COLD BEAM INJECTION IN RELATIVISTIC EMEC WAVE FOR KAPPA DISTRIBUTION FUNCTION WITH AC FIELD FOR MAGNETO-PLASMA

The effect of cold electron beam on electromagnetic electron cyclotron (EMEC) wave has been studied by using the unperturbed Lorentzian (Kappa) distribution in the magnetosphere for relativistic plasma. The dispersion relation is obtained by using the method of characteristic solutions and kinetic approach. An expression for the growth rate of a system has been calculated. It is inferred that in addition to the relativistic plasma obliquity and effect of cold electron beam modifies the growth rate and it also shifts the wave band significantly. The relativistic electrons by increasing the growth rate and widening the bandwidth may explain a wide frequency range of EMEC wave emissions in the magnetosphere.


INTODUCTION
There is strong evidence that the high latitude ionosphere is an important source of magnetospheric ions, which is confirmed by the observations of the upward flowing accelerated ionospheric ions in the auroral zone, These up flowing accelerated ionospheric ions can be characterized as either a beam like distribution of particles or conic like distribution depending upon their pitch angles.
In the last decade some authors have reported the presence of upward and downward flow of ions in the ionosphere [1][2].Evidence for ion heating exists in the F-region where an intense super thermal ion beam was reported in confirmation with electrostatic ion-cyclotron wave observed near an auroral arc and was interpreted interms of ion heating.Moreover different satellite observation have confirmed the presence of energetic (keV) up streaming ion beams and electrostic waves in the magnetospheric plasma also.Penetration of ion beam is background plasma has been observed by various satellites and space probes.The ions accelerated towards the Earth's magneto tail far away from the earth are reported to be of ionospheres origin.
Theoretical studies of whistler wave excitation have generally adopted anisotropic Maxwellian distributions to describe the resonant population.The dispersion relation can then be evaluated in terms of the plasma dispersion function [3].However, in the natural space environment of planetary magnetospheres, astrophysical plasmas and the solar wind, plasmas are generally observed to possess a non-Maxwellian high energy tail component [4].The origin of the high energy tail is not well understood, but once generated; the high energy tail persists in the collisionless magnetospheric environment.A useful distribution function to model such plasmas is the generalized Lorentzian (Kappa) distribution [5][6] explained by a spectral index  and valid for  > 3/2.The modified dispersion function (Kappa distribution) approaches the plasma dispersion function (Maxwellian) in the limit as

  κ
In the recent past whistler mode energetic electrons interaction assuming bi-Lorentzian (Kappa) particle distribution was studied by Thorne and summers [12].These authors derived the dispersion relation in the absence of AC electric field.These studies require the understanding of plasma properties subjected to fields of oscillating nature.In addition to injection of AC fields into space [13], electric field measurements at magnetospheric heights and in shock regions have given values of AC field along and perpendicular to Earth's magnetic field [14][15].The behaviour of plasma in high frequency parallel and perpendicular AC fields have also been studied by a large number of investigators.
Motivated by these studies, in the present chapter, whistler mode instabilities have been analyzed for an anisotropic plasma having a generalized Lorentzian (Kappa) distribution function having spectral index , reducible to Maxwellian distribution in the limit    in the presence of parallel AC electric field by method of characteristics solution.Using details of particle trajectories, dispersion relation and growth rate have been derived in analytical form and are evaluated for plasma parameters suited to the Earth's magnetosphere.Results have been discussed and compared with that obtained by earlier workers using Maxwellian distribution.It is observed that the Maxwellian and Kappa distributions differ substantially in the high energy tail, but the differences become less significant as  increases.

MATHEMATICAL FORMULATION:
A spatially homogeneous anisotropic, collisionless plasma subjected to external magnetic field Where force is given as The particle trajectories are obtained by solving equation of motion defined in equation (3) and S(r,v,t) is defined as: where s denotes species and E1, B1 and fs1 are perturbed quantities and are assumed to have harmonic dependence in E1, B1 and fs1=exp i(k.r− ωt).The method of characteristic solution is used to determine the perturbed distribution function, fs1, which is obtained from Eq. ( 2) by The phase space coordinate system has been transformed from      5), ( 6) and the Bessel identity and performing the time integration, following the technique and method of Pandey and Kaur [17], the perturbed distribution function is found after some lengthy algebraic simplifications as : and the Bessel function arguments are defined as The conductivity tensor is written as (10) J a n u a r y 0 7 , 2 0 1 5 By using these in the Maxwell's equations we get the dielectric tensor, For parallel propagating whistler mode instability, the general dispersion relation reduces to The unperturbed Lorentzian-Kappa distribution function is: And associated parallel and perpendicular effective thermal speeds are Applying the approximation in electron-cyclotron range of frequencies.In this case, ion temperature are assumed

 
for background plasma.Therefore Equation ( 13) becomes the following, as a sum of background and injected cold beam plasma:  21) is the modified plasma dispersion function defined by Summers and Thorne [8].
Applying condition with p=1, n=1 and q=0 we can get the growth rate and real frequency using When EMEC waves propagate parallel to magnetic field direction, the expression of growth rate and real frequency becomes: This satisfied the condition of electromagnetic waves.Figure 3 shows the variation of growth rate and real frequency with k ~ for various values of ratio of number density of cold electrons to hot electrons.The number density of hot background plasma is assumed variable as 5x10 6 m -3 , 2.5x10 6 m -3 and 1.6x10 6 m -3 , thus giving nc/nw as 10, 20 and 30 (approximately).The growth rate is 0.0051 for nc/nw =10 at k ~=0.24, the growth rate is 0.0058 when nc/nw =20 at k ~=0.28 and growth rate is This satisfied the condition of electromagnetic waves.Results can be compared with Pandey et al. [18] Lorentzian/Kappa plasma series expansion brings change in thermal velocity (perpendicular), affecting terms of temperature anisotropy.Temperature anisotropy being the primary source of instability gets further modified by Kappa distribution function, giving rise to further increase in growth rate.The theory of kappa distribution also explains that suprathermal electron in Kappa distribution modifies the intensity and Doppler frequency of electron plasma.The inclusion of temperature anisotropy in Lorentzian (Kappa) plasma can explain the observed higher frequencies spectrum of whistler waves [17,19].This satisfied the condition of electromagnetic waves.
to get dispersion relation.In this case, linearized Vlasov-Maxwell equations, obtained after neglecting higher order terms and separating the equilibrium and non-equilibrium parts, following the technique of Pandey et al.[16], are given as below:

.
The particle trajectories which are obtained by solving eq.(3) for the given external field and wave propagation,

P
 and Pz denote momenta perpendicular and parallel to the magnetic field.Using equations (


Due to the phase factor the solution is possible when m = n.Here for relativistic case with perpendicular AC electric field for g= 0, p = 1, n = 1 is written as:

Figure 1
Figure 1 shows the variation of growth rate and real frequency with respect to k ~ for various values of temperature anisotropy AT of background plasma and other fixed parameters as listed in figure caption.Since AT = [(T  /T||)-1], AT becomes 0.25, 0.50 and 0.75.For T  /T||=1.25,1.5 and 1.75, the growth rate is 0.00389, 0.0053 and 0.0062 respectively.The maxima changes from k ~=0.22 to k ~=0.23 and k ~=0.23.As the value of increases, growth rate increases.In this graph the growth rate increases with increase of temperature anisotropy and maxima is shifted significantly towards the higher k values.It is clear from the figure that the temperature anisotropy is the main source of energy to drive the for nc/nw =30 at k ~=0.33.This shows that as the number density of hot electrons decreases, that is, as the ratio of nc/nw increases from 10 to 30, growth rate increases.The increase in bandwidth and significant shift in k ~valueis also seen in graph from 0.24 to 0.33 for kappa distribution index κ= 2. The real frequency increases with increasing value of k ~.

Figure 4
shows the variation of growth rate and real frequency with respect to k ~ for various values of relativistic factor of background plasma and other fixed parameters as listed in figure caption.In this graph the growth rate increases with increase of relativistic factor and maxima is fixed for k ~ values.It is clear from the figure that the relativistic factor is the source of energy to drive the excitation of the wave.The real frequency increases with increasing value of k ~.

Fig 1 .
Fig 1. Variation of Growth Rate and Real Frequency with respect to k ~ for various values

Fig 2 .Fig. 3 .
Fig 2. Variation of Growth Rate and Real Frequency with respect to k ~ for various values of AC

Fig 4 .
Fig 4. Variation of Growth Rate and Real Frequency with respect to k ~ for various values of  at nc/nw=10,