Revisiting some analytical and numerical interpretations of Cairns and Kappa–Cairns distribution functions

Aman-ur-Rehman, Aman-ur-Rehman; Ahmad, Mushtaq; Shahzad, Muhammad Ahsan

doi:10.1063/5.0018906

In this tutorial, three-dimensional (3D) Cairns and Kappa-Cairns distribution functions are re-examined both analytically and numerically. The difference between one-dimensional (1D) and (3D) Cairns distribution functions (CDF) has been explained by deriving reduced 1D Cairns distribution function. It is noted that expressions of 1D and reduced 1D distributions such as Maxwellian and Kappa distributions are similar to each other, while the plots of 1D and reduced 1D CDF are significantly different from each other. The effect of non-thermality parameter on the 3D CDF is also studied by plotting it as a function of magnitude of the 3D velocity. It shows that the peak of the distribution function shifts toward higher speeds with an increase in the value of non-thermality parameter. The longitudinal dielectric response function is derived by using the 3D CDF for studying kinetic behavior of electrostatic waves in an unmagnetized isotropic plasma. The dielectric function is written in terms of plasma dispersion function and is then used to derive the dispersion relations and Landau damping increments for electron plasma waves, ion acoustic waves, and dust acoustic waves in a Cairns distributed plasma. The expressions of the dispersion relation and Landau damping rate of Cairns distributed plasma change into the corresponding expressions of the Maxwellian distributed plasma when the nonthermality parameter is taken equal to zero. Mathematical manipulations have been done for 3D Kappa-Cairns distribution function (KCDF) to find the correct normalization factor. An appropriate and valid range of values of the spectral index κ is obtained by calculating second moment of the velocity by integrating KCDF over 3D velocity space. The effect of non-thermality parameter and the spectral index κ on the 3D KCDF is studied by plotting it as a function of magnitude of the 3D velocity. It is also seen that the velocities where the tails of KCDF exists are much higher compared to the velocities where the tail of Kappa distribution occurs. For both 3D CDF and KCDF, it has been observed that the difference in the distribution functions becomes negligible when the value of the non-thermality parameter becomes more than 0.5.

I. INTRODUCTION

The distribution function of plasma particles deviates from the Maxwellian distribution function when the high energy particles make collisions much less often as compared to the collisions made by slower particles. Such high energy particles have mean free path which is proportional to $v^{4}$ ⁠, where $v$ is the velocity of the particles and the distribution does not relax to the Maxwellian distribution. Velocity distribution functions that have a power law-like tail, with an excess of superthermal particles, are often observed in space and astrophysical environments e.g., the solar flares,^1–3 the solar wind,⁴ the galactic cosmic ray distribution,⁵ and plasma in a superthermal radiation field.⁶ Some nonthermal populations that have been evidenced in and around the Earth's bowshock and foreshock,^7,8 the upper Martian ionosphere,⁹ and the vicinity of the Moon.¹⁰

The presence of ion and electron populations which are not in thermodynamic equilibrium in space plasma observations led to model these effects by simplest analytical way. Often, these nonthermal velocity distributions include a ring structure. One such non-thermal distribution function was introduced by Cairns et al.¹¹ to explain the reverse-polarity structures observed in space plasma. They used one dimensional nonthermal Cairns distribution function (CDF) to study the existence of nonlinear structures like those observed by the Freja and Viking satellites. It was shown that this distribution (with a non-thermal electron population) resulted in the coexistence of both positive and negative potential solitons, which could not be prevailed with Maxwellian electrons. Although the Cairns distribution has not been employed to fit observed velocity distribution data, it still can be used as a useful model for the non-Maxwellian plasmas. This nonthermal velocity distribution shows an enhanced high energy tail, superposed on a Maxwellian-like low energy portion. The effect of nonthermal electrons and hot ions on the ion acoustic waves was studied by Mamun¹² and Tang et al.¹³ Bahache et al.¹⁴ analysed the expansion of an intense laser produced plasma into vacuum by using Cairns distribution. It, therefore, serves as a useful theoretical model for the family of non-Maxwellian or non-thermal space plasmas and has been utilized by a number of researchers, e.g., Refs. 15–19.

Recently, electrostatic waves in an isotropic unmagnetized plasma has been studied by Hadi et al.²⁰ and Hadi and Ata-ur-Rahman²¹ using the Cairns distribution function. To study the kinetic behavior of electrostatic waves in an isotropic unmagnetized plasma, one requires to use a three-dimensional (3D) distribution function or reduced one-dimensional (1D) distribution functions.^22–27 The reduced 1D distribution function is obtained by averaging our two directions in the velocity space and leaving the dependence of the distribution on only one velocity component. Now it is very interesting to note that the reduced 1D Cairns distribution function obtained from an isotropic 3D Cairns distribution function is significantly different from the 1D Cairns distribution function reported by Cairns et al..¹¹ This difference in the functional dependence of these 1D distribution functions leads to misleading results if the derivations of the dielectric function are not performed carefully.

On the other hand, another approach to model the non-Maxwellian like behavior is provided by the κ distribution function^28–32 that was for the first time introduced by Vasyliunas³² to fit phenomenologically the power law-like dependence of electron distribution functions observed in space plasma. Summers and Thorne³³ introduced a new plasma dispersion function, based on this distribution function, but they restricted κ to strictly integer values. Effects of superthermal particles on electrostatic and electromagnetic waves with Kappa distribution³⁴ and Langmuir modes with hybrid Kappa-Maxwellian velocity distribution³⁵ in magnetized space plasmas have been investigated in strongly and weakly magnetized regimes. Like Cairns distribution, the Kappa distribution is also a generalization of the Maxwellian distribution. As will be explained in more explicit way in the coming section, by taking certain limits of Cairns and Kappa distributions functions, these will reduce to a Maxwellian distribution.

In addition to this, Abid et al.³⁶ have proposed a hybrid 3D distribution named as Kappa-Cairns (also called Vasyliunas–Cairns) distribution by combining Cairns distribution and Kappa distribution functions. This distribution depends on two parameters i.e., non-thermality parameter α and the spectral index $κ$ ⁠. A careful analysis shows that 3D versions of the Cairns and Kappa-Cairns distributions have a different range of values of nonthermality and spectral index parameters for which these distributions are valid. Our analysis shows that moving from 1D Cairns distribution to 3D distribution is not straightforward, and if care is not taking, we might arrive at erroneous and wrong conclusions.

This paper is organized in the following order. In Sec. II, 3D isotropic Cairns distribution is examined critically by comparing it with other isotropic distributions such as Maxwellian and Kappa. This is followed in Sec. III by presenting a derivation of the dielectric function for electrostatic waves in a Cairns distributed plasma. Langmuir, ion acoustic, and dust acoustic waves for a Cairns distributed plasma are studied in Sec. IV. Normalization and range of validity of spectral indices for Kappa-Cairns distribution are presented in Sec. V. Summary of the paper is given in Sec. VI.

II. 3D CAIRNS DISTRIBUTION FUNCTION AND ITS CORRECT VALIDATION

As aforementioned, the handling of electrostatic waves based on kinetic theory in Cairns distributed plasma requires 3D distribution or reduced 1D distribution functions. However, reduction of a 3D distribution to a 1D distribution function is not straightforward. Here, it is very interesting to note that for all other distributions, such as Maxwellian, Kappa, and q-nonextensive distribution, the dependence of the distribution on the velocity is same for both 1D and reduced 1D distribution. Actually, the 1D distribution for these distributions is the reduced distribution derived from corresponding 3D distribution functions. For example, 1D and 3D Maxwellian distribution functions are given by

f_{1}^{M} (v_{z}) = \frac{n_{0}}{{(2 π v_{t h}^{2})}^{\frac{1}{2}}} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(1)

f_{3}^{M} (v_{x}, v_{y}, v_{z}) = \frac{n_{0}}{{(2 π v_{t h}^{2})}^{\frac{3}{2}}} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}},

(2)

where

v_{t h} = \sqrt{k_{B} T / m}

is the thermal velocity. In Eq. ,

v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

⁠. Reduced 1D distribution depending on

v_{z}

only can be obtained by integrating over the other two dimensions of the velocity space. The reduced 1D distribution function represented by

F_{1}^{M} (v_{z})

is then defined as

F_{1}^{M} (v_{z}) = \iint f_{3}^{M} (v_{x}, v_{y}, v_{z}) d v_{x} d v_{y} .

(3)

By using Eq. , one gets

F_{1}^{M} (v_{z}) = \frac{n_{0}}{{(2 π v_{t h}^{2})}^{\frac{1}{2}}} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(4)

Comparison of Eqs. and shows that 1D and reduced 1D Maxwellian distribution functions are exactly the same.

Now, as another example, take 1D Kappa distribution given by³⁷^,

f_{1}^{K} (v_{z}) = \frac{n_{0} Γ (κ + 1)}{\sqrt{π} κ^{1 / 2} θ Γ (κ + 1 / 2)} {[1 + \frac{v_{z}^{2}}{κ θ_{t h}^{2}}]}^{- κ - 1},

(5)

where in 1D case,

θ_{t h} = \sqrt{(\frac{2 κ - 1}{κ}) (\frac{k_{B} T}{m})}

and

κ > \frac{1}{2}

for non-negative temperatures and number densities. The 3D Kappa distribution function is given by³³^,

f_{3}^{K} (v_{x}, v_{y}, v_{z}) = \frac{n_{0} Γ (κ + 1)}{{(π κ)}^{3 / 2} θ^{3} Γ (κ - 1 / 2)} {[1 + \frac{v^{2}}{κ θ_{t h}^{2}}]}^{- κ - 1},

(6)

where in 3D case,

θ_{t h} = \sqrt{(\frac{2 κ - 3}{κ}) (\frac{k_{B} T}{m})}

and

κ > \frac{3}{2}

for non-negative temperatures and number densities. In this expression,

v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

⁠. The reduced 1D distribution function represented by

F_{1}^{K} (v_{z})

is given by³³

F_{1}^{K} (v_{z}) = \frac{n_{0} Γ (κ + 1)}{\sqrt{π} κ^{3 / 2} θ Γ (κ - 1 / 2)} {[1 + \frac{v_{z}^{2}}{κ θ_{t h}^{2}}]}^{- κ} .

(7)

Comparison of Eqs. and shows that the dependence on spectral index

κ

is different; however, the dependence on velocity

v_{z}

is same. Equations and can be shown to be same if the spectral index

κ

is considered to be the functions of system's dimensionality. Livadiotis and McComes have studied the connection of kappa values with various degrees of freedom.³⁷ There they have shown that

κ_{D} - \frac{D}{2} = constant

⁠, where

D

is the number of dimensions in the problem.

κ_{D} - \frac{D}{2} = constant

⁠, which implies that

κ_{1} - \frac{1}{2} = κ_{3} - \frac{3}{2}

or

κ_{1} = κ_{3} - 1

⁠. As shown in Appendix A, Eqs. and are equivalent. Similarly, it can be shown that the 1D and reduced 1D q-nonextensive distributions are equivalent.

Now if we come to the Cairns distribution function, it was the 1D version of this distribution that was initially proposed.¹¹ 1D Cairns distribution can be written as¹¹^,

f_{1}^{C} (v_{z}) = \frac{1}{\sqrt{2 π} v_{t h} (1 + 3 α)} [1 + α \frac{v_{z}^{4}}{v_{t h}^{4}}] e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}} .

(8)

However, some authors have straightforwardly extended it into three dimensions even without paying attention to its normalization factor.^20,21 Properly normalized 3D Cairns distribution function is given by³⁸

f_{3}^{C} (v_{x}, v_{y}, v_{z}) = \frac{n_{0}}{(1 + 15 α) {(2 π v_{t h}^{2})}^{\frac{3}{2}}} (1 + α \frac{v^{4}}{v_{t h}^{4}}) e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}},

(9)

where

0 \leq α \leq 1

is the nonthermality parameter in both the distribution functions. In this expression again,

v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

⁠. The reduced 1D distribution detail derivation is given in Appendix B] is given by

F_{1}^{C} (v_{z}) = \frac{1}{\sqrt{2 π} v_{ths} (1 + 15 α)} [1 + α (\frac{v_{z}^{4}}{v_{ths}^{4}} + 4 \frac{v_{z}^{2}}{v_{ths}^{2}} + 8)] e^{- \frac{v_{z}^{2}}{{2 v}_{ths}^{2}}} .

(10)

Comparison of Eqs. and shows that 1D and reduced 1D Cairns distribution functions are different from each other. The plots of Eqs. and show that there is a significant and substantial difference between these two kinds of distribution functions as shown in Fig. 1. Panel (a) of Fig. 1 shows the plots of 1D Cairns distribution function, while panel (b) shows the plots of reduced 1D Cairns distribution function for various values of the non-thermality parameter α. This comparison shows that these two distributions are significantly different from each other. The plots of 1D Cairns distribution evolve from single-humped to multi-humped as one increases the value of α, while the plots of reduced 1D Cairns distribution remain single-humped for all the values of α. The plots of reduced 1D Cairns distribution function show that with an increase in the value of α, the peak of the distribution decreases the and width of the plateau region increases. The high energy tail is also increased with an increase in the value of α. One important observation is that when the value of α is increased beyond 0.5, there is no significant change in the 1D Cairns distribution for

0.5 < α < 1

⁠.

FIG. 1.

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One-dimensional Cairns (panel a) and reduced One-dimensional Cairns (panel b) for various values of nonthermality parameter, i.e., $α = 0.0$ (black), $α = 0.1$ (magenta), $α = 0.2$ (green), $α = 0.3$ (purple), $α = 0.4$ (yellow), $α = 0.5$ (cyan), $α = 07$ (blue), and $α = 0.9$ (red).

Another inconsistency that is found in the literature is the plot of 3D Cairns distribution function. The plots of 3D distribution function shown in some published literature are similar to 1D.^11,12,28 In order to plot 3D distributions, one needs to reduce the number of dimensions by averaging over the directions that are to be eliminated. This procedure gives us the reduced distribution which is shown in panel (b) of Fig. 1. However, for isotropic distributions, one can define another function

g (v)

which is a function of the magnitude of the 3D velocity such that^26,27

\int_{0}^{\infty} g (v) d v = ∭_{- \infty}^{\infty} f (v) d^{3} v .

(11)

By using the procedure described in Refs. 26 and 27 for Cairns distribution, one can write

g^{C} (v)

as

g^{C} (v) = \sqrt{\frac{2}{π}} \frac{n_{0} v^{2}}{(1 + 15 α) v_{t h}^{3}} (1 + α \frac{v^{4}}{v_{t h}^{4}}) e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} .

(12)

Figure 2 shows the plot of

g^{C} (v)

for various values of non-thermality parameter α. This figure shows that peak of the speed dependent distribution function shifts toward higher values of the normalized velocity with an increase in the value of α. The peak exists at

{(\frac{v}{v_{t h}})}_{max} = \sqrt{2 - \frac{0.6934}{s (α)} + \frac{8.3203 α}{s (α)} + \frac{0.48075 s (α)}{α}},

(13)

where

s (α) = {(72 α^{3} + 1.73205 \sqrt{α^{3} - 36 α^{4} + 432 α^{5}})}^{1 / 3}

⁠. Figure 2 also shows that there is no significant change in the plots of 3D Cairns distribution for

0.5 < α < 1

⁠. This behavior shows that the effect of non-thermality parameter almost saturates when its value goes beyond 0.5. This can be further confirmed by plotting Eq. as a function of non-thermality parameter in Fig. 3, which shows that the value of

{(\frac{v}{v_{t h}})}_{max}

saturates when the value of

α

crosses 0.5.

FIG. 2.

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Three-dimensional Cairns distribution function as a function of the magnitude of three-dimensional velocity, for various values of nonthermality parameter, i.e., $α = 0.0$ (black), $α = 0.1$ (magenta), $α = 0.2$ (green), $α = 0.3$ (purple), $α = 0.4$ (yellow), $α = 0.5$ (brown), $α = 0.7$ (blue), and $α = 0.9$ (red).

FIG. 3.

Plot of v v t h max as a function of non-thermality parameter for Cairns distribution function.

View large Download slide

Plot of ${(\frac{v}{v_{t h}})}_{max}$ as a function of non-thermality parameter for Cairns distribution function.

III. DERIVATION OF DIELECTRIC FUNCTION

The dielectric response function in the case of an isotropic plasma for electrostatic waves is obtained from coupled system of Poisson and Vlasov equations. For electrostatic waves propagating parallel to the

z - axis

⁠, the dispersion relation can be written as

ϵ (k, ω) = 1 - \sum_{s} \frac{q_{s}^{2}}{{ε_{0} m_{s} k}^{2}} ∭ \frac{\partial f_{0 s} / \partial v_{z}}{(v_{z} - ω / k)} d^{3} v .

(14)

This expression can be written in terms of reduced distribution function by performing integration over

v_{x} - axis and v_{y} - axis

in the velocity space

ϵ (k, ω) = 1 - \sum_{s} \frac{ω_{p s}^{2}}{k^{2}} \int_{- \infty}^{\infty} \frac{\frac{\partial F_{0 s} (v_{z})}{\partial v_{z}}}{(v_{z} - \frac{ω}{k})} d v_{z},

(15)

where

ω_{p s}^{2} = \frac{n_{0 s} q_{s}^{2}}{ε_{0} m_{s}}

is the square of the plasma frequency and

F_{0 s} (v_{z})

is the reduced one dimensional distribution function and is given by Eq. . Derivative of

F_{0 s} (v_{z})

with respect to

v_{z}

can be written as

\frac{\partial F_{0 s} (v_{z})}{\partial v_{z}} = - \frac{1}{\sqrt{2 π} v_{ths}^{3} (1 + 15 α)} [α \frac{v_{z}^{5}}{v_{ths}^{4}} + v_{z}] e^{- \frac{v_{z}^{2}}{{2 v}_{ths}^{2}}} .

(16)

The dielectric function for the electrostatic waves in an unmagnetized plasma can be written by using the value of derivative of reduced distribution function in Eq. as

ϵ (k, ω) = 1 + \sum_{s} \frac{ω_{p s}^{2}}{(1 + 15 α) k^{2} v_{ths}^{2}} [1 + α (3 + 2 ζ_{s}^{2} + 4 ζ_{s}^{4}) + (ζ_{s} + 4 α ζ_{s}^{5}) {Z (ζ}_{s})] = 0,

(17)

where

ζ_{s} = \frac{ω}{\sqrt{2} k v_{ths}},

with

v_{ths} = \sqrt{T_{s} / m_{s}}

which is the thermal velocity and

{Z (ζ}_{s}) = 1 / \sqrt{π} \int_{- \infty}^{\infty} \frac{e^{- x^{2}}}{x - ζ_{s}} d x

is the plasma dispersion function.^39,40

IV. ANALYTICAL EXPRESSIONS OF DISPERSION RELATION AND DAMPING RATE

A. Electron plasma waves in cairns distributed plasma

Electron plasma (Langmuir) waves can be studied in plasma composed of warm electrons and immobile ions using the following dielectric function:

ϵ (k, ω) = 1 + \frac{ω_{p e}^{2}}{(1 + 15 α) k^{2} v_{the}^{2}} [1 + α (3 + 2 ζ_{e}^{2} + 4 ζ_{e}^{4}) + (ζ_{e} + 4 α ζ_{e}^{5}) {Z (ζ}_{e})] = 0.

(18)

Under the condition

ω / k > v_{the}

⁠, the argument of the plasma dispersion function satisfies the condition

ζ_{e} ≫ 1

⁠, and the plasma dispersion function can be written as^39,40

{Z (ζ}_{e}) = i \sqrt{π} e^{- ζ_{e}^{2}} - \frac{1}{ζ_{s}} - \frac{1}{2 ζ_{e}^{3}} - \frac{3}{4 ζ_{e}^{5}} - \frac{15}{8 ζ_{e}^{7}} - \frac{105}{16 ζ_{e}^{9}} - \dots .

(19)

Using this large argument expansion in Eq. , real and imaginary parts of the dielectric function can be written as

ϵ_{r} (k, ω) = 1 - \frac{1}{k^{2} λ_{D e}^{2}} [\frac{1}{2 ζ_{e}^{2}} + \frac{3}{4 ζ_{e}^{4}} \frac{(1 + 35 α)}{(1 + 15 α)}],

(20)

ϵ_{i} (k, ω) = \frac{\sqrt{π}}{(1 + 15 α) k^{2} λ_{D e}^{2}} {(ζ_{e} + 4 α ζ_{e}^{5}) e}^{- ζ_{e}^{2}},

(21)

where

ζ_{e} = \frac{ω}{\sqrt{2} k v_{the}}

⁠. Now by taking

ϵ_{r} (k, ω) = 0

⁠, and using the successive substitution method, the real part of the frequency can be written as

ω_{r}^{2} = ω_{p e}^{2} + 3 (\frac{1 + 35 α}{1 + 15 α}) k^{2} v_{the}^{2} .

(22)

The imaginary part of the frequency can be obtained by using

γ = - \frac{ϵ_{i} (k, ω_{r})}{\partial ϵ_{r} (k, ω_{r}) / \partial ω_{r}}

and can be written as

γ = - \sqrt{\frac{π}{8}} \frac{ω_{p e}}{k^{3} λ_{D e}^{3} (1 + 15 α)} [1 + α {\{\frac{1}{k^{2} λ_{D e}^{2}} + 3 (\frac{1 + 35 α}{1 + 15 α})\}}^{2}] e^{- \frac{1}{{2 k}^{2} λ_{D e}^{2}} - \frac{3}{2} (\frac{1 + 35 α}{1 + 15 α})} .

(23)

In the limit when the nonthermality parameter

α = 0

⁠, Eqs. and become, respectively,

ω_{r}^{2} = ω_{p e}^{2} + 3 k^{2} v_{the}^{2},

(24)

γ = - \sqrt{\frac{π}{8}} \frac{ω_{p e}}{k^{3} λ_{D e}^{3}} e^{- \frac{1}{{2 k}^{2} λ_{D e}^{2}} - \frac{3}{2}} .

(25)

These are the same expressions for the frequency and damping rates that are obtained when the Maxwellian distribution function is used to study Landau damping of electron plasma waves.^23–25 Figure 4 shows the dispersion curve (panel a) and damping rate (panel b) of Langmuir waves (LWs) in Cairns distributed plasma. This figure shows that the slop of the dispersion curve increases with an increase in the values of the non-thermality parameter. The damping rate of the Maxwellian plasma is less as compared to the damping rate of the corresponding nonthermal plasma. However, an increase in value of the non-thermality parameter

α

results in a trivial decrease in the damping rate of the LWs. This pattern of the damping rate can be explained in terms of plots of the reduced distribution function shown in panel (b) of Fig. 1. In the weak growth approximation and with applications of Plemelj formulation in Eq. , one gets²⁵

γ = π \frac{k}{|k|} \frac{ω_{p}^{2}}{k^{2} \partial ϵ_{r} / \partial ω_{r}} {\frac{\partial F_{0} (v_{z})}{\partial v_{z}} |}_{v_{z} = \frac{ω_{r}}{k}} .

(26)

FIG. 4.

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Normalized frequency as a function of the normalized wave number (panel a) and normalized Landau damping rate as a function of the normalized wave number (panel b) of the Langmuir waves in a nonthermal plasma for various values of nonthermality parameter, i.e., $α = 0.0$ (black), $α = 0.1$ (blue), $α = 0.2$ (magenta), $α = 0.3$ (cyan), $α = 0.5$ (purple), and $α = 0.8$ (red).

This expression shows that the damping rate is proportional to the slope of the 1D reduced distribution function evaluated at the velocities which is equal to the phase velocity of the wave.²⁵ Owing to their high frequency, LWs have high phase velocity. As shown in Fig. 1(b), at high velocities, the slope of the Maxwellian distribution is less than the slope of the corresponding nonthermal distribution functions. This results in lower damping rate of LWs for a Maxwellian plasma as compared to the nonthermal plasma. Figure 1(b) also shows that the slope of the nonthermal distribution function decreases slightly with an increase in the value of the nonthermality parameter. This slight decrease in the slope of the reduced distribution function results in a slight decrease in the damping rate of the LWs when the value of the nonthermality parameter is increased. The damping rates of the electrostatic waves are plotted for a shorter range of the wavenumber $k$ ⁠. This is because while deriving analytical results one assumes that $ω / k > v_{ths}$ ⁠. For large values of the wavenumber $k$ ⁠, this approximation breaks down and the damping rates obtained from analytical expressions are not valid. The relevance of present studies in non-Maxwellian plasma may be useful for those LWs which are excited by different ways such as by the streaming of energetic particles, for example, in solar flares that can affect the distribution function leading to a spectrum of LWs in solar wind, which is a non-Maxwellian plasma.^41,42

B. Ion acoustic waves in cairns distributed plasma

Ion acoustic waves (IAWs) are low frequency waves, for which ion motion cannot be neglected. In this case, the dielectric response function can be written as

ϵ (k, ω) = 1 + \sum_{s = e, i} \frac{ω_{p s}^{2}}{(1 + 15 α) k^{2} v_{ths}^{2}} [1 + α (3 + 2 ζ_{s}^{2} + 4 ζ_{s}^{4}) + (ζ_{s} + 4 α ζ_{s}^{5}) {Z (ζ}_{s})] = 0.

(27)

In the case of IAWs, the velocity of the wave is such that

v_{thi} < \frac{ω}{k} < v_{the}

⁠. This implies that

ζ_{e} ≪ 1

⁠, and

ζ_{i} ≫ 1

⁠, and the plasma dispersion function for electrons and ions can be expanded as^{23–27,39,40,43,44}

{Z (ζ}_{i}) = i \sqrt{π} e^{- ζ_{i}^{2}} - \frac{1}{ζ_{i}} - \frac{1}{2 ζ_{i}^{3}} - \frac{3}{4 ζ_{i}^{5}} - \frac{15}{8 ζ_{i}^{7}} - \frac{105}{16 ζ_{i}^{9}} - \dots,

(28)

{Z (ζ}_{e}) = i \sqrt{π} e^{- ζ_{e}^{2}} - 2 ζ_{s} [1 - \frac{2 ζ_{e}^{2}}{3} + \frac{4 ζ_{e}^{4}}{15} - \frac{8 ζ_{e}^{6}}{105} + - \dots] .

(29)

Using Eqs. and in Eq. , the real and imaginary parts of the of the dielectric function can be written as

ϵ_{r} (k, ω) = 1 + \frac{(1 + 3 α)}{(1 + 15 α) k^{2} λ_{D e}^{2}} - \frac{1}{k^{2} λ_{D i}^{2}} [\frac{1}{2 ζ_{i}^{2}} + \frac{3}{4 ζ_{i}^{4}} \frac{(1 + 35 α)}{(1 + 15 α)}],

(30)

ϵ_{i} (k, ω) = \frac{\sqrt{π}}{(1 + 15 α) k^{2} λ_{D e}^{2}} ζ_{e} + {\frac{\sqrt{π}}{(1 + 15 α) k^{2} λ_{D i}^{2}} (ζ_{i} + 4 α ζ_{i}^{5}) e}^{- ζ_{i}^{2}} .

(31)

By taking

ϵ_{r} (k, ω) = 0

⁠, and using the successive substitution method, the real part of the frequency can be written as

ω_{r}^{2} = k^{2} c_{s}^{2} [\frac{1}{{(\frac{1 + 3 α}{1 + 15 α}) + k}^{2} λ_{D e}^{2}} + 3 (\frac{1 + 35 α}{1 + 15 α}) \frac{T_{i}}{T_{e}}],

(32)

where

c_{s} = \sqrt{T_{e} / m_{i}}

is the ion acoustic speed. Using again

γ = - \frac{ϵ_{i} (k, ω_{r})}{\partial ϵ_{r} (k, ω_{r}) / \partial ω_{r}}

the imaginary part of the frequency can be written as

\frac{γ}{ω_{r}} = - \sqrt{\frac{π}{8}} \frac{1}{α_{0}} {(\frac{1}{{α_{1} + k}^{2} λ_{D e}^{2}})}^{\frac{3}{2}} [\sqrt{\frac{m_{e}}{m_{i}}} + {(\frac{T_{e}}{T_{i}})}^{\frac{3}{2}} \{1 + 4 α {(\frac{T_{e} / T_{i}}{2 ({α_{1} + k}^{2} λ_{D e}^{2})} + \frac{3}{2} α_{2})}^{2}\} e^{- \frac{T_{e} / T_{i}}{2 ({α_{1} + k}^{2} λ_{D e}^{2})} - \frac{3}{2} α_{2}}],

(33)

where

α_{0} = 1 + 15 α, α_{1} = \frac{1 + 3 α}{1 + 15 α}, and α_{2} = \frac{1 + 35 α}{1 + 15 α}

⁠. In the limit when the nonthermality parameter

α = 0

⁠,

α_{0} = α_{1} = α_{2} = 1

⁠, and expressions for real and imaginary parts of the frequency become what we get in the case of Maxwellian distribution and are given by

ω_{r}^{2} = k^{2} c_{s}^{2} [\frac{1}{{1 + k}^{2} λ_{D e}^{2}} + 3 \frac{T_{i}}{T_{e}}],

(34)

\frac{γ}{ω_{r}} = - \sqrt{\frac{π}{8}} {(\frac{1}{1 + k^{2} λ_{D e}^{2}})}^{\frac{3}{2}} [\sqrt{\frac{m_{e}}{m_{i}}} + {(\frac{T_{e}}{T_{i}})}^{\frac{3}{2}} e^{- \frac{T_{e} / T_{i}}{2 ({1 + k}^{2} λ_{D e}^{2})} - \frac{3}{2}}] .

(35)

These are the same expressions for the frequency and damping rate that are obtained when the Maxwellian distribution function is used to study Landau damping of IAWs.^25,43,44 Figure 5 shows the effect of nonthermality parameter on the real and imaginary parts of the ion acoustic wave (IAWs) for electron to ion temperature ratio $\frac{T_{e}}{T_{i}} = 10$ ⁠. Figure 5(a) shows that the phase velocity of the ion acoustic wave increases with an increase in the value of the nonthermality parameter. This behavior of the real frequency of the ion acoustic wave is similar to the real frequency of the Langmuir wave. Figure 5(b) shows the effect of nonthermality parameter on the damping rate of the ion acoustic waves. It shows that damping rate of the IAWs for a Maxwellian plasma is more as compare to damping rate of the corresponding nonthermal plasma. This behavior of the damping rate of IAWs is opposite to that of LWs. This behavior of the damping rate can be understood again by looking at Fig. 1(b). IAWs are low frequency waves resulting in a small value of the phase velocity $(\frac{ω_{r}}{k})$ of the IAWs. Figure 1(b) shows that for smaller values of the particle velocities, the slope of Maxwellian reduced distribution is large as compared to the slopes of the corresponding nonthermal reduced distribution functions. Again according to Eq. (26), the damping rate is proportional to the slope of the reduced distribution function. It means that for IAWs, the damping rate of the Maxwellian plasma is strong as compared to the damping rate of the corresponding nonthermal plasma. In the case of nonthermal plasma, in the resonance region, the slope of the reduced distribution function decreases with an increase in value of the nonthermality parameter. This decrease in the slope of the reduced distribution results in a decrease in the damping rate of the IAWs with an increase in the value of the nonthermality parameter. The dependence of Landau damping of IAWs on electrons or ions can be further understood by inspecting the analytically obtained Landau damping rate given by Eq. (33). This equation shows that Landau damping of IAWs is comprised of two parts. The first part which is proportional to $(\frac{1}{α_{0}} \sqrt{\frac{m_{e}}{m_{i}}})$ is called electron Landau damping term. The second part which is proportional to $(\frac{1}{α_{0}} {(\frac{T_{e}}{T_{i}})}^{\frac{3}{2}} \{1 + 4 α {(\frac{\frac{T_{e}}{T_{i}}}{2 ({α_{1} + k}^{2} λ_{De}^{2})} + \frac{3}{2} α_{2})}^{2}\} e^{- \frac{\frac{T_{e}}{T_{i}}}{2 ({α_{1} + k}^{2} λ_{De}^{2})} - \frac{3}{2} α_{2}})$ is called the ion Landau damping terms. Electrons and ions both contribute toward Landau damping when their temperatures are comparable. However, when $T_{e} ≫ T_{i}$ ⁠, the ion landau damping term vanishes and there is always a small Landau damping present due to electrons. The present IAW studies are applicable to comprehend the dispersion and damping features of these sorts of waves in space plasma systems where some nonthermality of the particles has been observed by the satellite missions of Freja and Viking, specifically in the upper part of the ionosphere and in the auroral regions.^45,46

FIG. 5.

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Normalized frequency as a function of the normalized wave number (panel a) and normalized Landau damping rate as a function of the normalized wave number (panel b) of the ion acoustic waves in a nonthermal plasma for various values of nonthermality parameter, i.e., $α = 0.0$ (black), $α = 0.1$ (blue), $α = 0.2$ (magenta), $α = 0.3$ (cyan), $α = 0.5$ (purple), and $α = 0.8$ (red). The values of other parameters are $\frac{T_{e}}{T_{i}} = 10$ and $\frac{m_{i}}{m_{e}} = 1836$ ⁠.

C. Dust acoustic waves in cairns distributed plasma

The dielectric function for the electrostatic waves in an unmagnetized Cairns distributed dusty plasma consisting of electrons, singly charged positive ions, and negatively charged dust particles can be written as

ϵ (k, ω) = 1 + \sum_{j = e, i, d} \frac{ω_{p j}^{2}}{(1 + 15 α) k^{2} v_{t j}^{2}} [1 + α (3 + 2 ζ_{j}^{2} + 4 ζ_{j}^{4}) + (ζ_{j} + 4 α ζ_{j}^{5}) {Z (ζ}_{j})] = 0.

(36)

If dust is considered to be Maxwellian, while electrons and ions are non- Maxwellian following the Cairns distribution, then the dielectric function then can be written as

ϵ (k, ω) = 1 + \sum_{j = e, i} \frac{ω_{p j}^{2}}{(1 + 15 α) k^{2} v_{t j}^{2}} [1 + α (3 + 2 ζ_{j}^{2} + 4 ζ_{j}^{4}) + (ζ_{j} + 4 α ζ_{j}^{5}) {Z (ζ}_{j})] + \frac{ω_{p d}^{2}}{k^{2} v_{t d}^{2}} [1 + ζ_{d} {Z (ζ}_{d})] = 0.

(37)

On the time scale of dust acoustic waves, the phase velocity of the wave is such that

v_{t d} ≪ {\frac{ω}{k} ≪ v}_{thi} ≪ v_{the}

⁠, which implies that

ζ_{e} ≪ 1

⁠,

ζ_{i} ≪ 1

⁠, and

ζ_{d} ≫ 1

⁠. This allows us to use asymptotic expansions of the plasma dispersion function for electrons, ions, and dust particle in Eq. . The real and imaginary parts of the dielectric function can be written as

ϵ_{r} (k, ω) = 1 + \frac{(1 + 3 α)}{(1 + 15 α) k^{2} λ_{D e}^{2}} + \frac{(1 + 3 α)}{(1 + 15 α) k^{2} λ_{D i}^{2}} - \frac{1}{k^{2} λ_{D d}^{2}} [\frac{1}{2 ζ_{d}^{2}} + \frac{3}{4 ζ_{d}^{4}}],

(38)

ϵ_{i} (k, ω) = {\frac{\sqrt{π}}{α_{0} k^{2} λ_{D e}^{2}} (ζ_{e} + 4 α ζ_{e}^{5}) e}^{- ζ_{e}^{2}} + {\frac{\sqrt{π}}{α_{0} k^{2} λ_{D i}^{2}} (ζ_{i} + 4 α ζ_{i}^{5}) e}^{- ζ_{i}^{2}} + {\frac{\sqrt{π}}{k^{2} λ_{D d}^{2}} ζ_{d} e}^{- ζ_{d}^{2}},

(39)

where

α_{0} = (1 + 15 α)

⁠. By taking

ϵ_{r} (k, ω) = 0

⁠, the real part of the frequency can be written as

ω_{r}^{2} = \frac{k^{2} C_{D A}^{2}}{α_{2} + k^{2} λ_{D}^{2}} + 3 k^{2} v_{t d}^{2},

(40)

where

λ_{D} = \frac{λ_{D e} λ_{D i}}{\sqrt{λ_{D e}^{2} + λ_{D i}^{2}}}

is the effective Debye length for a mixture of electron-ion plasma,

α_{2} = \frac{1 + 3 α}{1 + 15 α}

and

C_{D A} = ω_{p d} λ_{D}

is the dust acoustic speed. It is noted that the dispersion relation (40) is independent of polarization force effect as we consider locally uniform plasma background and also

a ≪ λ_{D}

by keeping the dimensionless charge

z = |Q| e / a T_{e} \sim 1

⁠. For such a steady state condition, the polarization force plays a negligible role and can be neglected;^43,44 here a is the dust size and Q is the charge of the dust grain. However, for locally non-uniform background of plasma where the ion temperature is coupled with neutral gas, these effects for low frequency complex wave modes (particularly for DAWS) are significant.^47–49

We are interested in dust perturbations in Cairns distributed plasma. In the present context of a complex plasma the dust grain charging processes, which are faster (typically of the order of 10⁻⁸s) than other characteristic processes (tens of milliseconds for micro sized dust grain) of the DA wave formation and propagation. Therefore, even on this kinetic time scale, the dust charge quickly reaches local equilibrium, which means that the dust grain charge will be fixed in each point of the DA perturbation and is determined there by the plasma parameters at this point. It follows that $ω_{p d} / ν_{c h} ≃ 0$ (where $ν_{c h}$ is the dust charging frequency), and in such case the dust charge instantaneously reaches its equilibrium value at each space-time point determined by the local electrostatic potential $ϕ (r, t)$ ⁠. Hence, it does not give rise to any dissipative effect, and this is known as adiabatic dust charge variation.^43,50

By using

γ = - \frac{ϵ_{i} (k, ω_{r})}{\partial ϵ_{r} (k, ω_{r}) / \partial ω_{r}}

and considering that

v_{t d} ≪ v_{ϕ}

⁠, the imaginary part of the frequency can be obtained as

γ = - \sqrt{\frac{π}{8}} \frac{{k C}_{D A}}{α_{0} {(α_{2} + k^{2} λ_{D}^{2})}^{2}} \frac{λ_{D}^{2}}{λ_{D i}^{2}} [\frac{λ_{D i}^{2}}{λ_{D e}^{2}} \frac{C_{D A}}{v_{t e}} \{1 + α \frac{v_{ϕ}^{4}}{v_{t e}^{4}}\} e^{- \frac{v_{ϕ}^{2}}{{2 v}_{t e}^{2}}} + \frac{C_{D A}}{v_{t i}} \{1 + α \frac{v_{ϕ}^{4}}{v_{t i}^{4}}\} e^{- \frac{v_{ϕ}^{2}}{{2 v}_{t i}^{2}}}] .

(41)

Here

α_{0} = 1 + 15 α, α_{2} = \frac{1 + 3 α}{1 + 15 α}

⁠, and

v_{ϕ} = \frac{ω_{r}}{k} = \sqrt{\frac{C_{D A}^{2}}{α_{2} + k^{2} λ_{D}^{2}} + 3 v_{t d}^{2}}

is the phase speed of the dust acoustic wave. In the limit when

α = 0

(with

α_{0} = α_{2} = 1

⁠), we get the same expressions for real and imaginary parts of dust acoustic wave as has been obtained by using the Maxwellian distribution function in Ref. 41, i.e.,

ω_{r} = \frac{k C_{D A}}{\sqrt{1 + k^{2} λ_{D}^{2}}},

(42)

γ = - \sqrt{\frac{π}{8}} \frac{{k C}_{D A}}{{(1 + k^{2} λ_{D}^{2})}^{2}} \frac{λ_{D}^{2}}{λ_{D i}^{2}} [\frac{λ_{D i}^{2}}{λ_{D e}^{2}} \frac{C_{D A}}{v_{t e}} e^{- \frac{v_{ϕ}^{2}}{{2 v}_{t e}^{2}}} + \frac{C_{D A}}{v_{t i}} e^{- \frac{v_{ϕ}^{2}}{{2 v}_{t i}^{2}}}] .

(43)

These are the same expressions which are given by Eq. (4.4.43) in Ref. 43 for real and imaginary parts of the dust acoustic waves in an unmagnetized Maxwellian plasma. Figure 6 shows the effect of nonthermality parameter on the dispersion curve and the damping rate of the dust acoustic waves (DAWs). For small values of the normalized wavenumber i.e., when

k λ_{D} ≪ 1

⁠, phase velocity of the DAWs can be written from Eq. as

v_{ϕ} = \frac{ω_{r}}{k} = (\sqrt{\frac{1 + 15 α}{1 + 3 α}}) ω_{pd} λ_{D} .

FIG. 6.

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Normalized frequency as a function of the normalized wave number (panel a) and normalized Landau damping rate as a function of the normalized wave number (panel b) of the dust acoustic waves in a nonthermal plasma for various values of nonthermality parameter, i.e., $α = 0.0$ (black), $α = 0.1$ (blue), $α = 0.2$ (magenta), $α = 0.3$ (cyan), $α = 0.5$ (purple), and $α = 0.8$ (red). The values of other parameters are $\frac{T_{e}}{T_{i}} = 10$ ⁠, $\frac{T_{d}}{T_{i}} = 0.1, \frac{n_{i o}}{n_{e 0}} = 10^{2}, \frac{n_{i o}}{n_{d 0}} = 10^{5}, \frac{m_{d}}{m_{i}} = 10^{5}$ ⁠, and $\frac{m_{i}}{m_{e}} = 1836$ ⁠.

This expression shows that the phase velocity of the DA wave increases with an increase in the value of the non-thermality parameter $α$ ⁠. Equation (40) also shows that when $k λ_{D} ≫ 1$ ⁠, the dispersion relation becomes independent of the wavenumber and gives $v_{ϕ} \approx 0$ and $ω_{r} = ω_{pd}$ ⁠. Both these two behaviors can be seen in panel (a) of Fig. 6. Panel (b) of Fig. 6 shows the dependence of normalized damping rate of DAWs on the nonthermality parameter $α$ ⁠. It shows that damping of DAWs increases with an increase in the value of $α$ ⁠. As we have just seen that phase velocity of the wave increases with $α$ ⁠. The slope of the reduced CDF is more for higher velocities as compare to lower velocities. The resonant particles will be those for which slope of the reduced 1D CDF is high. This will lead to a higher damping rate according to Eq. (26). These analytical and numerical results may be useful for understanding the propagation of DAWs in low-temperature laboratory plasmas where particles of plasma are far away from thermal equilibrium.⁵¹

V. KAPPA-CAIRNS DISTRIBUTION FUNCTION

Now consider 3D Vasyliunas–Cairns distribution function

f_{j}^{V C} (v_{x}, v_{y}, v_{z}) = A_{κ, α} (1 + α \frac{v^{4}}{v_{t j}^{4}}) {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1},

(44)

where

A_{κ, α}

is the normalization factor that can be determined by integrating Eq. over three-dimensional velocity space. In this equation,

\vec{v} = v_{x} \hat{i} + v_{y} \hat{j} + v_{z} \hat{k}

is the three dimensional velocity,

v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

and

v^{4} = {v_{x}^{4} + v_{y}^{4} + v_{z}^{4} + 2 v}_{x}^{2} v_{y}^{2} + {2 v_{x}^{2} v}_{z}^{2} + {2 v}_{y}^{2} v_{z}^{2}

⁠. The integration of Eq. over three-dimensional velocity space gives the total number of particles,

∭_{- \infty}^{\infty} f_{j}^{V C} (v_{x}, v_{y}, v_{z}) {d v}_{x} d v_{y} d v_{z} = n_{0 j} .

(45)

When integration over 3D velocity space is performed ( Appendix C), the normalization constant

A_{κ, α}

comes out be

A_{κ, α} = \frac{n_{0 j} Γ (κ + 1) / Γ (κ - 1 / 2)}{{(2 π)}^{\frac{3}{2}} {[1 + 15 α (\frac{κ - 3 / 2}{κ - 5 / 2})] (κ - 3 / 2)}^{\frac{3}{2}} v_{t j}^{3}} .

(46)

This expression shows that the factor

[1 + 15 α (\frac{κ - 3 / 2}{κ - 5 / 2})]

which depends on the spectral index

κ

is absent in the expression of Vasyliunas–Cairns distribution proposed by Abid et al. in Ref. 36. The presence of spectral index dependent factor

[1 + 15 α (\frac{κ - 3 / 2}{κ - 5 / 2})]

in the normalization factor makes this distribution different from the distribution given by Abid et al.³⁶

As pointed out by Lazar et al.,^52,53 there are two interpretations of temperature in the case of Kappa distribution. In first, the temperatures of the Kappa distribution and associated Maxwellian distribution are identical and thermal velocities are Kappa (spectral index) dependent. This approach has been advocated by Livadiotis.⁵⁴ In the second approach, thermal velocities of the Kappa distribution and associated Maxwellian distribution are identical and temperatures are Kappa (spectral index) dependent. This approach corresponds to the original definition by Olbert.⁵⁵ As the authors of Ref. 36 have used spectral index independent thermal velocity which means that thermal velocities of the Vasyliunas–Cairns and corresponding Maxwellian distribution must be the same. In means that the nonthermal temperature should be a function of nonthermality parameter

α

and spectral index

κ

⁠. The temperature depending on

α a nd κ

can be obtained by calculating the average energy of the particles. The average energy of the particles is given by

E_{avg} = \frac{1}{2} m ⟨v^{2}⟩ .

(47)

We also know that average energy of the particles has the following relationship with the effective temperature of the plasma species:

E_{avg} = \frac{3}{2} k_{B} T_{j}^{α, κ} .

(48)

Comparison of Eqs. and gives us an expression for an effective temperature that depends on spectral parameters on

α a nd κ

as

\frac{3}{2} k_{B} T_{j}^{α, κ} = \frac{1}{2} m ⟨v^{2}⟩,

(49)

where

k_{B}

is the Boltzmann constant and

T_{j}^{α, κ}

is the temperature of the jth species in Kelvin units

⟨v^{2}⟩ = \frac{1}{n_{0 j}} ∭_{- \infty}^{\infty} {v^{2} f}_{j}^{V C} (v_{x}, v_{y}, v_{z}) {d v}_{x} d v_{y} d v_{z} .

(50)

Using the value of Kappa-Cairns distribution in the above integral and performing all the necessary integrations ( Appendix D),

⟨v^{2}⟩

is given by Eq. and can be written as

⟨v^{2}⟩ = v_{t j}^{2} \frac{[3 (κ - 5 / 2) (κ - 7 / 2) + 105 α {(κ - 3 / 2)}^{2}]}{(κ - 7 / 2) [(κ - 5 / 2) + 15 α (κ - 3 / 2)]} .

Putting the value of

⟨v^{2}⟩

in Eq. gives

T_{j}^{α, κ} = \frac{m}{k_{B}} v_{t j}^{2} \frac{[(κ - 5 / 2) (κ - 7 / 2) + 35 α {(κ - 3 / 2)}^{2}]}{(κ - 7 / 2) [(κ - 5 / 2) + 15 α (κ - 3 / 2)]} .

(51)

Owing to the use of Kelvin as temperature unit

v_{t j} = \sqrt{\frac{k_{B} T_{j}^{M}}{m_{j}}}

with

T_{j}^{M}

being the temperature of the corresponding Maxwellian distribution in Eq. .This expression for temperature shows that the temperature

{(T_{j})}_{α, κ}

⁠; hence, the average energy of the particles becomes negative for values of the spectral index

κ < \frac{7}{2}

⁠. This shows that the range of validity of spectral index

κ

for Kappa–Cairns distribution is such that its value must be greater than

\frac{7}{2}

⁠.

Figure 7 shows the effect of non-thermality parameter $α$ and the spectral index $κ$ on 3D Kappa–Cairns distribution. Figure 7(a) depicts the effect of spectral index $κ$ when the value of $α = 0$ ⁠. This part of the figure shows that there is very little change in the behavior of the distribution with a change in value of the spectral index $κ$ ⁠. However, Figs. (7b)–(7f) show that when the value of $α$ is non-zero, the spectral index $κ$ has significant effect on the Kappa–Cairns distribution. This is because when $α = 0$ ⁠, the Kappa–Cairns distribution becomes the usual Kappa distribution as of Eq. (6). Figures (7b)–(7f) also show that when the value of the non-thermality parameter becomes non-zero, the distribution changes significantly with a change in the value of the spectral index $κ$ ⁠. It means that when both $α$ and $κ$ are non-zero, the peak of the distribution shifts to the higher values of $\frac{v}{θ_{t h}}$ ⁠, where $θ_{t h} = \sqrt{(\frac{2 κ - 3}{κ}) (\frac{k_{B} T}{m})}$ ⁠. It is important to note that in the absence of non-thermality parameter (i.e., when $α = 0$ ⁠), the behavior of the distribution function becomes nearly similar for all the values of spectral index $3.6 \leq α \leq 50$ ⁠. However, in the presence of the non-thermality parameter (i.e., when $α$ is non-zero), the behavior of the distribution function becomes different for different values of the spectral index when it is varied in the range $3.6 \leq κ \leq 50$ ⁠. The behavior of the Kappa–Cairns distribution for $κ = 3.6$ ⁠, with $α$ non-zero is similar to the behavior of Kappa distribution for $κ = 1.6$ except a shift in the peak of the distribution which depends on the value of $α$ ⁠. It means that the highest energy tail in the case of Kappa distribution exists when the value of spectral index is just above 1.5 and in the case of Kappa–Cairns distribution the highest energy tail exists when the value of spectral index is just above 3.5. The tail of Kappa–Cairns distribution is at much higher velocities/energies as compare to the tail of Kappa distribution function. This shows that in the presence of non-thermality the high energy tail corresponds to relatively higher values of the spectral index $κ$ ⁠. It is also observed that shift in the peak of the distribution due to the higher values of $α$ becomes insignificant when $α > 0.5$ ⁠. This can be seen more clearly in Fig. (8), which exhibits the effect of $α$ on 3D Vasyliunas–Cairns distribution for fixed values of spectral index $κ$ that's 3.6, 5, 10, and 50 in Figs. (8a)–(8d), respectively. It is evident that for a given value of the spectral index $κ$ ⁠, the change in the distribution with respect to $α$ becomes negligible when the value of $α$ becomes greater than 0.5. The saturation of 3D Kappa–Cairns distribution for values of $α$ greater than 0.5 can be further seen by plotting ${(\frac{v}{θ_{t h}})}_{max}$ as a function of $α$ for various values of the spectral index $κ$ ⁠. Figure 9 shows the plots of ${(\frac{v}{θ_{t h}})}_{max}$ Vs $α$ for $κ = 4, 5, 6, 10, and 50$ ⁠. This figure shows that for given value of $κ$ ⁠, the difference in value of $\frac{v}{θ_{t h}}$ where maximum of the Kappa-Cairns distribution occurs becomes negligible when the value of $α$ becomes greater than 0.5. The numerical results presented in this section are in good agreement with the recently derived simulation results of Darian³⁸ for Kappa–Cairns distributed space plasma.

FIG. 7.

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Three-dimensional Kappa–Cairns distribution function as a function of the magnitude of normalized three-dimensional velocity, for various values of nonthermality parameter $α$ and the spectral index $κ$ ⁠. In each panel of this figure $α$ is fixed and $κ$ is varied.

FIG. 8.

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Three-dimensional Kappa–Cairns distribution function as a function of the magnitude of normalized three-dimensional velocity, for various values of nonthermality parameter $α$ and the spectral index $κ$ ⁠. In each panel of this figure $κ$ is fixed and $α$ is varied.

FIG. 9.

Plot of v θ t h max as a function of non-thermality parameter for Kappa–Cairns distribution function for various values of the spectral index κ.

View large Download slide

Plot of ${(\frac{v}{θ_{t h}})}_{max}$ as a function of non-thermality parameter for Kappa–Cairns distribution function for various values of the spectral index $κ$ ⁠.

VI. SUMMARY

In this tutorial paper we have revisited the correct mathematical and numerical analysis of Cairns and Vasyliunas–Cairns distribution functions using their appropriate normalization constant. It is found that 1D Cairns and reduced 1D Cairns distribution functions are not similar. Their dependence on one-dimensional velocity is significantly different from each other. This behavior of CDF is different from other distribution functions such as Maxwellian, Kappa and q-nonextensive distribution. In the case of Maxwellian, Kappa, and q-nonextensive distributions, the dependence on the one-dimensional velocity is the same: either it is reduced 1D or directly written 1D distribution. The effect of non-thermality parameter on 3D CDF is studied by plotting it as a function of magnitude of the 3D velocity. These plots show that the dependence on the non-thermality parameter $α$ saturates when taking $α > 0.5$ ⁠. Dielectric response function has also been derived by using properly normalized 3D Cairns distribution function. The dielectric function has been deduced in terms of plasma dispersion function using the reduced distribution function. This dielectric function can easily be converted into the corresponding dielectric function of the Maxwellian distribution function just by taking $α = 0$ ⁠. It is observed that the 3D Cairns distribution function used in Refs. 20 and 21 is not properly normalized. It is also observed that the reduced distribution used in Refs. 20 and 21 is not properly deduced. The wrong normalization and wrong calculation of the reduced distribution function has led to wrong results and conclusions in Refs. 20 and 21. The dielectric function is used to find analytic expressions for the dispersion relations and Landau damping rates of Langmuir, ion acoustic, and dust acoustic waves in Cairns distributed plasmas.

It has been shown that the normalization factor of Kappa-Cairns (Vasyliunas–Cairns) distribution given in Ref. 36 is not properly calculated. When improperly normalized distribution is used, it may lead to erroneous plots and conclusions. Effective temperature that depends on non-thermality parameter and spectral index has been calculated by finding average energy of the particles. This effective temperature or the average energy tells us that for Vasyliunas–Cairns distribution the valid range of spectral index is such that $κ > \frac{7}{2}$ ⁠. In Ref. 36, it has been stated that the valid range of the value of spectral index is such that $κ > \frac{3}{2}$ which is misleading. 3D Kappa–Cairns distribution is also plotted as a function of magnitude of the 3D velocity. These plots show that in the presence of the non-thermality parameter, the peak of the distribution function is shifted to higher values of $\frac{v}{θ_{t h}}$ ⁠. It is also noted that the tail of Kappa–Cairns distribution exists at much higher velocities/energies as compare to the tail of Kappa distribution function. It is also seen that the difference between the 3D Kappa–Cairns distribution becomes negligible for values of $α > 0.5$ ⁠. A parameter ${(\frac{v}{θ_{t h}})}_{max}$ is defined which gives the value of $\frac{v}{θ_{t h}}$ ⁠, where maximum of the distribution occurs for given values of $α$ and $κ$ ⁠. The plots of ${(\frac{v}{θ_{t h}})}_{max}$ as a function of $α$ show that when $α < 0.5$ ⁠, the Vasyliunas–Cairns distribution functions are different from each other. However, when $α > 0.5$ ⁠, the difference becomes negligible.

APPENDIX A: EQUIVALENCE OF 1D AND 3D KAPPA DISTRIBUTION

In terms of

κ_{1}

⁠, Eq. can be written as

f_{1}^{K} (v_{z}) = \frac{n_{0} Γ (κ_{1} + 1)}{\sqrt{π} κ_{1}^{1 / 2} θ Γ (κ_{1} + 1 / 2)} {[1 + \frac{v_{z}^{2}}{κ_{1} θ_{t h}^{2}}]}^{- κ_{1} - 1},

(A1)

where

θ_{t h} = \sqrt{(\frac{2 κ_{1} - 1}{κ_{1}}) (\frac{k_{B} T}{m})}

and

κ_{1} > \frac{1}{2}

⁠. In terms of

κ_{3}

⁠, Eq. can be written as

F_{1}^{K} (v_{z}) = \frac{n_{0} Γ (κ_{3})}{\sqrt{π} {κ_{3}}^{1 / 2} θ Γ (κ_{3} + 1 / 2)} {[1 + \frac{v_{z}^{2}}{κ_{3} θ_{t h}^{2}}]}^{- κ_{3}},

(A2)

where

θ_{t h} = \sqrt{(\frac{2 κ_{3} - 3}{κ_{3}}) (\frac{k_{B} T}{m})}

and

κ_{3} > \frac{3}{2}

⁠. Using

κ_{1} = κ_{3} - 1

in Eq. , one gets

f_{1}^{K} (v_{z}) = \frac{n_{0} Γ (κ_{3})}{\sqrt{π} κ_{3}^{1 / 2} θ Γ (κ_{3} - 1 / 2)} {[1 + \frac{v_{z}^{2}}{κ_{3} θ_{t h}^{2}}]}^{- κ_{3}} = F_{1}^{K} (v_{z}) .

(A3)

APPENDIX B: DERIVATION OF EQ. (10)

By using Eq. , in Eq. one gets

F_{1}^{C} (v_{z}) = \frac{1}{(1 + 15 α) {(2 π v_{t h}^{2})}^{\frac{3}{2}}} [\iint_{- \infty}^{\infty} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} + \frac{α}{v_{t h}^{4}} \iint_{- \infty}^{\infty} v^{4} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y}] .

(B1)

In Eq. ,

v^{4} = {v_{x}^{4} + v_{y}^{4} + v_{z}^{4} + 2 v}_{x}^{2} v_{y}^{2} + {2 v_{x}^{2} v}_{z}^{2} + {2 v}_{y}^{2} v_{z}^{2}

⁠. This integral can be split into seven integrals such that

F_{1}^{C} (v_{z}) = \frac{1}{(1 + 15 α) {(2 π v_{t h}^{2})}^{\frac{3}{2}}} [I_{1} + \frac{α}{v_{t h}^{4}} (I_{2} + I_{3} + I_{4} + I_{5} + I_{6} + I_{7})],

(B2)

where

I_{1} = \iint_{- \infty}^{\infty} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} = 2 π v_{t h}^{2} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(B3)

I_{2} = \iint_{- \infty}^{\infty} v_{x}^{4} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} = 6 π v_{t h}^{6} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(B4)

I_{3} = \iint_{- \infty}^{\infty} v_{y}^{4} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} = 6 π v_{t h}^{6} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(B5)

I_{4} = v_{z}^{4} \iint_{- \infty}^{\infty} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} = 2 π v_{t h}^{2} v_{z}^{4} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(B6)

I_{5} = 2 \iint_{- \infty}^{\infty} v_{x}^{2} v_{y}^{2} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} = 4 π v_{t h}^{6} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(B7)

I_{6} = 2 v_{z}^{2} \iint_{- \infty}^{\infty} v_{x}^{2} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} = 4 π v_{t h}^{4} v_{z}^{2} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}},

(B8)

I_{7} = 2 v_{z}^{2} \iint_{- \infty}^{\infty} v_{y}^{2} e^{- \frac{v^{2}}{{2 v}_{t h}^{2}}} {d v}_{x} d v_{y} = 4 π v_{t h}^{4} v_{z}^{2} e^{- \frac{v_{z}^{2}}{{2 v}_{t h}^{2}}} .

(B9)

Putting the values of

I_{1} - I_{7}

in Eq. and after doing some algebraic manipulations one gets

F_{1}^{C} (v_{z}) = \frac{1}{\sqrt{2 π} v_{ths} (1 + 15 α)} [1 + α (\frac{v_{z}^{4}}{v_{ths}^{4}} + 4 \frac{v_{z}^{2}}{v_{ths}^{2}} + 8)] e^{- \frac{v_{z}^{2}}{{2 v}_{ths}^{2}}} .

(B10)

APPENDIX C: DERIVATION OF NORMALIZATION CONSTANT GIVEN BY EQ. (46)

By using Eq. in Eq. , one can write

n_{0 j} = A_{κ, α} [∭_{- \infty}^{\infty} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z} + \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} {v^{4} [1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z}] .

(C1)

In Eq.

, v^{4} = {v_{x}^{4} + v_{y}^{4} + v_{z}^{4} + 2 v}_{x}^{2} v_{y}^{2} + {2 v_{x}^{2} v}_{z}^{2} + {2 v}_{y}^{2} v_{z}^{2}

⁠. This integral can be split into seven integrals such that

n_{0 j} = A_{κ, α} [I_{1} + I_{2} + I_{3} + I_{4} + I_{5} + I_{6} + I_{7}],

(C2)

where

I_{1} = ∭_{- \infty}^{\infty} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{2} = \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{4} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{3} = \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{y}^{4} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{4} = \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{z}^{4} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{5} = \frac{2 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{2} v_{y}^{2} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{6} = \frac{2 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{2} v_{z}^{2} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{7} = \frac{2 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{y}^{2} v_{z}^{2} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z} .

Integration over the three dimensional velocity space gives

I_{1} = \frac{{(2 π)}^{\frac{3}{2}} {(κ - 3 / 2)}^{\frac{3}{2}} v_{t j}^{3} Γ (κ - 1 / 2)}{Γ (κ + 1)},

I_{2} = I_{3} = I_{4} = \frac{{3 α (π)}^{\frac{3}{2}} {(2 κ - 3)}^{\frac{7}{2}} v_{t j}^{3} Γ (κ - 5 / 2)}{4 Γ (κ + 1)},

I_{5} = I_{6} = I_{7} = \frac{{α (π)}^{\frac{3}{2}} {(2 κ - 3)}^{\frac{7}{2}} v_{t j}^{3} Γ (κ - 5 / 2)}{2 Γ (κ + 1)} .

Putting the value of

I_{1} to I_{7}

in Eq. and doing some algebraic manipulations give

A_{κ, α} = \frac{n_{0 j} Γ (κ + 1) / Γ (κ - 1 / 2)}{{(2 π)}^{\frac{3}{2}} {[1 + 15 α (\frac{κ - 3 / 2}{κ - 5 / 2})] (κ - 3 / 2)}^{\frac{3}{2}} v_{t j}^{3}} .

(C3)

APPENDIX D: CALCULATION OF (v²)

By using Eq. , Eq. can be written as

⟨v^{2}⟩ = A_{κ, α} [∭_{- \infty}^{\infty} {v^{2} [1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} d^{3} v + \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} {v^{6} [1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} d^{3} v] .

(D1)

In Eq. ,

d^{3} v = {d v}_{x} d v_{y} d v_{z}, v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}, and v^{6} = {v_{x}^{6} + v_{y}^{6} + v_{z}^{6} + 3 v}_{x}^{2} v_{y}^{4} + {3 v_{x}^{2} v}_{z}^{4} + {3 v}_{x}^{4} v_{y}^{2} + {3 v}_{x}^{4} v_{z}^{2} + {3 v_{y}^{2} v}_{z}^{4} + {3 v}_{y}^{4} v_{z}^{2} + 6 {v_{x}^{2} v}_{y}^{2} v_{z}^{2}

⁠. This integral can be split into thirteen integrals such that

⟨v^{2}⟩ = \frac{A_{κ, α}}{n_{0 j}} [I_{1} + I_{2} + I_{3} + I_{4} + I_{5} + I_{6} + I_{7} + I_{8} + I_{9} + I_{10} + I_{11} + I_{12} + I_{13}],

(D2)

where

I_{1} = ∭_{- \infty}^{\infty} {v_{x}^{2} [1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{2} = ∭_{- \infty}^{\infty} {v_{y}^{2} [1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{3} = ∭_{- \infty}^{\infty} {v_{z}^{2} [1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{4} = \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{6} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{5} = \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{y}^{6} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{6} = \frac{α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{z}^{6} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{7} = \frac{3 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{2} v_{y}^{4} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{8} = \frac{3 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{2} v_{z}^{4} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{9} = \frac{3 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{4} v_{y}^{2} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{10} = \frac{3 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{4} v_{z}^{2} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{11} = \frac{3 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{y}^{2} v_{z}^{4} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{12} = \frac{3 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{y}^{4} v_{z}^{2} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z},

I_{13} = \frac{6 α}{v_{t j}^{4}} ∭_{- \infty}^{\infty} v_{x}^{2} {v_{y}^{2} v}_{z}^{2} {[1 + \frac{v^{2}}{{2 κ_{1} v}_{t j}^{2}}]}^{- κ - 1} {d v}_{x} d v_{y} d v_{z} .

Integration over the three dimensional velocity space gives

I_{1} = I_{2} = I_{3} = \frac{{(π)}^{\frac{3}{2}} {(2 κ - 3)}^{\frac{5}{2}} v_{t j}^{5} Γ (κ - 3 / 2)}{2 Γ (κ + 1)},

I_{4} = I_{5} = I_{6} = \frac{{15 α (π)}^{\frac{3}{2}} {(2 κ - 3)}^{\frac{9}{2}} v_{t j}^{5} Γ (κ - 7 / 2)}{8 Γ (κ + 1)},

I_{7} = I_{8} = I_{9} = I_{10} = I_{11} = I_{12} = \frac{{9 α (π)}^{\frac{3}{2}} {(2 κ - 3)}^{\frac{9}{2}} v_{t j}^{5} Γ (κ - 7 / 2)}{8 Γ (κ + 1)},

I_{13} = \frac{{3 α (π)}^{\frac{3}{2}} {(2 κ - 3)}^{\frac{9}{2}} v_{t j}^{5} Γ (κ - 7 / 2)}{4 Γ (κ + 1)} .

Putting the value of

I_{1} to I_{13}

in Eq. and doing some algebraic manipulations give

⟨v^{2}⟩ = v_{t j}^{2} \frac{[3 (κ - 5 / 2) (κ - 7 / 2) + 105 α {(κ - 3 / 2)}^{2}]}{(κ - 7 / 2) [(κ - 5 / 2) + 15 α (κ - 3 / 2)]} .

(D3)

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2020

Author(s)

Revisiting some analytical and numerical interpretations of Cairns and Kappa–Cairns distribution functions

I. INTRODUCTION

II. 3D CAIRNS DISTRIBUTION FUNCTION AND ITS CORRECT VALIDATION

III. DERIVATION OF DIELECTRIC FUNCTION

IV. ANALYTICAL EXPRESSIONS OF DISPERSION RELATION AND DAMPING RATE

A. Electron plasma waves in cairns distributed plasma

B. Ion acoustic waves in cairns distributed plasma

C. Dust acoustic waves in cairns distributed plasma

V. KAPPA-CAIRNS DISTRIBUTION FUNCTION

VI. SUMMARY

APPENDIX A: EQUIVALENCE OF 1D AND 3D KAPPA DISTRIBUTION

APPENDIX B: DERIVATION OF EQ. (10)

APPENDIX C: DERIVATION OF NORMALIZATION CONSTANT GIVEN BY EQ. (46)

APPENDIX D: CALCULATION OF (v²)

REFERENCES

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Revisiting some analytical and numerical interpretations of Cairns and Kappa–Cairns distribution functions

I. INTRODUCTION

II. 3D CAIRNS DISTRIBUTION FUNCTION AND ITS CORRECT VALIDATION

III. DERIVATION OF DIELECTRIC FUNCTION

IV. ANALYTICAL EXPRESSIONS OF DISPERSION RELATION AND DAMPING RATE

A. Electron plasma waves in cairns distributed plasma

B. Ion acoustic waves in cairns distributed plasma

C. Dust acoustic waves in cairns distributed plasma

V. KAPPA-CAIRNS DISTRIBUTION FUNCTION

VI. SUMMARY

APPENDIX A: EQUIVALENCE OF 1D AND 3D KAPPA DISTRIBUTION

APPENDIX B: DERIVATION OF EQ. (10)

APPENDIX C: DERIVATION OF NORMALIZATION CONSTANT GIVEN BY EQ. (46)

APPENDIX D: CALCULATION OF (v2)

REFERENCES

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APPENDIX D: CALCULATION OF (v²)