## Abstract

Plasmas exhibit a vast variety of waves and oscillations in which moving charged particle produce fields which ultimately give rise to particle motion. These wave-particle effects are used in the acceleration heating methods of plasma particles, and in wave generation as well. Plasmas are often manipulated with EM waves, e.g., Alfvén waves are long-wavelength modes (drift-waves) where fluid theory is most reliable, while for short wavelength modes (e.g., Kinetic Alfvén waves), collisionless effects becomes important. In this chapter, the properties of kinetic Alfvén waves are aimed to study by employing two potential theory by taking particle streaming and Weibel instability with temperature anisotropy in a Lorentzian plasma.

### Keywords

- KAWs
- Lorentzian distribution
- streaming and temperature anisotropy
- dusty plasma

## 1. Introduction

This chapter addresses one of the intriguing topics of Astrophysics—the existence of kinetic Alfvén wave (KAW) and the important consequences for astrophysical and space science to explore and investigate the new avenues. Due to the fact that KAWs have non-zero electric field

The solar wind plasma is hot and weakly collisional, existing in a state far from thermal equilibrium [1] as observed in situ in the solar wind through its nonthermal characteristics of velocity distribution function (VDF). The electron VDFs measured at 1 AU have been used as boundary condition to determine the VDFs at different altitudes. It has been confirmed that for several solar radii, the suprathermal population of particles is present in the corona [2]. For low collision rates in such plasmas, the particles can develop temperature anisotropy and the VDFs become slanted and build up high energy tails and heat fluxes along the magnetic field direction especially in fast winds and energetic interplanetary shocks. Various processes in a collisionless solar wind plasma lead to the development of particle temperature anisotropy to generate plasma instabilities which are often kinetic in nature. The free energy sources associated with the deviation from the thermodynamical equilibrium distribution function could also excite plasma waves [3, 4, 5, 6, 7, 8, 9, 10, 11].

In general, the study of plasma waves and micro-instabilities in the solar wind shows that proton VDFs are prone to anisotropic instability and originate to be stable or marginally stable. Marsch [12] has discussed four significant electrostatic and electromagnetic wave modes and free energy sources to make them unstable. For example, the electrostatic ion acoustic wave may be destabilized by the ion beams and electrons and electron heat flux, [13] the electromagnetic ion Alfvén-cyclotron wave needs proton beam and temperature anisotropy, magnetosonic wave requires proton beam and ion differential streaming and whistler-mode and lower-hybrid wave [14] unstable solutions. Among several electromagnetic instabilities, the kinetic Alfvén wave instability is the most important one.

The satellite missions in space and astrophysical plasmas have confirmed the presence of non-Maxwellian high energy and velocity tails in the particle distribution function and found in the magnetosphere of Saturn, Mercury, Uranus and Earth [2, 15, 16, 17]. The non-Maxwellian distribution of charged particles has been observed to give a better fit to the thermal and superthermal part by employing kappa distribution, since it fits both thermal and suprathermal parts in the energy velocity spectra.

The subject area of this chapter involves the basic research of space plasma physics and in particular, focuses the investigations of electrostatic and electromagnetic waves in a multi-component dusty (complex) Maxwellian and non-Maxwellian plasmas. In the last few years, various power-law distribution functions (in velocity space), i.e., kappa and

When the intense radiations interact with plasmas, it ends up with many applications like instabilities, inertial confinement fusion [18], and pulsar emissions [19]. These instabilities further generate turbulent electromagnetic fields in plasma regimes. We can characterize instabilities as electrostatic as well as electromagnetic according to the conditions provided by nature [20]. In this chapter, we shall also discuss electromagnetic instability called Weibel instability in a Lorentzian plasma. The free energy source available for Weibel instability is temperature anisotropy and can be developed in magnetically confined and magnetic free plasma environment as well. First time Weibel [21] came up with the calculations of impulsive growing transverse waves with anisotropic velocity distribution function in 1958. This instability developed when the electrons in the fluctuating magnetic field generates momentum flux, this flux sequentially effects velocity

## 2. Model and methodology

In long-wavelength modes the fluid theory is most reliable, while for short wavelength modes (like KAWs), collisionless effects are important, for example, Landau damping due to finite ion Larmor radius explains observed damping rate and in dusty plasmas and charge fluctuations. Kinetic Alfvén waves (KAWs) are small scale dispersive Alfvén waves (AWs) which plays a significant role in particle acceleration and plasma heating. A coupling mechanism between small-scale KAWs and large-scale AWs in the presence of superthermal particles has been discussed which in turns giving rise to the excitation of KAWS in a solar/stellar wind plasma have been studied in the past. In this chapter, we intend to show the relationship between the growth rates of excited anisotropic KAWs and perpendicular wavelength by taking charge fluctuation and Landau damping variations into account. Moreover, when the perpendicular component of the wavelength, when comparable to the ion gyroradius, a magnetic field aligned electric field plays a significant role in the plasma acceleration/heating. Utilizing a two potential theory along with kinetic description, the properties of kinetic Alfvén waves are aimed to investigate different modes in low beta plasmas by incorporating the streaming effects. We present overview of electromagnetic KAW streaming instability in a collisionless dusty magnetoplasma, whose constituents are the electrons, ions and negatively charged dust particles. The interaction between monochromatic electron/ion beam with plasma is also discussed under various conditions. Further, to calculate the linear dispersion relation for Weibel instability in unmagnetized Lorentzian plasma, we shall employ linearized, nonrelativistic Vlasov equation.

### 2.1. Two potential theory

In a low beta plasma,

The linearized Poisson and Maxwell equations in terms of parallel and perpendicular operators can be expressed as

and

where

where

and

### 2.2. Number density and current density perturbations

Here, we may define the number density and current density as

## 3. Dispersion and damping of kinetic Alfvén waves (KAW)

Kinetic Alfvén waves (KAWs) are small scale dispersive Alfvén waves (AWs) which plays a significant role in particle acceleration, turbulence, wave particle interaction and plasma heating. Kinetic processes prevail in the regimes where plasma is dilute, multi-component, and non-uniform. A coupling mechanism between small-scale KAWs and large-scale AWs with superthermal plasma species which in turns gives rise to the excitation of KAWS in a solar/stellar wind plasma has proved dispersive Alfvén waves responsible for the solar wind turbulence especially when the turbulence cascade of these electromagnetic waves transfer from larger to smaller scale as compared to proton gyro radius. Moreover, from spacecraft observations in ionospheric plasma, it is evident that Alfvénic Poynting flux is responsible to transfer the energy for particle acceleration. All the energized auroral particles accelerate in ionosphere, initiate Joule heating phenomenon and stream out into the magnetosphere [25, 26, 27, 28].

There are number of studies to show the relationship between the growth rates of excited anisotropic KAWs and perpendicular wavelength by taking charge fluctuation and Landau damping variations into account. Moreover, the perpendicular component of wavelength, when comparable to ion gyroradius, a magnetic field aligned electric field plays a significant role in plasma acceleration/heating.

One of the important features in astrophysical plasma is the transportation of electromagnetic energy through the wave interaction with thermal plasma ions [29, 30, 31]. The KAW plays a vital role to transfer the wave energy through Landau damping (when thermal electrons travel along the magnetic field lines), which is regarded as collisionless damping of low-frequency waves and during this process the particles gain kinetic energy from the wave. This process can only happen when the distribution function has a negative slope which results in the heating of plasmas or acceleration of electrons along the magnetic field direction [24]. Recent studies also suggest the impact of non-Maxwellian distribution functions on the dynamics of solar wind and auroral plasma [32]. This study shows that the plateau formation in the parallel electron distribution functions minimize the Landau damping rate significantly.

In this chapter, the properties of kinetic Alfvén waves would be discussed by employing two potential theory, Maxwell equations and Vlasov model to study different plasma modes and by taking streaming of charged particles along and across the field direction in a Maxwellian and Lorentzian plasma.

### 3.1. Kinetic Alfvén waves in Maxwellian plasma

The propagation of kinetic Alfvén waves in a dusty plasma with finite Larmor radius effects will be discussed using a fluid-kinetic formulation by taking charge variations of dust particles. The coupling of Alfvén-acoustic mode results in the formations of kinetic Alfvén wave which would be discussed in forth coming subsections. In a magnetized plasma, we shall consider the electrons are thermal and strongly magnetized obeying an equilibrium Maxwellian distribution, while ions are hot and magnetized so that finite Larmor radius can be taken into account. For ions, we may employ Vlasov equation by utilizing guiding center approach to obtain the perturbed distribution function for an electromagnetic wave when the electric field and the wave vector

where

and

where

The dust component is considered to be cold and unmagnetized such that

and

To find the relation between

The coefficients in Eq. (15) are given by

where

where,

where

where,

### 3.2. Lorentzian distribution function

A number of processes in a space based plasma lead to the development of particle anisotropy through streaming or temperature and are responsible for plasma instabilities in collision-free plasma which are frequently kinetic in nature and their persistent features have been confirmed by many spacecraft measurements, e.g., the electron energy spectra and the near-earth environment observations have witnessed the presence of superthermal populations. It is a well-known fact that the equilibrium Maxwell-Boltzmann distributions are associated with the Boltzmann collision term, but on the large scale Fokker-Plank model is not appropriate due to strong interaction and correlation in a collisionless plasma. The kinetic foundations of generalized Lorentzian statistical mechanics has been remarkably established by [35] with the generalization of Boltzmann collision term that is not based on binary collisions. The long range correlation between particles vindicates that power law distributions posses a particular thermodynamical equilibrium state. The mathematical form on isotropic Lorentzian distribution function is given by

where

Due to the stated fact, the deviation from the Maxwellian equilibrium distribution function could also excite plasma waves by using free energy sources. Such distributions are frequently observed in solar and terrestrial environments and can be represented by anisotropy in temperature and velocity, i.e., [36]

where

In the limit

### 3.3. Lorentzian current and number density perturbations

Many space and astrophysical plasmas have been found to have generalized Lorentzian particle distribution functions. It is of some interest to observe the impact of the high energy tail on the current and number densities of plasma species. By using Eqs. (4), (7) and (20), we get the modified expressions of number and current densities based on kappa distribution function, i.e.,

and

where

### 3.4. KAW and instability in Lorentzian plasma

In a low

The longitudinal components of current density perturbation [7, 19, 37] is given by

where

By incorporating the values of

where

A visible modification can be noticed by the effect of superthermality via the kappa-modified plasma dispersion function and the appearance of dust lower hybrid frequency due to dust effects on the dispersion characteristics. Numerous standard wave modes can originate from the above dispersion equation by applying particular limits, i.e.,

(*i*)

In the limit

(*ii*)

where

where

where

where

### 3.5. Dust kinetic Alfvén waves (DKAWs)

DKAWs arise when the dispersion relation of ordinary Alfvén waves is modified by the finite Larmor radius effect of dust. This process is dominated by the collective dynamics of magnetized dust particles. We have investigated shear Alfvén waves and their coupling with dust-acoustic wave by considering magnetized dust and Lorentzian electrons and ions.

The perturbed current and number densities of cold and magnetized dust are obtained by using Eqs. (5) and (6)

The parallel component of perturbed dust current density turns out to be from Eq. (14)

In the limit

#### 3.5.1. Lorentzian-type charging currents

The charging equation containing Lorentzian electron and ion currents is

where the electron and ion currents are calculated using a surface integral through the dust grain surface of radius

and

where

The Lorentzian charging currents are derived by using Vlasov-kinetic model whose fluid version by Rubab and Murtaza [41] and in the limit

which clearly shows that charge fluctuation effects are insensitive to the form of the distribution function.

#### 3.5.2. Modified dust-acoustic wave

In the limit

where

#### 3.5.3. DKAW: Perpendicular streaming

We consider an electromagnetic dust kinetic Alfvén wave streaming instability in a collisionless electron-ion dusty magnetoplasma. The motion of DKAW is followed by considering thermal and magnetized Lorentzian electrons to be Maxwellian and Lorentzian ions drifting across the external magnetic field

where

The distribution function of Lorentzian ions where ions are streaming perpendicular to the field direction is given as,

where

The dispersion relation with the aid of Eq. (15) is obtained by using Eqs. (23), (24), (33) and (41) in Eqs. (1) and (2), i.e.,

which is the general dispersion relation of kinetic Alfvén waves in the presence of perpendicular streaming ions and cold and magnetized dust. In the above equation,

For parallel propagation and in the limit

where

where

The dispersion relation for the DKAW instability is found to be dependent on the spectral index

By using the same parameters as above, we have plotted the growth rates as the function of propagation vector for different values of kappa. We have seen that the cross-field streaming of superthermal ions inhibit the growth rate of instability as shown in Figure 7. Similarly,

## 4. Weibel instability in a Lorentzian plasma

The Weibel plasma instability has so many applications in astrophysical [43], and in laboratory plasmas as well [44]. The generation of magnetic field can be explained in the domain of gamma ray burst, galactic cosmic rays and supernovae [45, 48]. For the case of unmagnetized plasma, the Weibel instability [20] has been widely discussed in relativistic and nonrelativistic regimes. In 1989, Yoon [46, 47] generalized his work by using relativistic bi-Maxwellian plasma. Later, Schaerfer [48] have discussed this instability in relativistic regimes of plasma with arbitrary distributions and presented comparison with his previous works which was based on bi-Gaussian distribution functions. The Weibel instability was investigated by Califano [49, 50] with temperature anisotropy, produced by two counterstreaming electron populations. Davidson probed the multi species Weibel instability for the charged beam and intense ions in plasma [51].

In our work, we have derived the analytical expressions and compared the results numerically for the real and imaginary parts of the dielectric constant with the Maxwellian and kappa

By using kinetic model, the linear dispersion relation for Weibel instability in unmagnetized plasma has been derived after solving the linearized, nonrelativistic Vlasov equation as below [52],

where

To calculate Weibel instability in a Lorentzian plasma, we use Eq. (21), for zero streaming velocity of particle, i.e.,

where

Applying same procedure as above and again using Plemelj’s formula,

the integration of principal part yields

The dispersion relation will be solved under two following conditions

For

which shows the real part of Weibel instability is insensitive to the value of Lorentzian index and the imaginary part

For

The dispersion relation takes the form

Now, we define a new plasma dispersion function, i.e.,

the corresponding dispersion relation can be expressed as

We can Solve

So the three dispersion relations for the above three corresponding Z-functions are.

For

Similarly, for

and

Using the Vlasov model, we have derived new dispersion relations based on

## 5. Collisional Weibel instability with non-zero magnetic field

The dispersion relation of Weibel instability for transverse waves propagating parallel to magnetic field is obtained as

where

We notice that the final expression becomes independent of the spectral index

However, for

For

The imaginary

and

for

Considering

and

where

It is obvious from the above relation that collision frequency for particles obeying kappa distribution differs from that of Maxwellian distribution and is dependent on the value of specie of choice

## 6. Conclusion

In this chapter, we have described the electromagnetic waves and instabilities in a generalized Lorentzian plasma including particle streaming and finite and anisotropic thermal spread. It allows to grasp the practical understanding of a complex collisionless system in terms of spectra, bulk relative motion and instabilities. In particular, we have focused on kinetic Alfvén waves and instabilities in a dusty and Lorentzian plasma and several types of modes have been identified under various conditions. We have reviewed the kinetic waves and Weibel instabilities in a non-Maxwellian space and astrophysical plasmas by incorporating some basic concepts of dusty environments. We have found that dispersion characteristics involving kinetic Alfvén waves become significantly modified by superthermality effects and dust plasma parameters. The coupling of magnetized dust to the waves due to cyclotron resonance is shown to play a vital role on the wave dynamics. Moreover, the dust grain charging yield some additional plasma currents, which depends on the streaming velocity, Lorentzian index and plasma beta. The Lorentzian index is found to either enhance or quench the electromagnetic instabilities. The dust component is found to play an essential role in wave dynamics, i.e., introducing dust lower hybrid frequency when mobile dust particles are included in the plasma. We have seen that the temperature anisotropy in the distribution function has no effect on the wave characteristics, i.e., the employed model inhibits the temperature anisotropy, but supports the velocity anisotropy. Moreover, a brief analysis on Weibel instabilities in a non-Maxwellian plasma in is also presented.

Kinetic Alfvén turbulence are always present in the streaming solar wind near 1 AU and in situ measurements have confirmed the presence of non-Maxwellian proton distribution function. The present investigations show that the Lorentzian charged particle distributions in space lead to a essentially new physical situation as compared to the plasma with equilibrium distribution functions. Our results of the present analysis opens a new window of investigation to study various streaming and anisotropic modes in different plasma scenarios when Lorentzian distribution function is employed.