Kinetic Alfvén solitary waves in a low-β plasma with regularized kappa-distributed electrons

Alfvén waves (AWs) are classified as low-frequency transverse electromagnetic waves that propagate along the lines of the magnetic field (MF). These waves can be generated in electrically conducting fluids, such as plasma. Hanns Alfvén was the first person to find these waves in equations about fluid dynamics and electrically charged fluids. This happened in 1942.¹ These waves move through plasma and have recently caught the attention of many researchers. Their essential contribution lies in their ability to clarify variations in electromagnetic fluctuations, particle energization, and acceleration processes, and the observation of solitary waves (SWs) in plasmas found in Space, asteroids, and laboratory environments.^2,3 Since these waves were identified, they have become imperative to understand nonlinear phenomena in laboratory and Space plasmas.³ In the same way, AWs play a crucial role in making thermonuclear fusion possible in plasma systems that are magnetically confined. Kinetic AWs (KAWs) are a type of dispersive nonlinear waves that can propagate through plasma. Many written works go into more detail about how Kinetic Alfvén SWs (KASWs) are structured using either the Korteweg–de Vries (KdV)^4–9 or NLSE¹⁰ nonlinear differential equations. Hasegawa and Mima have observed Alfvén SWs with density humps of arbitrary amplitude.¹¹ It is important to remember that the Larmor radius has almost no effect on the waves because they are not dampened much. These changes reform waves into KAWs, which make it easier for the right energy to be transferred.^12,13 Many scholarly works have been published exploring the individual nature of KAWs. In many theoretical studies of AWs, the particle distribution is thought to be the Maxwellian particle distribution. However, much experimental evidence suggests that the Maxwellian distribution does not fully explain how particles are distributed in plasma systems. A small proportion of particles exhibit deviant behavior from the Maxwellian distribution. Recently, significant research efforts have been dedicated to investigating the influence of energetic particles on distribution functions in laboratories^14,15 and space plasmas.^16–20 When studying plasmas with particles moving at speeds other than thermal, it is necessary to use a modeling approach to look into the effects of these particles. The observed non-equilibrium distribution contains many suprathermal particles that move faster than the thermal velocity. For example, we can mention the distribution of particles in the planetary magnetosphere and solar wind plasmas.^21,22 One well-known distribution function that describes the role of energetic particles is the kappa distribution (KD) function.²³ This is a straightforward generalization of the non-Maxwellian distribution function.²⁴ The KD, an extension of the Maxwellian distribution, is widely used to analyze various nonlinear phenomena in different plasma models.^25–30 Although the standard KD (SKD) is a helpful function to describe particle distribution nonthermal, it has some restrictions mentioned in the following sentences: SKD cannot be used to calculate all velocity moments. Moreover, the efficiency of SKD is limited to an estimated spectral index value of more than 3/2. All of these restrictions in applying to the SKD can be solved by using the standard Kappa distribution (RKD), introduced for the non-relativistic case by Scherer et al.³¹ and for the relativistic case by Han-Thanh et al.³² Un-like SKD, RKD is an exponential power attenuation coefficient at high velocities. This exponential cut-off is because any acceleration process can occur on finite spatial and time scales. Therefore, it is not possible for a power law to continue indefinitely (as shown in the conventional kappa distribution), but rather it must have a limited range.

This section examines the impact of specific physical variables, such as the superthermality κ, the cut-off parameter α, the obliqueness parameter l_z, and β on the profile of the KASWs. Figure 1(a) illustrates the potential profile of KASWs as a function of $χ,κ$⁠. The results show that increasing κ reduces the amplitude and width of the compressive KASWs. In other words, the soliton wave narrows when the number of particles with higher energy decreases, and the system goes to equilibrium (Maxwell distribution). Remember that κ < 3/2 for RKD, while κ > 3/2 for SKD. Figure 1(b) shows the potential profile of the KASWs as a function of $χ,α$⁠. From Fig. 1(b), it is inferred that as the cut-off parameter α increases, the amplitude and width of the SWs grow. This means that, as α increases, the number of nonthermal particles in the tail of the distribution function increases. Thus, the increase in these particles leads to the increase in the amplitude of the KASWs, and considering that, in this system, the restoring force comes from the pressure of electrons. In contrast, the inertia force appears through ions. Therefore, with the increase of α, the number of energetic particles increases, and the pressure of electrons grows. Also, for α = 0, the SKD is recovered, and in this case, κ > 3/2. Figure 2(a) shows the effect of β on the profile of the KASWs. Remember that β only appeared in the dispersive coefficient B, which is responsible for the soliton width. Thus, we expect the soliton width to change with changing β while its amplitude will remain constant because the nonlinear coefficient A is not a function of β. As shown in Fig. 2(a), the soliton width grows with increasing β. Note that as β increases, the magnetic field decreases, and less confinement occurs. As a result, the soliton width is expected to increase. Figure 2(b) displays the profile of compressive KASWs as a function of χ and the obliqueness propagation l_z. It is evident that both the amplitude and width of the soliton decrease as the value of l_z increases. Since the l_z = cos θ, the KASWs become higher and narrower whenever the wave propagates in the direction of the external MF. This property can be easily understood from (18). According to Eq. (18), the coefficient A disappears for a large enough propagation angle, i.e., l_z = 0. As can be seen from Eq. (19), the coefficient B is proportional to sin² θ cos θ such that the soliton width is proportional to $w∝sin2θcosθ$⁠. In addition, when the propagation angle θ of the soliton becomes 0 or 90, the soliton can be exactly aligned or perpendicular to the MF. The dispersive and nonlinear coefficients of Eq. (17) as a function of α and different values of κ are represented in Figs. 3(a) and 3(b). According to these figures, the coefficients $A,B$ are always negative for all plasma parameters so that the width of SWs is imaginary for u₀ > 0. Therefore, for Eq. (17) to have a soliton solution, even though B is negative, u₀ < 0 must be considered. For u₀ < 0, Eq. (17) gives ψ > 0 and w > 0. Generally, u can be considered positive or negative, depending on the soliton speed. Considering u₀ < 0, the nonlinear coefficient A becomes positive. Therefore, this system only supports positive solitons. Considering u₀ < 0, the amplitude and width of the soliton become positive, and only compressive solitons can be propagated in this model. Figure 3(c) demonstrates the dispersive coefficient B as a function of α for various values of obliqueness propagation parameter l_z. Clearly, the dispersive coefficient B is always negative for all values of α. Therefore, for Eq. (17) to have a soliton solution, even though B is negative for all values of l_z, thus, u₀ < 0 must be considered. It is also worth noting that, with the increase of β, the dispersive coefficient B also increases.

To sum up, in this investigation, the propagation of kinetic Alfvén solitary waves (KASWs) in a normal magnetoplasma composed of heavy dynamical ions with regularized kappa-distributed electrons has been reported. The nonlinear KdV equation for kinetic solitary waves has been derived by employing the well-known reductive perturbation method. Due to the negativity of the coefficients of the evolution equation, only compressive kinetic Alfvén waves (KAWs) can be propagated in the current plasma model. We found that the dispersive coefficient depends on the cut-off parameter α, the obliqueness of propagation l_z, and the β of the plasma. The results show that the amplitude and width of the KASWs are highly dependent on the cut-off parameter α, which is related to non-equilibrium particles in the tail of the velocity distribution function. The numerical results indicate that the amplitude and width of the soliton propagated in the system increase with enhancing α. The insights obtained from this research help us understand the kinetic waves of electrons in plasma systems containing non-Maxwellian electrons. These findings make valuable contributions to understanding the mechanism of creating and propagating KAWs in non-Maxwellian electron-ion plasma, such as Saturn’s magnetospheres and laboratory, guiding future research in this field.

Future work: Consideration of additional physical effects, such as collisions between plasma charges with each other or with neutral particles, geometrical effects, and external periodic forces and perturbations, leads to more complicated evolution equations with higher order dispersion and additional terms. Following the same approach as previous studies that considered these effects, we can understand the propagation mechanism of nonplanar damped KAWs under these effects.^35–45 However, this endeavor remains a goal for future research.

The authors expressed their gratitude to Princess Nourah bint Abdulrahman University Researchers supporting Project No. (PNURSP2024R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

The authors have no conflicts to disclose.

Wedad Albalawi: Data curation (equal); Formal analysis (equal); Validation (equal); Visualization (equal). Muhammad Khalid: Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Visualization (equal); Writing – original draft (equal). C. G. L. Tiofack: Conceptualization (equal); Formal analysis (equal); Investigation (equal). S. A. El-Tantawy: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.