1.Introduction

Magnetosonic(MS)waves are low frequency and highly dispersive waves[1]which propagate perpendicularly to the ambient magnetic field[2].These waves are also sometimes called fast MS waves or equatorial noise when their frequency is above proton gyro frequency all the way up to the lower hybrid frequency[3].MS waves are of great importance because of their ubiquity in space plasmas.In the case of obliqueness,i.e.the presence of small parallel wavenumber,the waves exhibit wave–particle interaction(WPI),in which case the wave may either grow or become damped out depending on whether a free energy source is present or not.The WPI leads to mechanisms such as heating and particle acceleration in the solar wind[4],magneto sheath[5],plasma sheet[6],corona[7],magnetic equator[8],aurora[9],solar flares[10,11],inner magnetosphere[12]and Van Allen radiation belts[13].Based on the measurements of Van Allen Probe A,fast MS waves are believed to penetrate into the plasma sphere via the plasma pause and have been proven to be the best candidates for controlling the dynamics of electrons in the inner radiation belt and the unusual butter fly distribution of relativistic electrons in the geostationary orbit[3,13,14].

Depending on the nature of the plasma,WPI is affected by various effects such as the presence of dust[15–18],temperature anisotropy[19]and density inhomogeneity[20].In Maxwellian distributed plasmas,studies have shown that increasing the density inhomogeneity or temperature anisotropy withenhances the growth rate[20],whereas the presence of dust reduces it[15].

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Numerous research studies have been conducted to study MS waves in non-Maxwellian plasma.The WPI of MS waves under the assumption　ω　≪　ωi has been investigated in a loss cone distribution[21]as well as in an anti-loss cone one[22].For the loss-cone distribution,depending upon the choice of the loss-cone parameters,the wave growth may either be increased or decreased[21].However,in the case of the antiloss cone,the wave is always damped and is smaller as compared to the Maxwellian distribution[22].

Apart from the aforesaid distributions,the non-Maxwellian character,especially the presence of suprathermal particles,can be modeled effectively by kappa-type distribution functions.Several kappa-type distributions appear in the literature:non-relativistic kappa[23,24],relativistic kappa[25–28],kappa-Maxwellian[29]and many more.In the nonrelativistic limit,the values of kappa from 2 to 6 provide a good fit for many regions in the space plasma[23].However,at very high energy,the non-relativistic kappa distribution deviates from the observed particle distribution;a relativistic kappa distribution seems to fit the data very well[25–28].The kappa-Maxwellian distribution is advantageous both physically and mathematically.Usually,in space magnetoplasma,particles are accelerated along the magnetic field direction,leading to a power law distribution;however,in the perpendicular direction,there is equilibrium and is otropization　[29,30].From　the　mathematical standpoint,the perpendicular integration in the generalized dielectric tensor can be handled with ease.Depending on the environment,each distribution has its own importance,and all the environments discussed in the first paragraph where the MS waves play an important role in WPI have been con firmed to fit with the kappa-type distribution.

In this paper,we discuss the damping rate of the obliquely propagating MS waves in a suprathermal plasma modeled by kappa-Maxwellian distribution.To the best of our knowledge,ours is the first report that addresses this issue.The layout of this paper is as follows.In section 2,we present the mathematical model and derive the dispersion relation along with the analytical expressions for the real frequency and the damping rate.Then,we discuss the exact numerical analysis in section 3.Finally,we summarize the results and conclude the paper in section 4.

To manipulate χ ,we choose the following kappa-Maxwellian distribution function[29]

and

2.Mathematical model

with and

where we have assumed integer values of κ.With this choice ofZκ*(ζ0α),we extract the real part of the dielectric tensor from equation(1)as

We assume low frequency waves,　ω　≪　Ωi ,in a uniform low beta plasma,in which the real wave vector k lies in the x-z plane.The dispersion relation in such a plasma can be written as[15,19,20,31,32]

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whereThe perpendicular and parallel thermal velocities are related to their corresponding temperatures byandwithThe spectral index κ controls the number of high energy particles in a system.The smaller the value of κ,the larger the number of high energy particles.Moreover,in the limit κ　→ ∞,the Maxwellian distribution can be retrieved.

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Following the same procedure adopted by the authors in references[15,19,20,31,32],we get a simplified expression for∈as

whereis the gyro radius and　　ζ0α=　Generally,　we　assume　　ω　to　be　complex,satisfying the condition　γ　≪　ωr .Consequently,the modified plasma dispersion function is given by[23,24]

with the supposition that .As we see,is a complex-valued function which,in turn,makes the dielectric tensor,equation(1),a complex-valued function that can be separated　into　real　and　imaginary　parts:　D( k,ω)=Invoking Taylor’s series with the assumption　　it　can　easily　be　shown　thatevaluated at　ω=ωr .

For MS waves,as we know,the parallel wavenumber is small compared to the perpendicular wavenumber which means　ζ0α　≫　1should be used in equation(4),in which case,the asymptotic form of the plasma dispersion function needs to be taken into account,i.e.,[23,24]

where

where the sound velocity and Alfvén velocity are,respectively,given　by and IfD　r=　0,then it is straightforward to obtain

which means that the real frequency is independent of κ,in much the same way as the real frequency of electrostatic waves.This has to do with the structural similarities of the two dielectric tensors(equations(40)and(3)of references[24]and[33],respectively).

Substituting the above expression in we obtain

For the WPI,as is known,we take into account the imaginary part of the plasma dispersion function with which the resulting imaginary part of the dielectric tensor becomes

which clearly shows that the wave must always be damped;a negative sign cannot be generated from any values defined above.Also,it should be noted that the termgets pretty small for large,ensuring a reduced damping rate.

Since an obliquely propagating mode and its WPI in the solar wind have been observed[34],hence we choose the following parameters appropriate for the solar wind at about 1 AU[35]:10-5G and　β=　0.0057.We numerically solve equation(1),

3.Numerical results and discussion

keeping the plasma dispersion function unexpanded.For the exact numerical analysis,we consider low integer values of the spectral index κ,i.e., κ =2,3 and 4,which were experimentally observed in the aforementioned environment[36].The other reason for taking low integer values is that they give closed-form expressions for the plasma dispersion function[23]

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Figure 1.Panel(a)shows the exact numerical plots of equation(1)for the normalized real frequencyagainst normalized wavenumberfor different values of the spectral index κ.In panel(b),the frequency is in the large wavenumber regimes.The Maxwellian curve(solid line)is obtained using equation(6)to make Comparison.

These expressions make the numerical calculations much easier as compared to the cases of non-integer values of κ.We also note that there are no closed-form expressions for both the Maxwellian distributions and the non-integer values of κ.

As shown figure 1(a),the real frequency is almost independent of the parameter κ if we take to be small.Even for moderate values of (see figure 1(b)),the real frequencies for different κ are indistinguishable.The reason is that for the MS mode,which implies thatζ0α must be larger than 1.When we carefully observe the behavior of Zκ*(ζ0α)for large ζ0α (see equation(5)),we see that the expression in the parentheses in the real dispersion relation(see equation(4))for　ζ0α　≫　1is not a sensitive function of κ unless higher-order terms(greater than five)are invoked.

Figure 2.The normalized damping ratebased on equation(8)vs normalized wavenumberfor different values of the spectral index κ.

As is evident from equation(8),κ does not provide the free energy source;the wave is always damped,and the damping rate strongly depends on the spectral index κ.Figure 2 is the plot of equation(8),while figures 3(a)and 3(b)are based on the exact numerical analysis.As expected,for small values of the wave is heavily damped,when κ is low,although the difference is minuscule in magnitude.Whenincreases,the wave behaves otherwise,i.e.,it now becomes heavily damped for large values of κ,and the difference in the magnitude is relatively substantial.The underlying mechanism behind this is the participation of resonant particles.From the given data,we have.As the values of become small,ζ0α gets large and is shifted towards the tail of the distribution(see figure 4(c))where the number of resonant particles is large for a low spectral index.Moreover,the curves are quite close to each other,and thus the difference between the two damping rates is not appreciable.Table 1 depicts the numerical values of the damping rate for different values of the spectral index κ at the given small and large ζ0e.Obviously,the values of from 0.03 to 0.06 demand ζ0eto be in the range of 7.23–3.62.

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Whenincreases,　ζ0e　,still being larger than 1,is shifted to the left where the resonant particles are more in number for high values of κ.Figure 4(b)shows the marked difference between the curves which explains Significant damping.From table 2,it is clear that for or above,the values ofζ0α ranges from 1.44 to 1.14 wherein the large number of resonant particles for higher values of κ are seen from figure 4(b).

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4.Summary and conclusion

Figure 3.The exact numerical plot of equation(1)for the normalized damping ratevs normalized wavenumberfor different κ values.In(a),the damping rate is in the small wavenumber regimes which is too small to be seen in(b)which portrays both small as well as large wavenumber regimes.

Suprathermal particles have been regularly observed in laboratory as well as in space plasmas.That ambience affects the dynamics of the system has astonishing consequences.In the present work,we have considered a kappa-Maxwellian distribution with κ being an integer.We have shown that the real dispersion relation is not affected considerably by κ variation,however,the damping rate is.We have used the data from the solar wind at about 1AU and have found that for small values of the parallel wavenumber,the high energy resonant particles participate in the WPI,leading to a high damping rate.When parallel wavenumber increases,the resonant particles are mainly in the low energy part of the distribution function.But in the low energy part,there are more resonant particles for higher values of κ,contrary to what is observed in the high energy tail.

The results we have obtained may find fruitful applications in various suprathermal plasma environments,such as the solar corona,magnetosphere,aurora,plasma sheets and magnetosheath.Some of these environments may also contain suprathermal particles with non-integer values.However,following Mace[24]and Summer[37],we can extend our analysis by reconsidering the modified plasma dispersion function for non-integers.Furthermore,in addition to the presence of high energy particles in the aforesaid space environments,they may contain streaming particles or have temperature anisotropy or are inhomogeneous in density.By incorporating these effects,our calculations can be further extended,which will be the focus of our attention in our future research work.

Figure 4.In panel(a),parallel part of the normalized distribution functionagainst normalized velocityfor different κ values is shown.The middle panel(b)shows that when κ decreases,the number of suprathermal particles decreases whenis between 1 and 2.If is greater than 3,then,as shown in panel(c),suprathermal particles are large in number for smallκ.

Table 1.Values of normalized damping rate for different ,ζ0e and κ.The normalized damping rates are expressed in

k ∣∣k⊥　ζ0e k vAκ=γ　⊥,　2k vA　　κ=γ　⊥,　3k vA　　κ=γ　⊥,　4 0.03　7.23　　−0.36　　−0.094　　−0.022 0.04　5.43　　−1.46　　−0.644　　−0.246 0.05　4.43　　−4.33　　−2.758　　−1.485 0.06　3.62　　−10.00　　−8.702　　−5.982

Table 2.Values of normalized damping rate for different ζ0e and κ.The normalized damping rates are expressed in10-3.

k⊥　ζ0e k vA⊥k vA　⊥kvA⊥0.15　1.44　　−0.57　　−0.94　　−1.11 0.16　1.33　　−0.73　　−1.19　　−1.38 0.18　1.20　　−1.11　　−1.73　　−1.98 0.19　1.14　　−1.33　　−2.02　　−2.28

IAK is grateful to the Salam Chair in Physics for providing a vital research environment.

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