investigate the properties of nonlinear and supernonlinear Alfvénic waves, we propose a three-component magnetized plasma model. This model is composed of the DSED and positively and negatively charged ions immersed in an ambient magnetic field in the z-direction, i.e. ${B}_{0}={B}_{0}\hat{z}$. At equilibrium, the quasineutrality condition reads ${n}_{{0}_{+}}={n}_{{0}_{-}}+{n}_{{{\rm{e}}}_{0}}$ or δ + μ = 1, where $\delta =\tfrac{{n}_{{{\rm{e}}}_{0}}}{{n}_{{0}_{+}}}$, $\mu =\tfrac{{n}_{{0}_{-}}}{{n}_{{0}_{+}}}$, and ${n}_{{0}_{-}}$ and ${n}_{{{\rm{e}}}_{0}}$ represent the unperturbed number densities of negative ions and electrons, respectively. The wave propagation was considered to be in the x–z plane. In view of the two-potential theory and for low-β plasma, the parallel and perpendicular electrostatic potentials, $\Psi$ and φ, were determined through the following relations: ${E}_{x}=-\frac{\partial \phi }{\partial x}$, ${E}_{z}=-\tfrac{\partial {\rm{\Psi }}}{\partial z}$, and Ey = 0 [43]. The set of normalized governing equations before normalization is presented in appendix A, which describes the proposed system as follows:
where n+ and n− denote the number densities of positive and negative ions, respectively. Additionally, (${v}_{{x}_{+}}$, ${v}_{{z}_{+}}$) represent the positive ion velocity components, whereas (${v}_{{x}_{-}}$, ${v}_{{z}_{-}}$) represent the negative ion velocity components. The drift velocities of positive and negative ions are given as follows:
The final formula for the electron number density is given in appendix B. The DSD has two spectral indices that follow the conditions q > 1 and $q(r+1)\gt \tfrac{5}{2}$ [46]. The DSD function shows that by varying the value of r and fixing the value of q, there are notable changes in the shoulders, as well as a reduction in the contribution of the plasma energetic particles on the tail. Likewise, by varying the value of q and fixing the value of r, the width of the DSD function increases, and the tail still contains high-energy particles. The DSD function turns to kappa distribution when r = 0 and q ⟶ κ + 1, and to Maxwellian distribution when r = 0 and q ⟶ ∞ [45].
The previous physical quantities were normalized as follows. The number densities of electrons and positive and negative ions, ne, n+, and n−, were normalized by the equilibrium number density of positive ions, ${n}_{{+}_{0}}$. The velocity components $({v}_{{x}_{+}},{v}_{{x}_{-}},{v}_{{z}_{+}}$, and ${v}_{{z}_{-}}$) were scaled using Alfvén wave speed, VA. In addition, the time variable was normalized using the reciprocal of ${{\rm{\Omega }}}_{{{\rm{i}}}_{+}}=\tfrac{{{eB}}_{0}}{{m}_{+}c}$, where ${{\rm{\Omega }}}_{{{\rm{i}}}_{+}}$ represents the positive ion cyclotron frequency. The space variable was scaled using ${V}_{{\rm{A}}}/{{\rm{\Omega }}}_{{{\rm{i}}}_{+}}$ and the potentials (φ, $\Psi$) using Te/e.
3. Sagdeev pseudopotential analysis
To investigate the proposed plasma model, the Sagdeev potential technique was employed to examine the creation of nonlinear and supernonlinear Alfvénic wave structures. A comoving frame was employed, ξ = kxx + kzz − Mt, where ξ is a space coordinate normalized by ${V}_{{\rm{A}}}/{{\rm{\Omega }}}_{{{\rm{i}}}_{+}}$; M represents the Mach number normalized by VA; and ${k}_{x}=k\sin \theta $ and ${k}_{z}=k\cos \theta $ denote the directional cosines of the x and z directions, respectively, where θ and k denote the angle of propagation with respect to the magnetic field and the unit length vector, k = 1. The condition ${k}_{x}^{2}+{k}_{z}^{2}={k}^{2}$ was satisfied. Applying ξ transformation to the set of equations (1)–(7), we obtained the following system of equations:
To obtain the Sagdeev equation, we used the neutrality condition, n+ = n− + ne. After using some mathematical manipulations, we obtained the Sagdeev potential equation:
The structures of the Alfvénic double-layer solution (DLS) and supersolitons are discussed in the next section.
4. Nonlinear wave analysis
The numerical values of the physical parameters were chosen as β− = 0.175−0.189, β+ = 0.0499−0.0525, and θ = 26° [13, 47], and the values of r and q follow the constraints q > 1 and $q(r+1)\gt \tfrac{5}{2}$. We investigated the structure of all the nonlinear waves that could be found in our proposed plasma model through figure 1, which shows the phase portrait ($\Psi$, d$\Psi$/dξ) related to equation (18). This approach helped us identify the separatrices and trajectories associated with the various types of nonlinear waves. Figure 1 shows four types of nonlinear waves: DLS, supersoliton, periodic, and superperiodic waves. The phase portrait ($\Psi$, d$\Psi$/dξ) contains various trajectories, with each trajectory corresponding to one of the aforementioned structures. To classify trajectories, the nonlinear and supernonlinear periodic orbits are denoted as POm;n and SPOm;n, respectively, whereas the nonlinear and supernonlinear homoclinic orbits are denoted as HOm;n and SHOm;n, respectively, where m refers to the number of central points enveloped by the orbit, and n denotes the number of separatrices contained by the orbit [22].
Figure 1. Phase portrait (d$\Psi$/dξ) of the nonlinear and supernonlinear orbits at M = 0.0405, r = 1.9, β+ = 0.0499, β− = 0.18, θ = 26°, and q = 4.
The outer red separatrix (SHO2;1) that envelops the others represents a higher-amplitude solitary wave called supersoliton. In addition, in figure 1, the orange (HO1;0) curve corresponds to the DLS. Moreover, the orbit PO1;0 that surrounds the central points without any self-intersections is formed. This orbit corresponds to the periodic waves. In addition, there are larger-amplitude periodic trajectories SPO2;1 that surround the two equilibrium points as well. These orbits, SPO2;1, refer to the existence of superperiodic waves. Generally, the existence of closed orbits in the phase portrait proves that there is an invariant energy in the dynamic system that has a minimum corresponding to the location of the equilibrium points [18]. The corresponding Sagdeev potential of the supersoliton solution is presented in figure 2(a), which has two subwells and four fixed points. The two central points, P0($\Psi$0, 0) and P1($\Psi$1, 0), are located at the minima of the potential curve, and the two saddle points, P2($\Psi$2, 0) and P3($\Psi$3, 0), are located at the maxima of the potential curve. The shapes (shallowness and deepness) of the two subwells, related two supersoliton potential, depend on the proposed system [22, 48, 49]. On the other hand, figure 2(b) shows the Sagdeev potential profile of the soliton wave structure.
Figure 2. (a) Sagdeev potential of the supersoliton with M = 0.0405, r = 1.9, β+ = 0.0499, β− = 0.18, θ = 26°, and q = 4. (b) Sagdeev profile for the soliton with M = 0.08, r = 1.7, β+ = 0.022, β− = 0.1, θ = 26°, and q = 1.5.
Figure 3 shows the profile of the supersoliton and its electric field, E (= −∇$\Psi$). Figure 3(a) shows that the supersoliton structure undergoes a deformation compared to the regular soliton structure [48]. To clarify this confusion, the supersoliton electric field represents a distinctive feature of the supersoliton existence. As depicted in figure 3(b), the electric field structure of the supersoliton has wiggles on both sides of the configurations [48], and this shape was recorded in space [18]. Clearly, it is easy to notice that when the potential energy curves have many maxima and minima, the probability of the existence of supernonlinear waves in a soliton or periodic form increases. Figure 4 shows the domains in which the supersoliton, DLS, and periodic waves could be existing. In figure 4(a), by varying the value of M, a certain structure of nonlinear waves is developed at a certain value of M. For example, at MDLS = 0.0399, the DLS structure is formed, whereas the supersoliton structure is formed by moving to higher values of M than MDLS. Additionally, a periodic structure is formed by moving to a lower value of M than MDLS. This means that the DLS may be regarded as the lower limit of the existence range of supersolitons. This leads to the conclusion that the nonlinear structures (periodic, DL, and supersoliton) are highly sensitive to the Mach number and exist only in a distinct range. Figure 4(b) presents a comparison between the Sagdeev potentials of the supersoliton, DLS, and periodic waves. For small values of β−, i.e. β− = 0.179, the periodic Sagdeev potential (blue solid curve) was found, and by further increasing β−, the supersoliton Sagdeev potential (dotted–dashed curve) was created at β− = 0.189, passing by the structure of DLS (dashed curve) at β− = 0.184. This means that decreasing the mass of negative ions enhances the existence of supersolitons. Physically, reducing the mass of negative ions contributes to structuring the supersoliton, which has a larger amplitude than a regular soliton, as discussed previously. This is because a decrease in mass helps the waves interact and propagate easily. In addition, it enables the wave to overcome damping related to the collision process. This helps the amplitude of the waves to grow significantly until it reaches the supersoliton structure.
Figure 3. (a) Electrostatic potential $\Psi$(ξ) vs ξ. (b) Electric field structure of the corresponding supersoliton profile. Both panels are at M = 0.0406, r = 1.9, β+ = 0.0499, β− = 0.18, θ = 26°, and q = 4.
Figure 4. Sagdeev potential of supersoliton (dotted–dashed red curve), DLS (orange dashed curve), and periodic (blue solid curve) waves. (a) Different structures of nonlinear waves at different values of M with r = 1.9, q = 4, β− = 0.18, β+ = 0.0499, and θ = 26°. (b) Sagdeev potential at different values of β− with M = 0.04, r = 1.9, β+ = 0.0499, θ = 26°, and q = 4.
The periodic profile is shown in figure 5. Figure 5(a) shows that when the number of electrons that populate the tail of the distribution function (increasing q) increased, the amplitude of the periodic waves decreased. Figure 5(b) shows that by decreasing the value of β+, the amplitude and width increased, and by decreasing, the width increased rapidly and the amplitude became spikier, indicating the existence of superperiodicity. This implies that decreasing the temperature causes distortion in the periodic profile and helps in creating superperiodic waves in the proposed plasma medium. Generally, decreasing the electron temperature contributes to the structuring of supernonlinear waves in the space plasma. Supernonlinear waves are known to have very high amplitudes and very long wavelengths [17, 18]. The probability of the existence of such waves is extremely high in plasmas with low electron temperatures. This is because electron temperature is a measure of the kinetic energy of electrons in a plasma [50]. When the electron temperature decreased, the electrons possessed less energy and started to move more slowly. This implies that they are less likely to interact with each other and scatter waves. This manages the waves to propagate without disturbance and helps them grow to an extremely large amplitude. Moreover, figure 6 shows that enhancing the flat part of the distribution function (increasing r) increased the width of the superperiodic structure and slightly decreased its amplitude.
Figure 5. (a) Profile of the periodic Alfvén waves ($\Psi$ versus ξ) at different values of q with M = 0.038, r = 1.7, β− = 0.18, θ = 26°, and β+ = 0.047. (b) Profile of the periodic and superperiodic waves at different values of β+ with M = 0.04,β− = 0.18, θ = 26°, q = 4, and r = 1.2.
Figure 6. Profile of the superperiodic waves at different values of r with M = 0.04, β+ = 0.0499, β− = 0.18, θ = 26°, and q = 4.
5. Conclusions
In this study, the features of nonlinear and supernonlinear Alfvénic waves were investigated using a three-component plasma model. The proposed plasma medium is composed of positive and negative ions and DSED. Using the Sagdeev potential method and phase portrait plots, the existence of various types of nonlinear and supernonlinear waves, such as DLS, periodic, supersoliton, and superperiodic waves, was revealed. The essential points of our findings are as follows:
(a) At different values of Mach number, M, there are different types of nonlinear waves. At M > MDLS, the supersoliton exists, whereas at M = 0.0399, the DL wave structure was observed. This is shown in figure 4.
(b) The profile of the supersoliton and its attached wiggling electric field are shown in figure 3.
(c) The domains in which various nonlinear structures were formed are defined in figure 4. A comparison between the Sagdeev potentials of the supersoliton, DLS, and periodic waves is presented in figure 4. The effects of increasing the electron temperature (β−) on structuring the supersoliton Sagdeev potential were investigated. It was found that by increasing the electron temperature, the Sagdeev structure was converted from a periodic structure to a supersoliton structure passing by the DL structure.
(d) The periodic wave structure was found to decrease with increasing number of electrons that populate the tail of the distribution function. In addition, it was found that by decreasing the electron temperature (β+), a superperiodic structure appeared clearly, as shown in figure 5.
(e) Finally, it was found that the width of the superperiodic waves increased by broadening the flat part of the DSD function (increasing r), as shown in figure 6.
The obtained theoretical results are helpful in interpreting some nonlinear and supernonlinear excitations [51] of Alfvén waves that propagate in the ionosphere [47, 52], in which positive and negative ions and non-Maxwellian particles could be present, especially in the D region. [32, 35, 53–55].
Compliance with ethical standards
The authors confirm that all ethical standards have been adhered. This includes declarations regarding conflicts of interest, ethical approval, and animal welfare, as applicable.
Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have influenced the study reported in this paper.
Authorship statement
All persons who meet the authorship criteria are listed as authors, and all authors certify that they have participated sufficiently in the study to take public responsibility for the content, including participation in the concept, analysis, writing, or revision of the paper.
Ethical approval
This paper does not include any studies with human participants or animals performed by any of the authors.
Funding information
The authors declare that there was no funding received.
Informed consent statement
The authors declare that no human subjects were involved in the experiments.
Appendix A
The set of equations describing the ion fluid dynamics before normalization is as follows:
where ${\bar{n}}_{+},{\bar{n}}_{-},$ c $,\bar{{\rm{\Psi }}}$, and $\bar{\phi }$ represent the number density of positive ions, number density of negative ions, speed of light, and parallel and perpendicular electrostatic potentials, respectively; $({\bar{v}}_{{x}_{+}},{\bar{v}}_{{z}_{+}})$ represent the positive ion velocity components, and $({\bar{v}}_{{x}_{-}},{\bar{v}}_{{z}_{-}})$ denote the negative ion velocity components.
More mathematical details regarding the drift velocity are presented as follows, starting from the classical momentum equation:
where vi is the ion fluid velocity; i = + or − for positive or negative ions, respectively; mi is the ion mass; E is the electric field; c is the speed of light; B is the magnetic field; and q = 1 (–1) for positive (negative) ions. Thus,
After normalization, we get equations (5) and (6), which represent the drift velocities for positive and negative ions, respectively. Kaur and Saini [56] introduced a complete derivation of drift velocity.
Appendix B
Here are more mathematical details about Faraday's and Ampere's equation.
where J represents the displacement current. The second term on the rhs is ignored because its value is smaller than unity $(\tfrac{1}{c}\tfrac{\partial E}{\partial t}\ll 1)$. After applying the vector product and some mathematical simplifications, we get
where ${J}_{z}=e({n}_{+}{v}_{{z}_{+}}-{n}_{-}{v}_{{z}_{-}}-{n}_{{\rm{e}}}{v}_{{\rm{e}}z})$ represents the parallel current density. Faraday's equation states that
The following electron number density is obtained after integrating the DSD function over space velocity, i.e $\iiint $frq(v)d3v. Expanding the outcome in limit ψ ≪ 1, we get the total number density as follows [22, 45]:
HasegawaA, UberoiC1982Alfven wave. DOE critical review series U.S dept of Energy, Washington, D.C
4
LeeL C, JohnsonJ R, MaZ W1994 Kinetic Alfvén waves as a source of plasma transport at the dayside magnetopause J. Geophys. Res.: Space Phys.99 17405 17411
MäkeläJ S, MälkkiA, KoskinenH, BoehmM, HolbackB, EliassonL1998 Observations of mesoscale auroral plasma cavity crossings with the Freja satellite J. Geophys. Res.: Space Phys.103 9391 9404
AbulwafaE M, ElhanbalyA M, BedeirA M, MahmoudA A2022 Formation of double-layers and super-solitons in a six-component cometary dusty plasma Eur. Phys. J. D76 120
PickettJ S, ChenL-J, KahlerS W, SantolikO, GurnettD A, TsurutaniB T, BaloghA2004 Isolated electrostatic structures observed throughout the cluster orbit: relationship to magnetic field strength Ann. Geophys.22 2515 2523
TahaR M, El-TaibanyW F2020 Bifurcation analysis of nonlinear and supernonlinear dust–acoustic waves in a dusty plasma using the generalized (r, q) distribution function for ions and electrons Contrib. Plasma Phys.60 e202000022
El-TantawyS A2016 Rogue waves in electronegative space plasmas: the link between the family of the KdV equations and the nonlinear Schrödinger equation Astrophys. Space Sci.361 1 9
AtteyaA2018 S Sultana, and R Schlickeiser. Dust-ion-acoustic solitary waves in magnetized plasmas with positive and negative ions: the role of electrons superthermality Chin. J. Phys.56 1931 1939
GoslingJ T, AsbridgeJ R, BameS J, FeldmanW C, ZwicklR D, PaschmannG, SckopkeN, HyndsR J1981 Interplanetary ions during an energetic storm particle event: the distribution function from solar wind thermal energies to 1.6 MeV J. Geophys. Res.: Space Phys.86 547 554
ZettergrenM, LynchK, HamptonD, NicollsM, WrightB, CondeM, MoenJ, LessardM, MiceliR, PowellS2014 Auroral ionospheric F region density cavity formation and evolution: MICA campaign results J. Geophys. Res.: Space Phys.119 3162 3178
LivadiotisG2017Kappa distributions: Theory and applications in plasmas Elsevier
37
MasoodW, QureshiM N S, YoonP H, ShahH A2015 Nonlinear kinetic Alfvén waves with non-Maxwellian electron population in space plasmas J. Geophys. Res.: Space Phys.120 101 112
PaulS N, RoychowdhuryA2016 Modulational instability and on the existence of rogue wave in electron-ion-positron plasma with kappa distributed electrons Chaos Solitons Fractals91 406 413
QureshiM N S, ShahK H, ShiJ, MasoodW, ShahH A2020 Investigation of cubic non-linearity-driven electrostatic structures in the presence of double spectral index distribution function Contrib. Plasma Phys.60 e201900065
UllahS, MasoodW, SiddiqM2020 Modulation of ion sound excitations in electron–ion plasmas with double spectral index distribution function Contrib. Plasma Phys.60 e201900182
QureshiM N S, ShahH A, MurtazaG, SchwartzS J, MahmoodF2004 Parallel propagating electromagnetic modes with the generalized (r, q) distribution function Phys. Plasmas11 3819 3829
PakhotinI P2018 Diagnosing the role of Alfvén waves in magnetosphere-ionosphere coupling: swarm observations of large amplitude nonstationary magnetic perturbations during an interval of northward IMF J. Geophys. Res.: Space Phys.123 326 340
TamangJ, SahaA2020 Bifurcations of small-amplitude supernonlinear waves of the mKdV and modified Gardner equations in a three-component electron-ion plasma Phys. Plasmas27 012105
Yanguas-GilA, CotrinoJ, González-ElipeA R2006 Measuring the electron temperature by optical emission spectroscopy in two temperature plasmas at atmospheric pressure: a critical approach J. Appl. Phys.99 033104
NarcisiR S1973 Mass spectrometer measurements in the ionosphere Physics and Chemistry of Upper Atmosphere: Proc. Symp. Organized by the Summer Advanced Study Institute (University of Orléans, Orléans, France, 31 July–11 August 1972) Springer 171 183
54
ArnoldF, ViggianoA A, FergusonE E1982 Combined mass spectrometric composition measurements of positive and negative ions in the lower ionosphere–II. Negative ions Planet. Space Sci.30 1307 1314
KaepplerS R, SanchezE, VarneyR H, IrvinR J, MarshallR A, BortnikJ, AshtonS R, ReyesP M2020 Incoherent scatter radar observations of 10–100 keV precipitation: review and outlook The dynamic loss of Earth's radiation belts Elsevier 145 197
56
KaurN, SainiN S2016 Ion acoustic kinetic Alfvén rogue waves in two temperature electrons superthermal plasmas Astrophys. Space Sci.361 331