© The Authors 2025

1. Introduction

In space plasma environments such as the solar wind, solar corona, planetary magnetospheres, and the interstellar medium, turbulence plays a critical role in the dissipation of energy and charged particle acceleration (Jokipii 1966; Bieber et al. 1996; Farmer & Goldreich 2004; Engelbrecht & Burger 2013). The dissipation and particle acceleration processes are central to understanding how energy is transferred across different scales, from large fluid-like motions down to smaller kinetic scales. The dissipation of turbulent energy helps to explain key phenomena, including the nonadiabatic temperature profiles observed in the solar wind and planetary magnetospheres (Williams et al. 1995; Matthaeus et al. 1999; Adhikari et al. 2015a,b), which increase with distance from the Sun and the central planet, respectively (Richardson et al. 1995; Saur 2004; Bagenal & Delamere 2011; Von Papen et al. 2014). Among the various scaling mechanisms observed in different space plasma environments, the Kolmogorov scaling (k^−5/3) (Matthaeus et al. 1982; Bale et al. 2005), ion scales (such as the ion Larmor radius and ion inertial length) (Leamon et al. 1999; Chen et al. 2014), sub-ion range (Alexandrova et al. 2009; Chen et al. 2010), and electron range (Denskat et al. 1984), to name a few, are particularly important. Although observations at these small scales are relatively scarce and often subject to varying interpretations, they provide crucial insights into the turbulent fluctuations within the dissipation range (Denskat et al. 1984). At scales comparable to the ion Larmor radius, Alfvén waves are thought to be slightly damped, indicating a transition to the dissipation range. Schreiner & Saur (2017) presented an analytical dissipation model for solar wind turbulence at electron scales, combining energy transport from large to small scales with collisionless damping. The model suggests that wave-particle interactions involving kinetic Alfvén waves (KAWs) are the primary damping mechanism. Our research further supports this by analytically demonstrating that Alfvén waves dissipate more rapidly at very small, kinetic scales. Over time, interest in the heating effects of Alfvén waves in the solar corona has grown, with numerous studies underscoring their significance (for details, see Matthaeus et al. 1999; Zank et al. 2018b, 2021; Yalim et al. 2024, and references therein).

Despite advances, our understanding of the kinetic processes responsible for turbulent dissipation, wave heating, and energy transfer remains incomplete. This knowledge is essential for explaining phenomena such as solar wind acceleration and coronal heating (Parker 1958; Richardson et al. 1995; Fox et al. 2016), as well as analogous astrophysical processes. More recently, Bowen et al. (2024) investigated cyclotron resonant heating of the turbulent solar wind at ion-kinetic scales, finding that ion cyclotron heating rates correlate strongly with the turbulent energy cascade rate. This correlation implies that cyclotron heating is an important dissipation mechanism in the solar wind.

The several promising frequently used dissipation mechanisms of Alfvén waves are: the resonant heating in coronal loops (Hollweg 1978, 1981; Schwartz et al. 1984), turbulent heating due to Kelvin-Helmholtz instabilities (Heyvaerts & Priest 1983), turbulence cascade (Williams et al. 1995; Matthaeus et al. 1999), phase mixing caused by Alfvén velocity gradients (Heyvaerts & Priest 1983; Abdelatif 1987), spatial resonance – that is, resonant mode conversion (Ionson 1978; Lee & Roberts 1986), nonlinear mode coupling (Chin & Wentzel 1972; Uchida & Kaburaki 1974; Wentzel 1974), and kinetic Landau damping (Stéfant 1970; Hollweg 1971), to name a few. In a microphysical context, the competition among these mechanisms is largely influenced by the presence of smaller-scale Alfvén waves. Generally, the dissipation rate of Alfvén waves is proportional to the square of the wave number (k²). For the heating rate resulting from Alfvén wave dissipation to balance the energy loss rate in coronal plasmas, these small-scale Alfvén waves (i.e., KAWs) must have scales close to the particle kinetic scales typical of coronal plasmas, such as the ion gyroradius (ρ_i) or the electron inertial length (λ_e), both of which are significantly smaller than the mean free path of coronal plasmas. At these particle kinetic scales, Alfvén waves transition into dispersive KAWs. The wave-particle interactions, particularly through Landau damping, become critically significant in the context of kinetic dissipation of wave energy and plasma particle heating. In this regard, one of the most promising candidates for a coronal heating mechanism is the kinetic dissipation of KAWs via wave-particle interactions through the Landau damping mechanism. This approach offers a comprehensive framework for applying KAWs to the coronal heating problem, including chromospheric heating and solar wind heating. Consequently, we utilize this mechanism within the framework of the most generalized Vlasov-Maxwellian model, situated in the domain of kinetic theory in plasma physics.

The first detection of Alfvén waves (Alfvén 1942) in the solar corona, as was reported by Tomczyk et al. (2007), opened up a new avenue of research in space physics. However, their observational evidence suggested that these waves are weak and might not significantly energize the solar wind and corona. De Pontieu et al. (2007), followed Tomczyk et al. (2007), used 3D radiative MHD simulations and found that Alfvén waves are strong enough to accelerate and heat the solar wind and the solar corona. These waves are thought to play a crucial role in the heating mechanisms of the solar corona (Del Zanna & Velli 2002; Khan et al. 2020). Davila (1987) explored solar corona heating through the resonant absorption of Alfvén waves. He suggested that this method is a viable mechanism for heating the corona of the Sun and other late-type stars. Alfvén waves can energize the solar wind through the work done by the Alfvén waves pressure force, which directly accelerates plasma away from the Sun (Hollweg 1973; Wang 1993). The significant interest in the heating phenomena of Alfvén waves in the solar corona has evolved over time, highlighted by various studies (e.g., Matthaeus et al. 1999; Zank et al. 2018b, 2021). Several spacecraft observations, including in situ measurements, have demonstrated the presence of large-amplitude outward-propagating Alfvén waves in the interplanetary medium (Belcher & Davis 1971; Tu & Marsch 1995; Bruno & Carbone 2013). Moreover, remote observations have identified Alfvén wave-like motions in the low corona that carry an energy flux sufficient to power the solar wind (De Pontieu et al. 2007). Several considerations suggest that much of this heating results from the dissipation of MHD “wave/turbulence-driven” (W/T) models or the “reconnection/loop-opening” (RLO) models (Cranmer & Van Ballegooijen 2010), further explored by Zank et al. (2018b, 2021). Alfvén waves have interested scientists for many decades due to their remarkable role in transporting energy across space and astrophysical environments (see Matthaeus et al. 1999; Cramer 2011; Zank et al. 2018b, 2023, 2024; Wu & Chen 2020, and references therein). These waves have practical applications in laboratory plasma, particularly in transferring energy within fusion devices (Hasegawa & Uberoi 1982).

Observations have confirmed the presence of short-wavelength Alfvén waves (also known as KAWs) and electron beams, as has been demonstrated by Kerr et al. (2016) using the RAdiative hydro DYNamics code (RADYN). Their work synthesized various chromospheric spectral lines, revealing an intriguing distinction: Mg II profiles in wave-heated simulations differed from those in electron beam simulations, with the wave-heated model more closely matching IRIS observations. This suggests that detailed studies of spectral lines formed in the mid to upper chromosphere may help distinguish between heating mechanisms. The high-resolution in situ measurements of electromagnetic (EM) fluctuations and plasma distribution functions from satellites, including the Magnetospheric Multiscale (MMS) mission, Cluster, and the PSP, have created unprecedented opportunities to investigate the complex plasma dynamics within the sub-ion scale of turbulence (for details, see Sahraoui et al. 2009; Alexandrova et al. 2009, 2012, 2013; Chen et al. 2019). The primary aim of these studies is to unravel the mechanisms of energy dissipation in weakly collisional plasmas and to understand how electrons and ions gain energy in these environments (Parashar et al. 2015). Hui & Seyler (1992) conducted hybrid simulations showing that large-amplitude, short-wavelength Alfvén waves can accelerate electrons to velocities exceeding the Alfvén speed. Observational support for small perpendicular-wavelength Alfvén waves has been obtained through sounding rocket missions (Boehm et al. 1990) and the Freja satellite (Wahlund et al. 1994; Louarn et al. 1994; Boehm et al. 1995).

Very recently, in situ measurements by Rivera et al. (2024) using the PSP and Solar Orbiter spacecraft have provided some of the most compelling evidence yet on Alfvén wave heating in the solar corona and solar wind. These observations reveal that after leaving the Sun’s corona, the solar wind continues to accelerate and cool at a slower rate than expected for a freely expanding adiabatic gas. Alfvén waves, which are perturbations in the interplanetary magnetic field, transport energy and are shown to induce both heating and acceleration in the plasma between the corona’s outer edge and near Venus’s orbit. The study indicates that the damping and mechanical work performed by these large-amplitude Alfvén waves are sufficient to power the heating and acceleration of the fast solar wind in the inner heliosphere. Early analyses by Chaston et al. (2014) estimated the heating rate of KAWs and found that the heating occurs primarily in the perpendicular direction. Observations from various spacecraft also confirm that KAWs dissipate and contribute to plasma heating during propagation (Wygant et al. 2002; Lysak & Song 2003; Gershman 2017), highlighting their essential role in transport, heating, and acceleration processes in space and astrophysical plasmas. The high-resolution imaging and spectroscopy of flare ribbons provided by the Daniel K. Inouye Solar Telescope (DKIST) are crucial for understanding energy transport in solar flares. According to 1D radiative hydrodynamics simulations, the lower solar atmosphere responds distinctly to heating by Alfvénic waves compared to heating by electron beams, though there are notable similarities as well (Kerr et al. 2016; Reep & Russell 2016; Reep et al. 2018). Localized electric fields associated with KAWs are often observed to accelerate electrons, forming electron beams (e.g., Artemyev et al. 2015a; Damiano et al. 2015, 2016). Electron beams are ubiquitous in space and astrophysical plasmas, detected remotely through radio observations and in situ by numerous satellites. Examples include beams associated with solar bursts, Earth’s foreshocks, and Earth’s magnetosphere. In the context of our model, which explores KAWs in a Kappa-distributed plasma via kinetic plasma theory, we focus on the ion gyro-radius and the electron beam aligned with the electric field. This configuration can indeed excite and heat the coronal region, highlighting the strong connection between KAW-driven electron beams and particle acceleration activity in such plasma environments.

Hasegawa & Chen (1975) and Hasegawa (1977) were the first to study Alfvén waves by incorporating the effect of a finite perpendicular wavelength – also known as gyroradius corrections – leading to the identification of kinetic Alfvén waves (KAWs). Hasegawa & Chen (1975) added gyroradius correction terms and derived the dispersion relation for obliquely propagating Alfvén waves, now recognized as the dispersion relation for KAWs. They utilized the Vlasov model and assumed a Maxwellian distribution function to derive the following dispersion relation for KAWs:

$$\begin{array}{c}\hfill {\omega }^2=k_{\Vert }^2{\mathrm{v}}_{\mathrm{A}}^2(1+\frac{3}{4}{k}_{\perp }^2{\rho }_i^2+\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }^2{\rho }_i^2),\end{array}$$

where k_⊥ and k_∥ are the perpendicular and parallel wavevectors, the symbols, ⊥ and ∥, denote the directions perpendicular and parallel to the mean/ambient magnetic field B₀, ρ_i is the ion gyroradius, and v_A is the Alfvén speed, respectively. In the expression, the first term $\frac{3}{4}{k}_{\perp }^2{\rho }_i^2$ is the gyro-radius correction, and the second term $\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }^2{\rho }_i^2$ arises due to the wave’s obliqueness. Following Hasegawa’s pioneering work, KAWs garnered significant attention. Lysak & Lotko (1996) further explored KAWs by incorporating the full kinetic effects of electrons and ions, rather than just the gyroradius correction terms. They demonstrated that the parallel phase velocity (ω/k_∥) of these waves exceeds the thermal speed of electrons in the presence of hot ions (Štverák et al. 2009).

Naim et al. (2014) extended Lysak & Lotko (1996)’s work by considering density inhomogeneity in the bi-Maxwellian distribution function to study KAWs instability. They found that density inhomogeneity destabilizes KAWs, but temperature anisotropy can stabilize them. KAWs have been extensively studied using various velocity distributions within kinetic plasma theory (Shukla et al. 2009; Vranjes & Poedts 2010; Liu et al. 2014; Khan et al. 2019a; Ayaz et al. 2020; Wu & Chen 2020; Barik et al. 2020, 2021). In our previous work (Ayaz et al. 2019), we examined the dispersion and damping relations of KAWs in the Cairns-distributed plasmas, finding that nonthermal particles significantly influence the real phase velocity and damping rate of KAWs. Khan et al. (2019a, 2020) studied KAWs in Kappa and non-extensive velocity distribution functions, finding that suprathermal particles and non-extensive parameters, characterized by the index κ and q, significantly affect the waves’ characteristics, enhancing their heating and energy transport over extended spatial distance in space.

KAWs have been extensively investigated by Barik et al. (2020, 2023) in space plasmas with κ-electrons, finding that the ion beam velocity can potentially excite the growth rate of KAWs. More recently, they examined the excitation of KAWs by multiple free energy sources, concluding that a combination of positive velocity shear with counter-streaming beam ions or parallel streaming beam ions with negative velocity shear favors KAW excitation (Barik et al. 2021). KAWs are likely to play a significant role in particle acceleration and energy transporting to the coronal plasma in the form of heat (Khan et al. 2019b). These waves are considered one of the primary agents responsible for transporting energy from the distant magnetosphere to the auroral ionosphere (Louarn et al. 1994; Seyler et al. 1995; Chaston et al. 2005, 2006; Ergun et al. 2005; Artemyev et al. 2015b; Gershman 2017), in particle energization at the formation of depolarization fronts (Ganguli et al. 2018; Ukhorskiy et al. 2022), and in electron trapping and acceleration in solar flares (Artemyev et al. 2016), and the counterpart (see Che et al. 2023).

KAWs have garnered significant attention over the past few decades due to their potential role in various space environments. They are proposed as an energy source for planetary auroras (Chaston et al. 2003; Keiling et al. 2003) and for driving atmospheric escape induced by space weather (Chaston et al. 2006). KAWs are key to understanding space and astrophysical turbulence (Boldyrev & Perez 2012), potentially solving the plasma heating mystery in the solar corona, solar wind, magnetospheres, and the interstellar medium (Liu et al. 2023). KAWs are the most rapidly damped waves (Shukla et al. 2009) that efficiently transfer energy to plasma particles through Landau damping (Khan et al. 2020; Ayaz et al. 2020).

Temerin et al. (2001) provided the first evidence of small-scale KAWs within the plasma sheet and plasma sheet boundary layer (PSBL). These waves likely play a crucial role in the local heating of the plasma sheet, potentially generating earthward-moving, magnetic field-aligned electron beams, and transversely heated ions. Extensive observational studies support the existence of KAWs and their significant role in energy transport across various regions of Earth’s magnetosphere, including the magnetopause (Johnson et al. 2001; Chaston et al. 2005), magnetosheath and auroral regions (Boehm et al. 1990; Louarn et al. 1994; Wahlund et al. 1994), magnetotail and PSBL (Wygant et al. 2002; Keiling et al. 2003; Duan et al. 2012), and central plasma sheet (Keiling et al. 2001). Wygant et al. (2002), presented observational evidence showing that the increase in Poynting flux in the PSBL during the substorm expansion phase is related to KAWs. Several researchers have extensively studied these small fluctuating waves (Lee et al. 1994; Shukla et al. 2009; Wu & Chen 2020; Khan et al. 2020; Liu et al. 2023; Barik et al. 2023; Ayaz et al. 2024a), highlighted their significant role in energy transport and particle acceleration in space plasmas.

In this proposed research study, we aim to address the role of perturbed EM fields in identifying KAWs (Lysak & Song 2003; Khan et al. 2019b). In space plasmas, if the ratio of the perturbed EM fields equals Alfvén speed – the KAWs are most likely to be present in those regions. These fluctuating EM fields are crucial for energy transport in solar wind turbulence (Narita et al. 2020). Narita et al. (2020) revisited KAWs analytically, deriving transport ratios and scaling laws for the electric and magnetic fields to aid in wave mode identification in spacecraft observations. This is particularly important for missions like the PSP, Solar Orbiter, and BepiColombo’s cruise to Mercury.

KAWs – defined by the condition where the perpendicular spatial scale of Alfvén wave approaches the ion gyroradius (ρ_i), can support significant electric and magnetic field fluctuations in the parallel and perpendicular directions (Xiang et al. 2022). These fluctuations facilitate energy transfer between the wave field and plasma particles via Landau or transit-time interactions (Barnes 1966; Hasegawa & Chen 1976; Gershman 2017). Recently, Xunaira et al. (2023) studied the instability and energy transport of KAWs in the solar corona using the field-aligned drift velocity distribution function, finding that the field-aligned drift velocity significantly influenced the EM fields of the KAWs. Khan et al. (2019a) investigated the perturbations of EM field ratios for different values of the non-extensive parameter q. They found nearly similar results observed by polar spacecraft (Wygant et al. 2002). The knowledge of these perturbed EM fields is pivotal in wave-particle interactions through Landau resonance and can be used to understand how KAWs transport energy in the solar corona (Ayaz et al. 2024a).

Hollweg (1999) provides a comprehensive analysis of KAW physics. Many observational studies of the solar wind (Bale et al. 2005; Sahraoui et al. 2012; Podesta & TenBarge 2012; Salem et al. 2012; Chen & Wu 2012; Kiyani et al. 2012), theories (Howes et al. 2008; Schekochihin et al. 2009), and numerical simulations (Gary & Nishimura 2004; TenBarge & Howes 2012) have suggested that KAWs play a significant role in the dissipation of turbulent energy. Specifically, understanding the perturbations in EM fields is important for comprehending how KAWs transport energy in space. The EM fields can be further used in the expressions of the Poynting flux vector, which quantifies the energy per unit time per unit area and evolves according to the Poynting theorem (Khan et al. 2019b; Ayaz et al. 2024a). Essentially, the Poynting flux vector reveals whether a wave can effectively carry or deliver its energy over long distances from its source. Lysak & Song (2003) investigated the Poynting flux of KAWs in a Maxwellian-distributed plasma. Extending this research, Khan et al. (2020) analyzed KAWs in a more generalized Kappa-distribution and discovered that the Poynting flux vector of KAWs is rapidly damped in the presence of fewer suprathermal particles. Shay et al. (2011) examined the Poynting flux of KAWs associated with quadrupolar B_y using kinetic particle-in-cell (PIC) simulations and Cluster satellite observations, finding that KAWs can carry significant Poynting flux in the magnetosphere.

KAWs can be excited by various mechanisms such as phase mixing (Heyvaerts & Priest 1983), nonlinear decay from magnetohydrodynamic Alfvén waves (Zhao et al. 2011, 2013, 2014), turbulence cascade (Bian et al. 2010; Zhao et al. 2013), electron beams (Chen & Wu 2012), magnetic reconnection (Cranmer 2018), and photospheric fluctuations (Ulrich 1996). Near the Sun, these waves start their journey carrying a significant amount of Poynting flux, approximately 10⁵ W m⁻² (Srivastava et al. 2017). Although some energy is lost after partial reflection from the transition region (Zank et al. 2021), the waves can still carry a substantial amount of energy, around ∼10³ W m⁻² (Khan et al. 2020), as they travel through the corona. Observations indicate that a Poynting flux of 10²–10⁴ W m⁻² is sufficient to heat the corona to a million degrees Kelvin (Srivastava et al. 2017). These insights into the Poynting flux and energy transport mechanisms of KAWs showcase their significant role in space and astrophysical environments.

KAWs propagate obliquely to the ambient magnetic field B₀ with wavevectors such that k_⊥ ≫ k_∥ (Narita et al. 2020; Ayaz et al. 2024a). This characteristic means KAWs primarily carry energy along the magnetic field lines (Lysak & Song 2003), as is indicated by the Poynting flux vector (S_z) in the z-axis (i.e., in the parallel direction). Although KAWs also transport some energy in the perpendicular direction (S_x) (Khan et al. 2020), this is often less significant (S_x ≪ S_z) in regions like the aurora and plasma sheet boundary layers (Lysak & Song 2003; Khan et al. 2019a; Lysak 2023). In this study, we focus on KAWs in the solar coronal regions, particularly within solar flux tube loops, where both perpendicular and parallel energy transformations and heating are critical. More recently, we investigated the parallel and perpendicular Poynting fluxes of KAWs in the solar corona (Ayaz et al. 2024a). We found that the perpendicular Poynting vector (S_x) for KAWs dissipates rapidly, leading to plasma heating over short distances. The importance of perpendicular heating is also highlighted by Zank et al. (2018b), who investigated the heating of core solar coronal protons using a turbulence transport model and found significant perpendicular heating within 0.3 R_Sun. In this research, we utilize both the perpendicular and parallel Poynting vectors (S_x and S_z) to examine the total power transfer rate of KAWs in solar flux tube loops. This comprehensive approach helps to understand how KAWs contribute to energy transportation and heating in the solar corona.

In general, KAWs transfer energy to the plasma, causing variations in energy distribution as the wave propagates through space (Khan et al. 2020). This energy transfer significantly affects the behavior of charged particles, including their acceleration and speed, which remains a dynamic area of research. However, by employing a kinetic Vlasov-Maxwell model, we derived the net resonance velocity expressions of the particles, providing a comprehensive understanding of their behavior and interactions within the plasma. Moreover, Tiwari et al. (2008) studied the growth and damping length of KAWs in the PSBL using the loss-cone distribution function. Their findings indicate that the growth length of KAWs decreases with varying values of the loss cone parameter J ≥ 0. This suggests that more energetic particles are available to transfer energy to the waves through resonance wave-particle interactions. We extended this approach to a κ-distributed plasma to investigate how suprathermal particles influence the damping length of KAWs. These insights are pivotal for understanding the dynamics of KAWs and their role in energy transport and particle acceleration in various space plasma regions.

In this manuscript, we aim to address several key aspects of induced heating and acceleration of charged particles within the solar corona. These include:

• (a).

  The perturbed EM field ratios (imaginary parts).

• (b).

  The Poynting flux vectors of KAWs.

• (c).

  The total power rate of KAWs in the solar flux tube loop.

• (d).

  The net resonance velocity of the particles.

• (e).

  The group velocity of KAWs.

• (f).

  The characteristic damping length of KAWs.

We investigate these issues (a)–(f) using kinetic plasma theory in Kappa-distributed plasmas. Our analysis considers the influence of suprathermal particles characterized by the parameter κ, along with variations in T_e/T_i and the height (h) relative to the Sun’s radius (R_Sun). This comprehensive approach helps us understand how these elements affect the behavior and dynamics of KAWs and particles in the solar corona.

The manuscript is organized as follows. Section 2 provides the KAWs model that supports our analytical framework. Section 3 provides an in-depth numerical analysis to validate our theoretical results. Finally, Section 4 offers a comprehensive discussion and conclusion, summarizing our key findings and their broad implications.

2. Kinetic Alfvén wave model

We start with the dispersion relation described by Lysak & Lotko (1996), Lysak (1998) for a collisionless, homogeneous, and low-β (i.e., the ratio of thermal to magnetic pressure β = P/(B²/2μ₀)≪1; Zank et al. 2022) electrons/ions plasma:

$$\begin{array}{c}\hfill \left(\begin{array}{ccc}{ϵ}_{\mathit{xx}}-\frac{k_{\Vert }^2{c}^2}{{\omega }^2}& & \frac{k_{\perp }{k}_{\Vert }{c}^2}{{\omega }^2}\\ & & \\ \frac{k_{\perp }{k}_{\Vert }{c}^2}{{\omega }^2}& & {ϵ}_{\mathit{zz}}-\frac{k_{\perp }^2{c}^2}{{\omega }^2}\end{array}\right)\left(\begin{array}{c}{\mathrm{E}}_x\\ \\ {\mathrm{E}}_z\end{array}\right)=0.\end{array}$$(1)

In Eq. (1), ω is assumed to be a complex quantity – that is, ω = ω_r + iω_i with ω_i ≪ ω_r – and the subscripts r and i denote the real and imaginary, respectively. The term ck_⊥, ∥/ω is the perpendicular/parallel refractive indices, ϵ_xx, zz is the x,z-component of the permittivity tensors, and c is the speed of light.

In developing our model, specifically in deriving the expressions for the two permittivity tensors (ϵ_xx and ϵ_zz) given in Appendix A, we considered a constant mean/ambient magnetic field oriented along the z axis. This assumption ensures that the system remains homogeneous, so we have no inhomogeneity within our model. In the model, we assumed the magnetic field values, B, could range between 10 and 100 G, reflecting different regions of the solar corona at various heights or distances from the Sun (R_Sun). The specific value chosen for our model is simply a numerical selection from this range; however, it remains constant throughout our analysis. On the other hand, if we were to account for inhomogeneity in the system, the whole model, permittivity tensors (ϵ_xx and ϵ_zz) would require substantial modifications that are beyond the scope of the present study. Currently, we are uncertain whether introducing inhomogeneity would lead to significant findings.

Moreover, we assumed that the plasma is supporting low-frequency Alfvén waves, which means that the wave frequency is less than the ion gyrofrequency (ω ≫ Ω_i). The wave propagates in a cylindrical geometry where the ambient/mean magnetic field B₀ points in the z direction. The wave magnetic field, B, is in the y direction and the perturbed electric field, E, and the wave vector, k, are in the x-z plane (Fig. 1). Specifically, we are interested in the solar coronal region, however, we fit this obliquely propagating wave’s geometry in the solar flux tube loop, as is shown in Fig. 1 (right panel).

Fig. 1. Geometry of obliquely propagating KAWs. The left panel depicts the generation of the waves (KAWs) somewhere near the Sun’s surface and propagates into the solar corona. The right panel shows a detailed schematic of KAWs within a solar flux tube loop. The flux tube, with a height, h, and a circular crosssection of radius a, provides a structured pathway for wave propagation. This fitted geometry highlights how KAWs navigate through the solar atmosphere, emphasizing the importance of their spatial characteristics in understanding energy distribution and particle acceleration in the solar corona.

In addition to the above description, we assumed that the plasma possesses suprathermal high-energy particles. For the sake of simplicity, in this research work, we are considering the isotropic nature of the particles. To model such plasma configuration, we need a suitable velocity distribution function for which we employ the isotropic Kappa distribution function of the form (Summers & Thorne 1991):

$$\begin{array}{c}\hfill f_{0s}(v)=\frac{1}{{\pi }^{3/2}{\mathrm{v}}_{\mathit{Ts}}^3}\frac{\mathrm{\Gamma }(\kappa +1)}{\mathrm{\Gamma }(\kappa -1/2)}{(1+\frac{1}{\kappa }\frac{v^2}{{\mathrm{v}}_{\mathit{Ts}}^2})}^{-\kappa -1}.\end{array}$$(2)

In the distribution function, the symbol Γ denotes the gamma function, ${\mathrm{v}}_{\mathrm{Ts}}=\sqrt{\frac{2{\mathrm{T}}_s}{m_s}}$ is the thermal speed of s-species (s is ions or electrons) of mass m_s and temperature T_s. The spectral index κ should be greater than 3/2 and can take any values other than 3/2. Interestingly, the distribution function is reduced to the Maxwellian distribution when κ → ∞.

Using Eq. (2) in the two permittivity tensors (ϵ_xx and ϵ_zz) expressions and on substitute ϵ_xx and ϵ_zz in

$$\begin{array}{c}\hfill \mathrm{D}=({ϵ}_{\mathit{xx}}-n_{\Vert }^2){ϵ}_{\mathit{zz}}-{ϵ}_{\mathit{xx}}{n}_{\perp }^2,\end{array}$$

the real and imaginary frequencies of KAWs can be obtained (for instance, see the details in Appendix A). These dispersion relations are further used in Faraday’s law and employing Eq. (1), we get the real and imaginary parts of the perturbed EM field ratios, respectively. The mathematical steps for the derivation of these expressions are provided in the Appendix A.

Furthermore, we utilize these EM field expressions and get the parallel and perpendicular Poynting flux vectors (S_z and S_x) of KAWs and the net power transfer by KAWs through the solar coronal flux loop. Huang et al. (2018) studied KAWs and demonstrated that these waves are capable of efficiently transporting energy and play an important role in facilitating magnetic reconnection. Using particle-in-cell simulations, they examined Hall fields in the magnetic reconnection region and found that (1) the Hall electric field is balanced by the ion pressure gradient and (2) the ratio of Hall electric field to Hall magnetic field is on the order of Alfvén speed. These results are consistent with our current analysis of the perturbed EM field ratios. The simulation results of Huang et al. (2018) also indicate that KAWs are excited in the reconnection site and then transmitted along the separatrices. The wave Poynting flux propagates parallel to the magnetic field lines, carrying substantial energy. In our work, we also find that KAWs Poynting flux dissipates at a slower rate in the Kappa distribution demonstrating that KAWs carry significant energy along the magnetic field lines (i.e., in the parallel direction).

In general, when KAWs interact with ions and electrons in a resonant region, Landau resonance primarily generates field-aligned electron beams, while cyclotron resonance with ions results in cross-field ion heating. In the present study, we have electron-ion plasma, where KAWs interact with both species via the Landau damping mechanism. In our model, particularly during the derivation of the expressions for the permittivity tensors (ϵ_xx and ϵ_zz), we considered zero modes; in other words, n = 0. In this scenario, the species will experience Landau resonance when the condition ω = k_∥v_A is met. It is important to note that pure or Magnetohydrodynamic (MHD) Alfvén waves do not experience Landau resonance because they propagate strictly parallel to the ambient magnetic field (as is illustrated in Fig. 1, the left panel). In the parallel direction, there is no electric field component present, which is necessary for wave-particle interaction to occur. This distinction is crucial for understanding the dynamics of KAWs in electron-ion plasma, where the interaction mechanisms, driven by Landau damping, significantly influence energy transfer and heating processes within the plasma.

For the wave-particle interaction to occur, the wave should be obliquely propagating so that it has a non-vanishing electric field component in the direction of the ambient magnetic field. However, based on the n = 0 assumption in our model, we do not have the cyclotron resonance. On the other hand, in addition to Landau resonance, if other n ≠ 0 terms are considered; that is, n = 1 or other greater values of n, one can include cyclotron resonance and can get other modes (i.e., R and L waves).

As was stated in the introduction, Alfvén waves transition into dispersive KAWs in the particles’ kinetic scale. The wave-particle interactions via Landau damping become important in the context of kinetic dissipation of wave energy and plasma particle heating. The most promising candidate for the coronal heating mechanism is the kinetic dissipation of KAWs via wave-particle interactions through the Landau mechanism. We utilize this Landau mechanism and find the net resonance speed the particle gains from KAWs during the Landau mechanism.

Ultimately, dispersion relations are employed to determine both the group velocity and the characteristic damping length of KAWs. The group velocity quantifies the rate of energy propagation by KAWs, while the damping length reveals the extent to which KAWs transport energy within the solar corona before energy dissipation occurs. Detailed, step-by-step derivations for the Poynting flux vectors, net power deposition, resonance velocity of particles, group velocity, and characteristic damping length of KAWs are provided in Appendix B.

3. Numerical analysis

3.1. The perturbed EM-field ratios of KAWs

We begin by analyzing the normalized imaginary perturbed EM field Im(E_x/v_AB_y) of KAWs in the solar coronal region. Our interest in this expression (e.g., Eq. (A.14)) stems from our focus on a region where even small contributions play a crucial role. These minor contributions are often overlooked in the literature, but our work evaluates this field ratio for various values of the index parameter κ and the ratio of electron-to-ion temperature T_e/T_i.

Numerically, we first assumed T_e/T_i = 0.5 (Chandran 2010) in the solar corona, as is shown in Fig. 2 (left panel). We observed that the magnitude of the imaginary field ratio is enhanced for different values of κ and the normalized perpendicular wavenumber k_⊥ρ_i. The plots indicate that the difference between the Kappa and Maxwellian distributions becomes prominent at larger k_⊥ρ_i.

Fig. 2. Normalized imaginary EM field Im(Ex/vABy) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameters’ values appropriate for the solar coronal region are; vA ≈ 1.85 × 108 cm/sec, vTi ≈ 1.9 × 107 cm/sec, vTe ≈ 1.34 × 109 cm/sec, k⊥/k∥ = (100 − 115) (Chen & Wu 2012), magnetic field B = (50 − 100) G (Zirin 1996; Gary 2001), and temperature T > 106 Kelvin (De Moortel & Browning 2015), respectively. In the left panel, the ratio of electron-to-ion temperature is 0.5, and in the right panel, Te/Ti ≈ 0.4. The black curve (κ → ∞) represents the Maxwellian result.

Additionally, we evaluated the perturbed EM field at a constant T_e/T_i ≈ 0.4 (Mercier & Chambe 2015) to see how different temperatures affect the magnitude of Im(E_x/v_AB_y). As is shown in the right panel, it is evident that the magnitude of Im(E_x/v_AB_y) decreases when T_e/T_i = 0.4. One possible reason for this behavior is that according to Eq. (A.14), the temperature is directly linked to the field ratio. Consequently, the magnitude of Im(E_x/v_AB_y) decreases for smaller values of T_e/T_i and increases for larger values of T_e/T_i.

Moreover, Barik et al. (2020) studied the non-resonant instability of KAWs with κ-electrons using an electron-to-ion temperature ratio of T_e/T_i = 2. While their findings directly apply to the auroral region of Earth’s magnetosphere, they can also be extended to other magnetospheric regions, solar wind, and interplanetary medium, where ion beams, velocity shear, and non-Maxwellian electrons are present. We evaluated the normalized imaginary perturbed EM field Im(E_x/v_AB_y) using this temperature ratio (Fig. 3, left panel) and found that the magnitude of Im(E_x/v_AB_y) is significantly increased and decreases when T_e/T_i = 1/5 (right panel). These evaluations confirm that the temperature ratio directly influences the magnitude of the perturbed EM field.

Fig. 3. Normalized imaginary EM field Im(Ex/vABy) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameters’ values are the same as those in Fig. 2. In the left and right panels, Te/Ti = 2 and Te/Ti ≈ 0.2, respectively.

For clarity, we provide Table (Table 1) with the selected parameters relevant to the solar coronal region. These values are representative of the solar corona and solar wind conditions. We used these parameters to discuss and develop our model, ensuring that the analysis aligns with the physical properties of these environments.

In the present study, we focus less on the real part of the perturbed EM field ratio, which is extensively discussed by Khan et al. (2019b, 2020). Our findings show that the imaginary part of the EM field, while relatively small in magnitude in the suprathermal environment, plays a crucial role in the solar coronal region. Previous studies, such as Lysak & Song (2003), Khan et al. (2019a), Lysak (2023), often neglected these minor contributions due to their focus beyond the solar coronal regions. However, we find that these subtle contributions are significant in the warm plasma of the solar corona. Drawing from Lysak & Song (2003), we acknowledge that the small imaginary part of the Im(E_x/v_AB_y) ratio may appear inconsequential near the Sun. In previous work (Ayaz et al. 2024a), we evaluated this field ratio in a Cairns-distributed plasma and found that larger nonthermal parameters significantly influence the imaginary field ratio’s magnitude. Our current observations align with our prior work, indicating that for smaller κ values, the wave exhibits electrostatic characteristics consistent with Lysak (2023) findings. Additionally, our observations resonate with Polar observations by Wygant et al. (2002), which provide empirical evidence of the ion gyroradius effect amplifying the E_x/B_y ratio.

The normalized real EM field ratio $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ is evaluated for different values of κ and the electron-to-ion temperature ratio, as is illustrated in Fig. 4. Our analysis shows that $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ increases for smaller κ values, with the difference becoming more pronounced at larger k_⊥ρ_i values. For smaller k_⊥ρ_i, the difference between Maxwellian and Kappa distributions diminishes. The suprathermal nature of the particles, characterized by the index κ, significantly influences the magnitude of $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$.

Fig. 4. Normalized real EM field $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ as a function of normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameters’ values are the same as those in Fig. 2. In the left and right panels, Te/Ti ≈ 0.5 and Te/Ti ≈ 0.4, respectively.

We also examined the EM field for different temperature ratios. At a higher electron-to-ion temperature ratio of 0.5 (left panel), the magnitude of the EM field is enhanced compared to lower temperature regions (right panel). Despite its relatively small magnitude, ×10⁻⁴, this part of $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ is typically neglected. However, our study highlights the importance of this small contribution to the solar coronal regime.

Fig. 5, illustrates the normalized imaginary $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Im}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ for different values of κ and T_e/T_i. For a given k_⊥ρ_i, the magnitude of this ratio increases gradually for smaller κ values. The difference between Maxwellian and Kappa distributions becomes noticeable only at larger k_⊥ρ_i values. Compared to the real part, $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ in Fig. 3, the imaginary part exhibits a similar trend with smaller κ values, but the magnitude of $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ is approximately ∼10³ times smaller.

Fig. 5. Normalized imaginary EM field $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Im}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ versus normalized perpendicular wavenumber k⊥ρi for different values of κ. The curves are generated using Eq. (A.17) with the parameters’ values the same as those in Fig. 2. In the left and right panels, Te/Ti = 0.5 and Te/Ti ≈ 0.4, respectively.

Furthermore, the trend of the curves for the imaginary part $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Im}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ is opposite to that of the real part (Fig. 4). Prior studies by Khan et al. (2019b) elaborate on these trends. While the detailed evaluation of $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Im}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ is extensively covered by Khan et al. (2020), we include these results here for consistency and comparison with our primary focus on Re(E_x/v_AB_y), which remains unexplored.

3.2. The Poynting flux vectors of KAWs

The Poynting flux vector is key to determining how much EM energy the waves transfer to the plasma as they propagate from z = 0. Figs. 6 and 7 show the Poynting flux vector of KAWs for various values at different κ, T_e/T_i, and heights h relative to the Sun’s radius (R_Sun). When T_e/T_i = 0.5 (Fig. 6, left panel), the Poynting flux rate S_z(z)/S_z(0) decreases moderately slower for smaller κ values. For larger κ values (κ → ∞), representing the Maxwellian case, the flux rate decays rapidly, causing the wave to damp quickly. Conversely, with a temperature ratio of 0.4 (Fig. 6, right panel), the wave damps at a slower rate, allowing it to transport energy over a more extended distance compared to when T_e/T_i = 0.5 with the same κ variation. This variation in S_z(z)/S_z(0) for different κ values has interesting implications. Smaller κ values, indicating the presence of suprathermal particles, enable the wave to transport energy over longer distances. When the temperature ratio is lower, the wave transports energy over greater distances compared to higher temperature ratios. This suggests that at higher temperatures, a larger number of particles resonate, causing the Poynting flux to decay quickly. In recent work by Khan et al. (2020), this phenomenon is well elaborated for both isotropic and anisotropic particle temperatures, with analytic results consistent with this study.

Fig. 6. Normalized Poynting flux Sz(z)/Sz(0) as a function of the radius of the Sun (RSun) for different values of κ. We assumed k⊥ρi ≈ 0.02, h = 0.05 RSun, and the other parameter values are the same ones that we used in Fig. 2. In the left panel, Te/Ti = 0.5 and in the right panel it is 0.4.

Fig. 7. Normalized Poynting flux Sz(z)/Sz(0) as a function of RSun for different values of κ at fixed h = 0.1 RSun. The parameter values are the same as those we used in Fig. 6 with the left panel, Te/Ti = 0.5, and 0.4 in the right panel.

Consider now a case in which we increased height h = 0.1 R_Sun and evaluate S_z(z)/S_z(0) for different κ values. The results show that S_z(z)/S_z(0) decreases rapidly for larger κ values, while for smaller κ values, the Poynting flux decays more gradually, as is depicted in Fig. 7. Different temperature ratios correspond to different spatial distances over which the wave transports energy. Specifically, with lower T_e/T_i ratios (right panel), the wave transports energy over a greater distance compared to higher T_e/T_i ratios. The flux rate also decays rapidly when h = 0.1 R_Sun.

This analysis highlights the significant differences between Maxwellian and non-Maxwellian plasmas within the small distance range of R_Sun, approximately (0.4–5) R_Sun. According to the Poynting flux expression (B.8), during energy transport, the wave converts its EM energy to Landau resonant electrons, with the conversion dependent on the stationary state described by the index κ. In Kappa-distributed plasma, compared to Maxwellian plasma, the wave transfers its energy to resonant electrons over larger distances. This means that in the small-κ regime, energy is distributed among a smaller number of suprathermal particles, making small κ values beneficial for heating and/or accelerating plasma particles over long distances. This behavior may be observed in the solar wind and corona, where the Polar spacecraft has reported κ ≤ 10 (Kletzing et al. 2003) that fit the data well compared to Maxwellian distribution.

The preceding analysis of S_z(z)/S_z(0) provides insights into the parallel energy transportation of KAWs. While this parallel Poynting flux and energy delivery by KAWs has been extensively studied by Lysak & Song (2003), Khan et al. (2019a, 2020), Lysak (2023), consider now the perpendicular energy transport of KAWs. Given that k_∥ ≪ k_⊥, KAWs typically carry less energy in the perpendicular direction. Figs. 8 and 9 illustrate how KAWs transport energy perpendicularly. The magnitude of the perpendicular Poynting flux S_x(z)/S_z(0) decreases for larger κ values and increases for smaller κ values. In the left panel, with a temperature ratio of T_e/T_i = 0.5, S_x(z)/S_z(0) decays faster compared to the right panel, where T_e/T_i = 0.4.

Fig. 8. Normalized perpendicular Poynting flux Sx(z)/Sz(0) versus RSun for different values of κ at a fixed height of h = 0.05 RSun. The parameter values are the same as those we used in Fig. 6 with the left panel, Te/Ti = 0.5, and 0.4 in the right panel.

For the case of h = 0.1 R_Sun, the magnitude of S_x(z)/S_z(0) follows the same trend observed previously when h = 0.05 R_Sun, for the same variations in κ and T_e/T_i, as is shown in Fig. 9. The primary difference lies in the distance (R_Sun) over which the wave transports energy. At h = 0.05 R_Sun, the wave carries energy over a longer distance (Fig. 8) compared to h = 0.1 R_Sun (Fig. 9). This difference in the spatial extent of energy transport is due to the contribution of the electric field in the x direction, as is described in Eq. (B.9). This electric field, combined with the strong magnetic field, significantly influences the energy transport of KAWs in the perpendicular direction.

Fig. 9. Normalized perpendicular Poynting flux Sx(z)/Sz(0) versus RSun for different values of κ at a fixed height of h = 0.1 RSun. The parameter values are the same as those we used in Fig. 6 with the left panel, Te/Ti = 0.5, and 0.4 in the right panel.

3.3. The net power delivered by KAWs

The Poynting flux expressions for parallel and perpendicular directions (S_x and S_z) are used to determine the total power transfer rate of KAWs in a solar flux tube loop (Fig. 1, right panel); that is, the total power transfer rate (I_x/I_z). The flux tube is characterized by its height, h, and radius or width a. The height – represented by the parallel Poynting flux vector S_z, indicates that more energy is transported in the parallel direction by the KAWs. Conversely, the width of the loop, characterized by the perpendicular Poynting flux S_x, shows that less energy is transported in the perpendicular direction. We need both S_x and S_z to determine the total power delivered within the flux tube loop.

Fig. 10 shows that the total power transfer rate (I_x/I_z) is significantly influenced by different values of κ. For larger κ values (∼ Maxwellian conditions), I_x/I_z increases sharply over a shorter distance, indicating that the wave delivers its power to the particles over a small spatial extent. For smaller κ values, indicative of more suprathermal particles, the power transfer rate increases more gradually over larger distances, meaning the wave transports power over a longer spatial extent. This occurs because the suprathermal environment facilitates more particle contributions to energy transport, allowing the wave to carry power over greater distances.

Fig. 10. The normalized power transfer rate (Ix/Iz) as a function of the normalized distance z × 0.0005 RSun, at a fixed height h = 0.1 RSun. The radius of the circular crosssection, a, is assumed to be 7 × 107 cm, with other parameters consistent with those used in Fig. 6. In the left panel, Te/Ti = 0.5, and in the right panel, Te/Ti = 0.4. The power transfer rate is significantly influenced by the suprathermal particles, characterized by κ.

We also evaluated I_x/I_z for different temperature ratios. Assuming T_e/T_i = 0.5 (left panel), the power transfer rate is lower compared to the case of T_e/T_i = 0.4 (right panel). At higher temperatures, the wave transports more power over a longer distance. When the temperature is lower, the wave transports less power over a shorter distance.

With h = 0.05 R_Sun and the same values for κ and the electron-to-ion temperature ratios, the normalized power transfer rate increases more rapidly, as is shown in Fig. 11. The power transfer rate accelerates more quickly in regimes with larger κ values, while it progresses more slowly in regimes with smaller κ values. We also observe that the power transfer rate in the h = 0.05 R_Sun case is faster compared to the h = 0.1 R_Sun scenario.

Fig. 11. The normalized Ix/Iz versus distance z × 0.0005 RSun at fixed h = 0.05 RSun. The parameters are the same as in Fig. 10. In the left panel, Te/Ti = 0.5, and 0.4 in the right panel.

In Ayaz et al. (2024a), we evaluated the power delivery rate of KAWs in a Cairns-distributed plasma and found that the power transfer rate is enhanced in the presence of a larger number of nonthermal particles. Our findings here are consistent with that, reaffirming that the power transfer rate is significantly influenced by suprathermal particles. Furthermore, our examination of the different heights implies that in regions closer to the Sun, where h is smaller, KAWs can transport energy more efficiently over shorter distances, especially in plasmas with higher suprathermal particle populations.

Besides variation in κ, we also evaluated the power transfer rate for different values of the normalized wavenumber k_⊥ρ_i, as is shown in Fig. 12. For simplicity, we assumed larger values of κ – the Maxwellian situation – and observed that I_x/I_z is significantly influenced by even minor changes in k_⊥ρ_i. This indicates that small changes in k_⊥ρ_i lead to an increased power delivery rate. Specifically, for larger k_⊥ρ_i values, the wave transports power more rapidly (the black curve). Conversely, for smaller k_⊥ρ_i values, the wave transport power is moderately weaker (the red curve).

Fig. 12. The power transfer rate (Ix/Iz) as a function of z × 0.0005 RSun at a fixed h = 0.05 RSun for different values of k⊥ρi. The parameters are the same as in Fig. 10. In both panels, we assumed h = 0.05 RSun and larger values of κ with Te/Ti = 0.5 (left), and Te/Ti = 0.4 (right panel).

Additionally, when we decreased the temperature ratio to 0.4 (right panel), the power transfer rate became sharper, and the wave transported power at an even faster rate. Physically, this implies that the efficiency of power transfer by KAWs is highly sensitive to the wavenumber and temperature ratios. In regions with higher k_⊥ρ_i or lower temperature ratios, KAWs can deliver power more effectively over shorter distances. The results align with our previous study (Ayaz et al. 2024a), reinforcing the significance of nonthermal particles in enhancing the power transfer rate of KAWs.

3.4. Net resonance velocity of the particles

Fig. 13, illustrates the normalized resonant velocity, v_net/v_A, for different values of κ and height h. The particle velocity, v_net/v_A, is significantly influenced by κ and decays at a moderately slower rate for smaller κ values. This provides insights into wave-particle interactions, showing that particles accelerate to higher speeds, enabling energy transport and heating over longer distances.

Fig. 13. Normalized parallel resonance velocity vnet/vA as a function of z(×10−4) RSun for different values of κ at fixed Te/Ti = 0.5. We assumed S(0) = 102 − 104 W cm–2 (Srivastava et al. 2017), density n0 ≈ 5 × 109 cm–3 (Morton et al. 2015), k⊥ρi ≈ 0.02, and the other parameters are the same as those in Fig. 12. In the left panel, h = 0.05 RSun and h = 0.1 RSun in the right panel.

We also evaluated v_net/v_A for both high and low temperatures and found no significant effect of temperature on the resonant velocity. The normalized v_net/v_A represents the heating and acceleration of charged particles in the parallel direction through Landau resonance. This process is described in expression (B.17), with corresponding plots shown in Fig. 13. We further investigated v_net/v_A for different heights h. At h = 0.05 R_Sun (left panel), particles accelerate and transport energy over longer distances compared to h = 0.1 R_Sun (right panel). For h = 0.1 R_Sun, particles heat the plasma over small distance. Nearby (0.05 R_Sun), particles accelerate over longer distances because more suprathermal particles contribute to wave-particle interactions, and facilitate extended heating distances. This detailed analysis gives the significant role of κ and height in the dynamics of energy transport and particle acceleration by KAWs in the solar corona.

Fig. 14 shows the normalized perpendicular net velocity v_net⊥/v_A of KAWs for different values of κ. KAWs carry less energy in perpendicular directions. We observe that v_net⊥/v_A decreases at a slower rate for small κ values and faster for larger κ values. In the case of smaller κ values, more suprathermal particles participate in wave-particle interactions, significantly affecting v_net⊥/v_A and accelerating and heating the plasma over larger distances. For larger κ values, the net resonant speed of particles decreases more rapidly, resulting in shorter acceleration distances.

Fig. 14. Normalized perpendicular resonance velocity, vnet⊥/vA, versus normalized distance z(×10−4) RSun for different κ values at a fixed Te/Ti = 0.5. The parameters are consistent with those in Fig. 13. In the left panel, h = 0.05 RSun and the right panel represents h = 0.1 RSun.

We also evaluated v_net⊥/v_A for different heights, h. When h = 0.05 R_Sun, particles heat the plasma over extended distances (left panel). For h = 0.1 R_Sun, the resonant speed of particles decays very quickly, limiting the acceleration distance (right panel). Furthermore, we assessed v_net⊥/v_A for different electron-to-ion temperature ratios and found no significant effect on the normalized perpendicular resonant speed. These variations in v_net⊥/v_A for different heights and suprathermal particles, characterized by the index κ, illustrate how resonant particles can heat and accelerate the plasma over extended distances.

Very recently, Rivera et al. (2024) investigated the heating and acceleration of solar wind plasma as it moves from the outer edge of the solar corona to the inner heliosphere, supported by observations from the PSP and Solar Orbiter. The study shows that large-amplitude Alfvén waves contribute significantly to both plasma heating and acceleration. This observation can be linked to our study of KAWs, which play a central role in transferring energy through wave-particle interactions, heating the plasma, and accelerating charged particles in the solar wind and solar corona. The analytical results are compared with Rivera et al. (2024) who found the Alfvén speed was approximately v_A ≈ 4.35 × 10⁷ cm/s at a distance of 13.5 R_Sun. To ensure consistency with this value, we assumed a magnetic field strength of B = 10 G in our analytical approach. Our calculations yielded an Alfvén speed of about v_A ≈ 3 × 10⁸ cm/s, which is largely consistent with Rivera et al.’s results, with the minor discrepancy may likely be due to the difference in distance, as we considered 0.05–0.1 R_Sun.

Furthermore, Rivera et al. reported a bulk particle speed (i.e., proton’s speed) of roughly v_α ≈ 3.96 × 10⁷ cm/s at the same 13.5 R_Sun region. However, our analytical results, particularly for the resonant particle speed, indicate that particles reach a net resonant speed around ∼3.15 × 10⁸ cm/s at 0.05–0.1 R_Sun. The findings of our research are typically consistent with the observational analysis, especially the net parallel and perpendicular resonance speed of the particles contributing to acceleration and heating processes in the solar wind and solar corona regions.

3.5. Group velocity and the damping length of KAWs

The analysis of group speed is also very important; it tells us how the energy flows. The normalized group velocity (v_G/v_A) of KAWs is evaluated for different values of the suprathermal parameter κ as is shown in Fig. 15. The results indicate a significant v_G/v_A enhancement for smaller κ values, in contrast to the Maxwellian case (κ → ∞). Lower κ values, and hence greater presence of suprathermal particles, collectively increase the group velocity of the wave.

Fig. 15. Normalized group velocity (vG/vA) as a function of the normalized wavenumber k⊥ρi for different values of κ. We assumed Te/Ti = 0.5 and the other parameter values are the same that we used in the previous figures. The normalized group velocity is enhanced for smaller values of κ.

Finally, we evaluate the characteristic damping length (L_G) – provides insight into how far KAWs and particles can transport energy before being damped, for different values of κ, shown in Fig. 16. L_G decreases for larger κ values and increases for smaller κ values. Waves (KAWs) damp quickly in the larger κ regimes and at a moderately slower rate when κ values are smaller. This implies that smaller κ values are advantageous for heating or accelerating plasma particles over long distances, which may be observed in the solar corona and solar wind regions.

Fig. 16. The damping length (LG) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The graphs are plotted using Eq. (B.22) with the same parameter values as those which we assumed in Fig. 1. In the left panel, Te/Ti = 0.4, and in the right panel, Te/Ti = 0.5. The damping length is significantly influenced by different values of electron-to-ion temperature ratios and κ.

We also investigated L_G under different temperature conditions. In the right panel, when the electron-to-ion temperature ratio is 0.4, the magnitude of L_G increased for the same variation in κ. This indicates that lower temperature values increase the damping length, allowing particles to transport energy for longer distances before damping. Conversely, in the higher temperature scenario (T_e/T_i = 0.5, left panel), the magnitude of L_G is reduced; that is, higher temperatures cause KAWs to decay faster and heat the plasma over shorter distances. This analysis provides a good approximation and estimate of how KAWs get damped and how far they can transport energy before damping.

The ratio of L_G and the parallel wavevector (k_∥) as a function of k_⊥ρ_i is evaluated for different values of κ (see Fig. 17). According to our chosen parameters and the required conditions of KAWs, we have k_∥ ≪ k_⊥. In this work, we have k_⊥/k_∥ ≈ (100–115), which corresponds to an angle of θ ≈ (89.5–90). Therefore, it is recommended to have the k_∥ 100 times smaller than k_⊥. Based on our model parameters, we have k_⊥ρ_i ≈ 0.01, which is consistent with Singh & Jatav (2019) who studied KAWs heating in the solar corona. Thus, in our model, k_∥ ≈ ×10⁻⁶ cm^–1. Substituting the parameter values in Eq. (B.22), we get L_G ≈ ×10¹⁰ – equivalent to R_Sun, which is shown in Fig. 16. The ratio of L_G and k_∥ gives ∼ × 10⁴. Based on this analysis, the magnitude of L_G/k_∥ is 10⁴ times smaller than L_G alone, as is illustrated in Fig. 17. This implies that KAWs can propagate over considerable distances with minimal damping. This efficiency in energy transport suggests that KAWs play a vital role in transferring energy (for example, released during magnetic reconnection) throughout the solar corona, enhancing particle heating and acceleration while highlighting the dominance of perpendicular structures in their dynamics.

Fig. 17. Damping length (LG) and the parallel wavevector (k∥) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameter values are the same as the ones that we assumed in Fig. 16. In the left panel, Te/Ti = 0.4, and in the right panel, Te/Ti = 0.5.

4. Discussion and conclusion

We present a detailed study of the energy transportation and heating phenomena of KAWs in the solar corona using Kappa-distribution function. Our focus is on the perturbed EM field ratios for different values of κ and the effects of electron-to-ion temperature ratios. Specifically, we investigate the imaginary field ratios (Im(E_x/v_AB_y) and k_⊥/k_∥Im(E_z/E_x)), which yields interesting results. For regions with larger wavenumbers and smaller suprathermal particle populations, the magnitude of the imaginary fields is slightly enhanced. This provides valuable insights into how EM perturbations are influenced by suprathermal particles and the electron-to-ion temperature.

We observe that the perturbed field ratios are significantly enhanced at higher temperatures. Our primary interest lies in determining how the energy stored in these EM fields is converted into the heating and acceleration of charged particles. By utilizing these field ratios in the Poynting flux expression, we gain knowledge about how KAWs transport energy to particles. Our analytical results align with previous studies by Lysak & Song (2003), Khan et al. (2020), and Lysak (2023). Additionally, our findings are consistent with observational data from Wygant et al. (2002) and the numerical investigation in our recent work (Ayaz et al. 2024a). We found that the perturbations in the imaginary field ratios are significantly affected by the larger nonthermal parameters. This comprehensive analysis advances our understanding of the role of KAWs in the solar corona’s energy dynamics.

Understanding the perturbed EM fields is important for confirming wave characteristics and determining how energy is stored and transported by KAWs in the coronal region (Khan et al. 2020). This study focuses on the Poynting flux vector, which provides essential information about how KAWs carry energy along and across the ambient magnetic field lines. Using an analytical approach (i.e., Eq. (B.8)), we demonstrate that KAWs transport significant energy parallel to these field lines. Our findings indicate that the Poynting flux rate decays more slowly for smaller values of κ. Physically, this suggests that an abundance of suprathermal particles contributes to slower energy decay, allowing the wave to transport energy over longer distances. The examination of the parallel Poynting vector (S_z(z)/S_z(0)) shows that KAWs transport more energy along the magnetic field lines. We also evaluated (S_z(z)/S_z(0)) by varying height and temperature. When h = 0.1 R_Sun, which is farther from the Sun, the wave’s energy transport decays more rapidly compared to h = 0.5 R_Sun, a closer distance. This behavior is expected, as waves transport energy over longer spatial extents in nearer regions and decay more quickly in farther regions.

Additionally, we evaluated the effect of different temperature ratios on the energy transportation of KAWs. Higher temperatures result in longer energy transport distances and slower decay rates, while lower temperatures lead to faster decay. This analysis aligns with previous studies by Khan et al. (2019a,b, 2020) who found that KAWs transport more energy when suprathermal particles are abundant. Our recent work (Ayaz et al. 2024a) also supports these findings, reinforcing the significance of suprathermal particles in the energy dynamics of KAWs in the solar corona.

Zank et al. (2018b) adopted the perspective of Markovskii et al. (2006) and studied perpendicular and parallel heating of plasma in the solar corona, showing that perpendicular heating is strongly dissipated. Building on this idea, we investigated the perpendicular Poynting flux to understand how KAWs transport energy in the perpendicular direction. Our findings reveal that the perpendicular Poynting flux (S_x(z)/S_z(0)) is indeed strongly dissipated. Focusing on the S_x(z)/S_z(0), we explored KAWs in the solar flux tube loop, where small amounts of energy are transported perpendicularly, forming a semi-circular loop. Our results indicate that the magnitude of S_x(z)/S_z(0) increases for smaller κ values and decreases for larger ones. This suggests that in environments with more suprathermal particles (smaller κ), the perpendicular energy transport is more significant, whereas in regions with fewer suprathermal particles (larger κ), the energy transport is reduced.

The two Poynting flux expressions (S_x and S_z) are of interest in estimating the total power of KAWs in the solar coronal loop. Our analysis shows that the net power transfer rate (I_x/I_z) of KAWs in the flux loop tube is significantly enhanced in suprathermal environments (smaller κ values) compared to Maxwellian distributions (larger κ values). In regions with a higher population of suprathermal particles, the wave transports power over extended distances. Specifically, in larger κ regimes, I_x/I_z increases rapidly, resulting in power being transported over shorter distances. Conversely, in smaller κ environments, I_x/I_z is only weakly enhanced, leading to power being transported over more extended distances in the solar corona. This power transfer rate provides crucial insights into how KAWs transport power and propagate through the flux tube. Our analytical observations align with our recent work (Ayaz et al. 2024a), where we found that the power rate is enhanced for higher values of the nonthermal parameters.

We investigated the parallel net resonant velocity, v_net, of the particles. We find that v_net decays slowly for smaller κ values and quickly for larger κ values. In smaller κ environments, particles travel at higher speeds and can heat the plasma over longer distances (R_Sun) before completely delivering their energy. This provides insights into how particles accelerate and heat the plasma in a parallel direction after resonating with the waves.

Moreover, we also evaluated the perpendicular resonant speed v_net⊥ of the particles for different κ values. We find the perpendicular speed of particles vanishes in small R_Sun regions compared to v_net in the parallel direction because particles primarily heat the plasma along the magnetic field lines, with much less heating occurring in the perpendicular direction and dissipating quickly. Consequently, heating and acceleration in the perpendicular direction are fast. This disparity highlights the acceleration and heating of the particles in KAWs, emphasizing the dominance of parallel processes. The parallel and perpendicular net speeds of the particles are compared to Rivera et al. (2024), and we find that these elements are apparently aligned with an observational analysis of the particles’ speed. Moreover, very recently, we studied KAWs and particle heating in the solar corona (Ayaz et al. 2024b). We found that the parallel and perpendicular net resonance speed of KAWs is significantly enhanced for larger nonthermal parameters.

The normalized group velocity, v_G/v_A, of KAWs was evaluated for both suprathermal particles and normalized perpendicular wavenumbers, k_⊥ρ_i. We find that the group velocity increases for smaller κ and larger k_⊥ρ_i values. At relatively small k_⊥ρ_i, the difference between Maxwellian and Kappa-distributed plasmas vanishes. The expression (B.11) of the group velocity was used to determine the characteristic damping length of KAWs. We find that the damping length of KAWs is moderately weaker in suprathermal plasma compared to Maxwellian plasma. In the Maxwellian situation (i.e., larger κ values), the damping length is shorter, causing the wave to damp completely over a smaller distance (R_sun). In contrast, in suprathermal plasma (i.e., smaller κ values), the damping length is longer, allowing the wave to transport energy over greater distances.

Finally, we analyzed the damping length for different electron-to-ion temperature ratios and found that lower temperatures slow the damping length. Physically, this means that in a cooler plasma environment, waves can transport energy further before damping completely, illustrating the importance of temperature in the energy transport dynamics of KAWs in the solar corona or solar wind regions.

Future extensions of this work could include studying the effects of temperature anisotropies and density homogeneity, or their combination. This approach would help us understand how parallel and perpendicular temperatures, along with density homogeneity, affect the dynamics of wave behavior in observed regions. Furthermore, the findings of this work can also be applied to other regions, such as the plasma sheet and the auroral zone, where the nonthermal nature of particles is frequently observed. This broader application could provide valuable insights into wave-particle interactions and energy transport mechanisms in various space-plasma environments.

Acknowledgments

SA acknowledges the support of an NSF grant 2149771 and GPZ the partial support of a NASA Parker Solar Probe contract SV4 - 84017 and an NSF EPSCoR RII - Track - 1 Cooperative Agreement OIA - 2148653.

References

Appendix A: Background mathematical formalism

The distribution function f_0s(v) given in Eq. (2) is used to find out the permittivity tensor components (ϵ_xx and ϵ_zz) in

$$\begin{array}{c}\hfill \mathrm{D}=({ϵ}_{\mathit{xx}}-n_{\Vert }^2){ϵ}_{\mathit{zz}}-{ϵ}_{\mathit{xx}}{n}_{\perp }^2,\end{array}$$(A.1)

which are related to the distribution function through

$$\begin{array}{c}\hfill {ϵ}_{\mathit{xx}}=1+{\displaystyle \sum _s}\frac{{\omega }_{\mathit{ps}}^2}{\omega }{\displaystyle \int d^3}v{\displaystyle \sum _{n=-\infty }^{\infty }}\frac{n^2}{{\zeta }_s^2}\frac{v_{\perp }{J}_n^2({\zeta }_s)}{(\omega -k_{\Vert }{v}_{\Vert }-n{\mathrm{\Omega }}_s)}\frac{\partial f_{0s}(v)}{\partial v_{\perp }},\end{array}$$(A.2)

and

$$\begin{array}{c}\hfill {ϵ}_{\mathit{zz}}=1+{\displaystyle \sum _s}\frac{{\omega }_{\mathit{ps}}^2}{\omega }{\displaystyle \int d^3}v{\displaystyle \sum _{n=-\infty }^{\infty }}\frac{J_n^2({\zeta }_s)}{(\omega -k_{\Vert }{v}_{\Vert }-n{\mathrm{\Omega }}_s)}\frac{v_{\Vert }^2}{v_{\perp }}\frac{\partial f_{0s}(v)}{\partial v_{\perp }}.\end{array}$$(A.3)

Here, the plasma and gyro frequencies are ${\omega }_{\mathit{ps}}=\sqrt{4\pi n_0{e}^2/m_s}$ and Ω_s = q_sB₀/m_sc, respectively. J_n(ζ_s) is the Bessel function with an argument ζ_s = k_⊥v_⊥/Ω_s. In Eqs. (A.2) and (A.3), using f_0s(v) and solving the parallel and perpendicular integrals with the assumption that the ion gyroradius is smaller than the perpendicular wavelength (i.e., k_⊥ρ_i ≪ 1), to get (Khan et al. 2019b, 2020)

$$\begin{array}{c}\hfill {ϵ}_{\mathit{xx}}=\frac{c^2}{{\mathrm{v}}_{\mathrm{A}}^2}(1-\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)\end{array}$$(A.4)

and

$$\begin{array}{c}\hfill {ϵ}_{\mathit{zz}}=\frac{2{\omega }_{\mathit{pe}}^2}{k_{\Vert }^2{\mathrm{v}}_{\mathit{Te}}^2}\frac{2\kappa -1}{2\kappa }+2i\frac{{\omega }_{\mathit{pe}}^2{\omega }_r}{k_{\Vert }^3{\mathrm{v}}_{\mathit{Te}}^3}\frac{1}{{\kappa }^{3/2}}\frac{\mathrm{\Gamma }(\kappa +1)}{\mathrm{\Gamma }(\kappa -1/2)}\sqrt{\pi },\end{array}$$(A.5)

where ${\mathrm{v}}_{\mathrm{A}}=\frac{{\mathrm{B}}_0}{\sqrt{4\pi n_0{m}_i}}$ is Alfvénic speed and ρ_i²(=v_Ti²/2Ω_i²) is the ion gyroradius.

Use Eqs. (A.4) and (A.5) in Eq. (A.1) and take D = 0 to obtain the real and imaginary frequencies of KAWs:

$$\begin{array}{c}\hfill {\omega }_r=k_{\Vert }{\mathrm{v}}_A[(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2{]}^{1/2},\end{array}$$(A.6)

and

$$\begin{array}{c}\hfill {\omega }_i=-k_{\Vert }{\mathrm{v}}_A\gamma ,\end{array}$$(A.7)

with

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \gamma =\frac{{\mathrm{v}}_{\mathrm{A}}}{{\mathrm{v}}_{\mathrm{T}\mathrm{e}}}(1+\frac{3}{4}\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2{)}^{-1}\\ \hfill \times [T_e{T}_i{k}_{\perp }^2{\rho }_i^2\frac{2{\kappa }^2}{{(2\kappa -1)}^2}\frac{\mathrm{\Gamma }(\kappa +1)}{\mathrm{\Gamma }(\kappa -1/2)}\sqrt{\pi }]\\ \hfill \times \{1+\sqrt{\frac{m_i}{m_e}}(T_e{T}_i{)}^{3/2}(1+\frac{1}{\kappa }\frac{{\omega }_r^2}{k_{\Vert }^2{\mathrm{v}}_{\mathit{Ti}}^2}{)}^{-\kappa -1}\}.\end{array}\end{array}$$(A.8)

The above expressions, particularly Eq. (A.8), illustrate the role of Landau damping. This equation, however, doesn’t explicitly illustrate how EM energy is converted into heat as the waves propagate through space. To understand this transformation, we must explore the relationship between the Poynting flux vector and the heating rate. The following sections explore this crucial element, providing a comprehensive analysis.

A.1. Derivation of the perturbed EM fields

Starting from the simple Faraday’s law

$$\begin{array}{c}\hfill \mathrm{\nabla }\times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}\end{array}$$

translates into

$$\begin{array}{c}\hfill \mathbf{k}\times \mathbf{E}=\omega \mathbf{B},\end{array}$$(A.9)

where we replaced ∇ by ik and $\frac{\partial }{\partial t}$ by −iω. As we assumed that k and E lie in the x-z plane and the B of the wave is along the y axis, Eq. (A.9) can also be written as

$$\begin{array}{c}\hfill E_x{B}_y=\frac{\omega }{k_{\Vert }-k_{\perp }{\mathrm{E}}_z/E_x}.\end{array}$$(A.10)

In Eq. (A.10), the ratio E_z/E_x in the denominator can be obtained by solving the first row of Eq. (1). Thus,

$$\begin{array}{c}\hfill E_z{E}_x=\frac{c^2{k}_{\Vert }^2-{\omega }^2{ϵ}_{\mathit{xx}}}{c^2{k}_{\Vert }{k}_{\perp }}.\end{array}$$(A.11)

Using Eq. (A.11) and Eqs. (A.4) - (A.8) with the assumption on ω = ω_r + iω_i, we get

$$\begin{array}{c}\hfill E_x{B}_y=\frac{c^2{k}_{\Vert }}{{ϵ}_{\mathit{xx}}}(\frac{{\omega }_r}{{\omega }_r^2+{\omega }_i^2}-i\frac{{\omega }_i}{{\omega }_r^2+{\omega }_i^2}).\end{array}$$(A.12)

In the assumption that ω_r ≫ ω_i, the real and imaginary parts in Eq. (A.12) yields

$$\begin{array}{c}\hfill \text{Re}(E_x{v}_A{B}_y)=\frac{(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)}{[(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2{]}^{1/2}}\end{array}$$(A.13)

and

$$\begin{array}{c}\hfill \text{Im}(E_x{v}_A{B}_y)=\frac{{\mathrm{v}}_{\mathrm{A}}}{{\mathrm{v}}_{\mathrm{Te}}}{C}_k\{1+\sqrt{\frac{m_i}{m_e}}(T_e{T}_i{)}^{3/2}(1+\frac{1}{\kappa }\frac{{\omega }_r^2}{k_{\Vert }^2{\mathrm{v}}_{\mathit{Ti}}^2}{)}^{-\kappa -1}\}\sqrt{\pi }.\end{array}$$(A.14)

Here, we define $C_k=[\frac{(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)(\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }^2{\rho }_i^2\frac{2{\kappa }^2}{{(2\kappa -1)}^2}\frac{\mathrm{\Gamma }(\kappa +1)}{\mathrm{\Gamma }(\kappa -1/2)})}{(1+\frac{3}{4}\frac{2\kappa }{2\kappa -1}\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }^2{\rho }_i^2)(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2+\frac{2\kappa }{2\kappa -1}\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }^2{\rho }_i^2)}]$.

Eq. (A.14) has often been overlooked due to its minimal contributions in certain regions of space, such as the auroral region (Lysak & Song 2003; Lysak 2023) and magnetopause (Khan et al. 2019a). However, in our study of KAWs in the solar corona, even these small contributions are crucial. Therefore, we emphasize the importance of not disregarding the minor components of the imaginary perturbed EM field ratio in our analysis.

Using the same algebra as previously, Eq. (A.11) gives

$$\begin{array}{c}\hfill E_z{E}_x=\frac{c^2{k}_{\Vert }^2-({\omega }_r^2+2i{\omega }_r{\omega }_i)\frac{c^2}{{\mathrm{v}}_{\mathrm{A}}^2}(1-\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)}{c^2{k}_{\Vert }{k}_{\perp }}.\end{array}$$(A.15)

The real and imaginary parts in Eq. (A.15) are

$$\begin{array}{cc}& \frac{k_{\perp }}{k_{\Vert }}\text{Re}(E_z{E}_x)=\hfill \\ \hfill & [1-(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2+\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2)(1-\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)]\hfill \end{array}$$(A.16)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{k_{\perp }}{k_{\Vert }}\text{Im}(E_z{E}_x)=2\gamma (1-\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)\\ \hfill \times [(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2{]}^{1/2},\end{array}\end{array}$$(A.17)

respectively.

Appendix B: Derivations of the Poynting flux vectors, power rate, resonant velocities, group speed, and the damping length expressions

B.1. The Poynting flux and heating rate of KAWs

We invoke the simplified EM field ratios (i.e., Eqs. A.10 - A.12) to find the Poynting flux vectors of KAWs. On averaging over a complete cycle, the differential term in the Poynting theorem

$$\begin{array}{c}\hfill \frac{\partial u}{\partial t}=-\mathbf{J}·\mathbf{E}-\mathrm{\nabla }·\mathbf{S}\end{array}$$(B.1)

(Lysak & Song 2003) is zero for sinusoidal EM perturbations. Simply put, we can write

$$\begin{array}{c}\hfill \mathrm{\nabla }·\mathbf{S}=-\mathrm{P},\end{array}$$(B.2)

with S being the Poynting flux vector given by

$$\begin{array}{c}\hfill \mathbf{S}=\text{Re}({\mathrm{E}}^{\ast }\times \mathbf{B})/2{\mu }_0,\end{array}$$

and P the power dissipation, given as

$$\begin{array}{c}\hfill \mathrm{P}=\text{Re}({\mathrm{J}}^{\ast }·\mathbf{E})/2.\end{array}$$

In the above expressions, μ₀ represents the permeability of free space, and J is the current density, which can be determined using Ampere’s law: J = (∇×B)/μ₀. The factor of 1/2 arises from averaging over a complete cycle.

As is shown in Fig. (1), the wave is propagating in the x-z plane, therefore, the y-component of the Poynting flux vector (S_y) is naturally zero. Following Ayaz et al. (2024a), we can write the Poynting flux vector of KAWs as

$$\begin{array}{c}\hfill \mathrm{\nabla }·\mathbf{S}=\frac{\partial }{\partial x}{\mathrm{S}}_x(x,z)+\frac{\partial }{\partial z}{\mathrm{S}}_z(x,z).\end{array}$$(B.3)

In Eq. (B.3), the first term represents the Poynting flux in the perpendicular direction and the second term gives the Poynting flux in the parallel direction. The averaged x and z components of the Poynting vectors alone in the above expression are

$$\begin{array}{c}\hfill {\mathrm{S}}_{\mathrm{x}}=-(E_z{E}_x)S_z\end{array}$$(B.4)

(Ayaz et al. 2024a) and

$$\begin{array}{c}\hfill {\mathrm{S}}_z=\text{Re}(E_x^{\ast }{B}_y)\end{array}$$(B.5)

(Lysak & Song 2003), respectively.

In general, KAWs, characterized by k_⊥ ≫ k_∥ (Narita et al. 2020), can carry some energy in the perpendicular direction (i.e., across the field lines), but that energy is typically small (Lysak & Song 2003; Khan et al. 2020). However, the perpendicular energy transport of KAWs, defined by the S_x expression, is often overlooked in certain environments, such as plasma sheaths, magnetopause, and auroral zones. Here, we focus on the solar coronal region and recognize the significance of even this small S_x contribution. This is crucial for fully understanding the energy transport mechanisms of KAWs in solar flux loop tubes, as is shown in Fig. 1 (right panel).

The parallel Poynting flux vector S_z can be simplified using the power dissipation expression

$$\begin{array}{c}\hfill \mathrm{P}=\text{Re}({\mathrm{J}}^{\ast }·\mathbf{E})/2=k_{\perp }\text{Re}(E_z{E}_x){\mathrm{S}}_{\mathrm{z}}=2\gamma k_{\Vert }{\mathrm{S}}_{\mathrm{z}}R,\end{array}$$(B.6)

with $R=(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)[(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2+\frac{2\kappa }{2\kappa -1}\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }^2{\rho }_i^2)]$.

For the z-component, the steady-state form of the Poynting flux theorem requires:

$$\begin{array}{c}\hfill \frac{\partial {\mathrm{S}}_{\mathrm{z}}(z)}{\partial z}=-\mathrm{P}=2\gamma k_{\Vert }{\mathrm{S}}_{\mathrm{z}}R.\end{array}$$(B.7)

The solution to the above equation is

$$\begin{array}{c}\hfill {\mathrm{S}}_{\mathrm{z}}(\mathrm{z})={\mathrm{S}}_{\mathrm{z}}(0)e^{-2\gamma k_{\Vert }\mathrm{z}R},\end{array}$$(B.8)

which describes the Poynting flux of KAWs in the parallel direction, highlighting how the waves transport energy as they propagate in space.

Similarly, substituting Eq. (B.8) in Eq. (B.4), we obtain the solution for S_x as

$$\begin{array}{c}\hfill {\mathrm{S}}_{\mathrm{z}}(\mathrm{x})=-(E_z{E}_x){\mathrm{S}}_{\mathrm{z}}(0)e^{-2\gamma k_{\Vert }\mathrm{z}R}.\end{array}$$(B.9)

Eqs. (B.8) and (B.9) provide a framework to quantify the conversion of EM energy into thermal energy within solar flux tube loops as the waves propagate from the initial reference point z = 0. At z = 0, where the waves are first excited, the Poynting flux has an initial magnitude of S(0). As these waves travel through the plasma, the Poynting flux decreases due to the negative imaginary part of the wave frequency, ω_i; that is, energy transfer from the waves to the plasma particles. Physically, this phenomenon results from wave-particle interactions, where plasma particles, including ions and electrons, absorb energy from the waves. This energy transfer causes the particles to gain kinetic energy, leading to plasma heating. The process is particularly significant in the solar corona, where such interactions can contribute to the high temperatures observed.

Solar flux tube loops, which are magnetic structures extending from the solar surface into the corona, play a critical role in energy transport and heating within the solar atmosphere. The plasma density within a coronal loop is typically higher than the plasma outside. The complex processes are occurring in these regions, particularly the mechanisms related to coronal heating including the turbulent cascade to very small dissipation length scales or in general kinetic scales (Marsch 2006; Browning 1991). Coronal heating remains an area of active research and debate.

Several seminal works, including those by Matthaeus et al. (1999), and Cranmer & Van Ballegooijen (2010), have laid the foundation for understanding the turbulent cascade in the solar corona. These studies describe how energy is transferred to progressively smaller perpendicular scales until it dissipates, typically at ion inertial or gyrofrequency scales. More recent investigations by Adhikari et al. (2024), Zank et al. (2018b, 2021), and Yalim et al. (2024) continue to explore the intricacies of solar coronal heating, shedding light on the complex interplay of turbulence and wave dissipation.

In particular, within coronal loops, Alfvén waves and quasi 2D fluctuations are generated by dynamic twisting and braiding motions at the loop’s footpoints on the photosphere (Zank et al. 2018a). These motions, caused by convective flows in intergranular lanes, distort magnetic flux tubes and create Alfvén waves that propagate outward along the magnetic field lines advected in 2D turbulence. As Alfvén waves travel through the chromosphere and corona, they dissipate their energy, contributing to the heating of the solar atmosphere. This process, while better understood today, continues to be a focal point of research, especially concerning the behavior of Alfvén waves (i.e., KAWs) at small kinetic scales and their role in turbulent energy cascade and dissipation.

B.2. The net power deposition by KAWs

The power transfer rate of KAWs in solar flux tube loops is a crucial indicator of energy distribution and absorption in the solar corona. As the waves propagate, their Poynting flux dissipates due to wave-particle interactions, where plasma particles absorb energy from the waves, leading to heating. This energy transfer is essential for explaining the high temperatures observed in the solar corona. It plays a significant role in solar wind acceleration, magnetic reconnection, and solar flares. Here, we quantify the power transfer rate of KAWs.

To derive the power transfer rate of KAWs, we substitute z = hθ in Eqs. (B.8) and (B.9) and integrate from 0 to π and π/2, to obtain

$$\begin{array}{c}\hfill {\mathrm{I}}_{\mathrm{x}}=a\left(E_z{E}_x\right)(\frac{-1+e^{-\gamma k_{\Vert }\mathrm{h}\pi R}}{\gamma k_{\Vert }R})\pi ,\end{array}$$(B.10)

and

$$\begin{array}{c}\hfill {\mathrm{I}}_{\mathrm{z}}=a^2(e^{-\gamma k_{\Vert }\mathrm{h}\pi R})\pi .\end{array}$$(B.11)

In Eqs. (B.10) and (B.11), a represents the crosssection of the semicircle, which ranges from 700 − 7000 km (Li et al. 2023), and h is the height of the flux tube loop. Effenberger et al. (2017) examined the height of 61 occulted flare loops, and we follow that study by assuming an h range of 0.05 − 0.1 R_Sun. The same range was considered in Li et al. (2023) to examine solar energetic neutral particles. By taking the ratio of Eqs. (B.10) and (B.11), the total power transfer rate emerging from the tube loop is

$$\begin{array}{c}\hfill \frac{{\mathrm{I}}_{\mathrm{x}}}{{\mathrm{I}}_{\mathrm{z}}}=-\left(E_z{E}_x\right)(\frac{1-e^{-\gamma k_{\Vert }\mathbf{h}\pi R}}{a\gamma k_{\Vert }R{e}^{-\gamma k_{\Vert }\mathrm{h}\pi R}}).\end{array}$$(B.12)

B.3. The resultant resonance velocity of the particles

Following Paraschiv et al. (2015), the kinetic energy flux passing is expressed as

$$\begin{array}{c}\hfill \mathrm{K}·\mathrm{E}=\frac{1}{2}\rho {\mathrm{v}}^3,\end{array}$$(B.13)

with ρ being the mass density and v be the speed of the particles, which can be written in terms of the Poynting flux vector (S_x, z) as

$$\begin{array}{c}\hfill {\mathrm{S}}_{x,z}=\frac{1}{2}\rho {\mathrm{v}}^3,\end{array}$$(B.14)

which gives

$$\begin{array}{c}\hfill \mathrm{v}={\left(\frac{2{\mathrm{S}}_{x,z}}{\rho }\right)}^{1/3}.\end{array}$$(B.15)

Eq. (B.15) represents the particle’s speed – the particles gained from the wave.

In general, when the particles interact with the wave, they receive energy from the wave to fulfill the resonance condition. In this situation, the wave delivers its energy to particles and is damped. The particles now have a net velocity that must be equal to the particle’s initial velocity and the energy that the particle gains from the wave giving

$$\begin{array}{c}\hfill {\mathrm{v}}_{\mathit{net}}=\frac{{\omega }_r}{k_{\Vert }}+(\frac{2{\mathrm{S}}_{x,z}}{\rho }{)}^{1/3}.\end{array}$$(B.16)

The first term in Eq. (B.16) represents the wave phase velocity at which particles resonate with the wave, while the second term quantifies the energy gained by the particles from the wave. Physically, this can be visualized as the wave giving a push to the particles under resonance conditions, resulting in an overall higher velocity for the particles after Landau resonance occurs. Consequently, the particle has a high speed and hence increased energy.

On substituting Eqs. (A.6) and (B.8) in Eq. (B.16), we obtain the expression for the normalized parallel net speed of the particles as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill v_{\mathit{net}}{v}_A=[(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2{]}^{1/2}\\ \hfill +\frac{1}{{\mathrm{v}}_{\mathrm{A}}}(\frac{2{\mathrm{S}}_z(0)e^{-2\gamma k_{\Vert }\mathrm{z}R}}{\rho }{)}^{1/3}.\end{array}\end{array}$$(B.17)

Similarly, employing Eqs. (A.6) and (B.9) in Eq. (B.16) to obtain the normalized perpendicular velocity expression as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill v_{net\perp }{v}_A=[(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2{]}^{1/2}\\ \hfill +\frac{1}{{\mathrm{v}}_{\mathrm{A}}}[-(E_z{E}_x)\frac{2{\mathrm{S}}_z(0)e^{-2\gamma k_{\Vert }\mathrm{z}R}}{\rho }{]}^{1/3}.\end{array}\end{array}$$(B.18)

The above expressions (B.17) and (B.18) yield the net parallel and perpendicular resonance velocity for the particles experiencing Landau damping in the solar corona. During this interaction, particles gain energy from the waves while the wave itself experiences damping.

In the parallel direction (i.e., Eq. (B.17)), the net velocity is boosted due to the efficient transfer of wave energy along the magnetic field lines, which is also shown in the numerical analysis section. This interaction primarily heats and accelerates the particles parallel to the ambient magnetic field. Similarly, although the contribution is generally smaller in the perpendicular direction (i.e., Eq. (B.18)), the particles still gain velocity due to wave-particle interactions, leading to perpendicular heating and acceleration. These dual mechanisms of energy transfer are crucial for understanding the dynamics within the solar corona.

This scenario highlights the significant role of KAWs in plasma heating and particle acceleration in the solar corona. The derived expressions for net velocity in both directions underscore how KAWs can contribute to the high temperatures observed in the corona and the efficient energy distribution throughout the solar atmosphere. This process not only elucidates the behavior of particles in the solar corona but also provides insights into broader astrophysical phenomena where similar wave-particle interactions are at play.

B.4. The group velocity and the characteristic damping length of KAWs

Following Tiwari et al. (2008), the damping length expression is

$$\begin{array}{c}\hfill {\mathrm{L}}_{\mathrm{G}}=\frac{{\mathrm{v}}_{\mathrm{G}}}{{\omega }_i},\end{array}$$(B.19)

where v_G is the group velocity defined as:

$$\begin{array}{c}\hfill {\mathrm{v}}_{\mathrm{G}}=\sqrt{(\frac{\partial {\omega }_r}{\partial k_{\perp }}{)}^2+(\frac{\partial {\omega }_r}{\partial k_{\Vert }}{)}^2}.\end{array}$$(B.20)

On evaluating Eq. (A.6) and substituting those values in Eq. (B.20), we obtain the normalized group velocity as

$$\begin{array}{c}\hfill \frac{{\mathrm{v}}_{\mathrm{G}}}{{\mathrm{v}}_{\mathrm{A}}}=[Q+(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}{T}_e{T}_i{k}_{\perp }^2{\rho }_i^2{]}^{1/2}.\end{array}$$(B.21)

Here we define, $Q=\frac{k_{\Vert }^2[(\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }{\rho }_i^2{]}^2}{(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{k}_{\perp }^2{\rho }_i^2)+\frac{2\kappa }{2\kappa -1}\frac{{\mathrm{T}}_{\mathrm{e}}}{{\mathrm{T}}_{\mathrm{i}}}{k}_{\perp }^2{\rho }_i^2}$.

Use of Eqs. (A.7) and (B.21) in (B.19) yields

$$\begin{array}{c}\hfill {\mathrm{L}}_{\mathrm{G}}=\frac{{\mathrm{v}}_{\mathrm{Te}}(1+\frac{3}{4}\frac{2\kappa }{2\kappa -1}{\mathrm{T}}_{\mathrm{e}}{\mathrm{T}}_{\mathrm{i}}{\mathrm{k}}_{\perp }^2{\rho }_{\mathrm{i}}^2)[\mathrm{Q}+(1+\frac{3}{4}\frac{2\kappa }{2\kappa -3}{\mathrm{k}}_{\perp }^2{\rho }_{\mathrm{i}}^2)+\frac{2\kappa }{2\kappa -1}{\mathrm{T}}_{\mathrm{e}}{\mathrm{T}}_{\mathrm{i}}{\mathrm{k}}_{\perp }^2{\rho }_{\mathrm{i}}^2{]}^{1/2}}{[T_e{T}_i{k}_{\perp }^2{\rho }_i^2\sqrt{\pi }\frac{2{\kappa }^2}{{(2\kappa -1)}^2}\frac{\mathrm{\Gamma }(\kappa +1)}{\mathrm{\Gamma }(\kappa -1/2)}]\times \{1+\sqrt{\frac{m_e}{m_i}}(T_e{T}_i{)}^{3/2}(1+\frac{1}{\kappa }\frac{{\omega }_r^2}{k_{\Vert }^2{\mathrm{v}}_{\mathit{Ti}}^2}{)}^{-\kappa -1}\}}.\end{array}$$(B.22)

All Tables

All Figures

Fig. 1. Geometry of obliquely propagating KAWs. The left panel depicts the generation of the waves (KAWs) somewhere near the Sun’s surface and propagates into the solar corona. The right panel shows a detailed schematic of KAWs within a solar flux tube loop. The flux tube, with a height, h, and a circular crosssection of radius a, provides a structured pathway for wave propagation. This fitted geometry highlights how KAWs navigate through the solar atmosphere, emphasizing the importance of their spatial characteristics in understanding energy distribution and particle acceleration in the solar corona.

In the text

Fig. 2. Normalized imaginary EM field Im(Ex/vABy) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameters’ values appropriate for the solar coronal region are; vA ≈ 1.85 × 108 cm/sec, vTi ≈ 1.9 × 107 cm/sec, vTe ≈ 1.34 × 109 cm/sec, k⊥/k∥ = (100 − 115) (Chen & Wu 2012), magnetic field B = (50 − 100) G (Zirin 1996; Gary 2001), and temperature T > 106 Kelvin (De Moortel & Browning 2015), respectively. In the left panel, the ratio of electron-to-ion temperature is 0.5, and in the right panel, Te/Ti ≈ 0.4. The black curve (κ → ∞) represents the Maxwellian result.

In the text

	Fig. 3. Normalized imaginary EM field Im(Ex/vABy) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameters’ values are the same as those in Fig. 2. In the left and right panels, Te/Ti = 2 and Te/Ti ≈ 0.2, respectively.
In the text

	Fig. 4. Normalized real EM field $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Re}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ as a function of normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameters’ values are the same as those in Fig. 2. In the left and right panels, Te/Ti ≈ 0.5 and Te/Ti ≈ 0.4, respectively.
In the text

	Fig. 5. Normalized imaginary EM field $\frac{k_{\perp }}{k_{\Vert }}\mathrm{Im}({\mathrm{E}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{x}})$ versus normalized perpendicular wavenumber k⊥ρi for different values of κ. The curves are generated using Eq. (A.17) with the parameters’ values the same as those in Fig. 2. In the left and right panels, Te/Ti = 0.5 and Te/Ti ≈ 0.4, respectively.
In the text

	Fig. 6. Normalized Poynting flux Sz(z)/Sz(0) as a function of the radius of the Sun (RSun) for different values of κ. We assumed k⊥ρi ≈ 0.02, h = 0.05 RSun, and the other parameter values are the same ones that we used in Fig. 2. In the left panel, Te/Ti = 0.5 and in the right panel it is 0.4.
In the text

	Fig. 7. Normalized Poynting flux Sz(z)/Sz(0) as a function of RSun for different values of κ at fixed h = 0.1 RSun. The parameter values are the same as those we used in Fig. 6 with the left panel, Te/Ti = 0.5, and 0.4 in the right panel.
In the text

	Fig. 8. Normalized perpendicular Poynting flux Sx(z)/Sz(0) versus RSun for different values of κ at a fixed height of h = 0.05 RSun. The parameter values are the same as those we used in Fig. 6 with the left panel, Te/Ti = 0.5, and 0.4 in the right panel.
In the text

	Fig. 9. Normalized perpendicular Poynting flux Sx(z)/Sz(0) versus RSun for different values of κ at a fixed height of h = 0.1 RSun. The parameter values are the same as those we used in Fig. 6 with the left panel, Te/Ti = 0.5, and 0.4 in the right panel.
In the text

Fig. 10. The normalized power transfer rate (Ix/Iz) as a function of the normalized distance z × 0.0005 RSun, at a fixed height h = 0.1 RSun. The radius of the circular crosssection, a, is assumed to be 7 × 107 cm, with other parameters consistent with those used in Fig. 6. In the left panel, Te/Ti = 0.5, and in the right panel, Te/Ti = 0.4. The power transfer rate is significantly influenced by the suprathermal particles, characterized by κ.

In the text

	Fig. 11. The normalized Ix/Iz versus distance z × 0.0005 RSun at fixed h = 0.05 RSun. The parameters are the same as in Fig. 10. In the left panel, Te/Ti = 0.5, and 0.4 in the right panel.
In the text

	Fig. 12. The power transfer rate (Ix/Iz) as a function of z × 0.0005 RSun at a fixed h = 0.05 RSun for different values of k⊥ρi. The parameters are the same as in Fig. 10. In both panels, we assumed h = 0.05 RSun and larger values of κ with Te/Ti = 0.5 (left), and Te/Ti = 0.4 (right panel).
In the text

	Fig. 13. Normalized parallel resonance velocity vnet/vA as a function of z(×10−4) RSun for different values of κ at fixed Te/Ti = 0.5. We assumed S(0) = 102 − 104 W cm–2 (Srivastava et al. 2017), density n0 ≈ 5 × 109 cm–3 (Morton et al. 2015), k⊥ρi ≈ 0.02, and the other parameters are the same as those in Fig. 12. In the left panel, h = 0.05 RSun and h = 0.1 RSun in the right panel.
In the text

	Fig. 14. Normalized perpendicular resonance velocity, vnet⊥/vA, versus normalized distance z(×10−4) RSun for different κ values at a fixed Te/Ti = 0.5. The parameters are consistent with those in Fig. 13. In the left panel, h = 0.05 RSun and the right panel represents h = 0.1 RSun.
In the text

	Fig. 15. Normalized group velocity (vG/vA) as a function of the normalized wavenumber k⊥ρi for different values of κ. We assumed Te/Ti = 0.5 and the other parameter values are the same that we used in the previous figures. The normalized group velocity is enhanced for smaller values of κ.
In the text

Fig. 16. The damping length (LG) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The graphs are plotted using Eq. (B.22) with the same parameter values as those which we assumed in Fig. 1. In the left panel, Te/Ti = 0.4, and in the right panel, Te/Ti = 0.5. The damping length is significantly influenced by different values of electron-to-ion temperature ratios and κ.

In the text

	Fig. 17. Damping length (LG) and the parallel wavevector (k∥) as a function of the normalized perpendicular wavenumber k⊥ρi for different values of κ. The parameter values are the same as the ones that we assumed in Fig. 16. In the left panel, Te/Ti = 0.4, and in the right panel, Te/Ti = 0.5.
In the text