Effect of Beam Velocity on Firehose Instability in Earth’s Magneto-plasma with General Loss-Cone Distribution Function

The dynamics of firehose instability (FHI) in the presence of beams is investigated by applying kinetic theory and general loss cone distribution (GLCD) function. It is explained how ion/electron beams affect the growth rate and length of the FHI in low beta homogeneous plasma. The energy of the transversely heated ions is found to be reduced by the ion beam, whereas the temperature anisotropy arranges free energy, to begin with, and accelerates the growth rate of the instability. It has been discovered that the firehose instability results from the extraction of energy from ions heated perpendicularly in the presence of upflowing ion beams in the asymmetrical magneto-plasma. In the auroral acceleration region, the findings are expressed in terms of the firehose instability. The findings may be checked and then may be applied in dusty and multi-component space plasma also.


Introduction
The basic concept to stimulate instability in plasma is the accessibility of unconfined energy.Instabilities do get up exclusively in the plasma when free energy is accessible in a thermodynamic system which is not in equilibrium order.Instability is generated due to the redistribution of energy which has gathered in a non-equilibrium state or it can be said that instabilities are the foremost reaction in the process of thermodynamic non-equilibrium conditions.Instabilities are the origin of all system transitions that occur as a structure switches from one state to another.Generally, instabilities can evolve unfamiliar compositions and occasionally they result in turbulence, a sort of intermediate condition in the middle of complete order and chaos.It can be understood from the above discussion that a plasma through instability can let out its free energy and will strive to gain thermodynamical equilibrium.[1,2].
In a regime with low plasma densities and strikes among the particles are almost nonexistent, such as in astrophysical and space environments, plasma particles interact with waves to gain energy and become superthermal.In the linear system, the wave-particle interaction mode is cyclotron resonance and Landau damping [3].Numerous observations in solar flares and solar wind have confirmed that these particles are accelerated and aligned to the enveloping magnetic field rather than perpendicularly.As a result, the particle distribution does not continue to be Maxwellian but rather takes the pattern of a quasi-Maxwellian distribution that is known as an anisotropic distribution.The temperature anisotropy (ratio of perpendicular temperature to parallel temperature) plays a part in starting FHI.The firehose instability arises due to free energy arranged by the anisotropic distribution of super-thermal particles [4,5].The orientation in the velocity space determines the direction of this distribution [6].A method to initiate instability is formed by the unconfined energy that permeates in the magnetic field's direction [7].A common example to understand the instability of firehose is intense water currents in a garden or fire hose in which a small disturbance can result in a fierce movement of the hose.Firehose instability's mass characteristics account for its non-resonant nature [8,9,10].
The dynamics of FHI are studied by anisotropic kappa-type distribution function by many researchers.In the current study, the nature of FHI is analysed through the general loss cone distribution (GLCD) function.The GLCD function is an applicable tool to explain the configuration of such plasma in which a superthermal population of plasma has mirror-like geometry as it is enveloped by varying guiding magnetic fields.This mirror-like geometrical pattern develops a cause to create anisotropy in velocity space.Since the GLCD function is widely used to describe many phenomena in plasma, hence given the name.For small values of V⊥ or small values of pitch angle α, where α= tan−1 (V⊥/ V∥) there is a lack of particles in this distribution [11].Loss cone distribution is the bestfitting function to describe such plasma configurations when there is a region of velocity space (V⊥, V∥ ) where ∂fa /∂V⊥ >0, [11][12][13][14][15].
In the past there are many types of waves and instabilities were studied by the GLCD function.The dynamics of the kinetic Alfven (KA) waves were studied by applying the GLCD function [16,17,18].The GLCD is also applied to study EMIC waves [13][14][15]19].
For the study of the influence of different beams (electron/ion) on firehose instability (FHI) here are some research papers regarding the observational evidence of beams and their impact on other instabilities and waves in a different part of the magnetosphere.The reaction of electron/ion beams on KA waves with GLCD function was studied in the PSBL region using kinetic approach [20].Researchers have discovered that waves that are excited above the acceleration area are held up by magnetic field gradients in the presence of ion and electron beams and transmit energy to the earth's ionospheric regions to exhibit aurora.
Effects of Ion and electron beams on drift wave instability with different distribution functions by particle aspect analysis are studied in the ionosphere [21].KA waves generated by ion beams and velocity shear in the Earth's magnetosphere are studied by Barik et al. [22].The dynamics of KA wave in the presence of ion and electron beams with GLCD function is studied by Dwivedi et al. [23].Also, they investigated how these beams affected KA waves in an irregular magnetosphere [24].
EMIC instability in the auroral acceleration region using the GLCD function and particle aspect approach is investigated by Patel et al. [25].Ion beams move at a speed that is the opposite of how waves move, and their density slows the growth and amplifies the energy loss from perpendicularly heated ions.Ahirwar et al. [12] used particle aspect analysis to study the influence of beams on EMIC waves with GLCD function in the anisotropic plasma.KA waves also have been investigated in the PSBL layer in the presence of electron beams [26].
It has been proven during the past ten years that auroral luminosity results from the collision of accelerated electron beams with the ionosphere and the up-flowing ion beam towards the magneto-tail at the same time [21,23,27,28,29].Based on FAST satellite measurements, it has recently been stated that the significance of electron and ions' dynamic effects in the acceleration of electrons in small-scale Alfven waves above the auroral oval [26,30,31].
The aforementioned study states that the impact of beams along with the GLCD function is applied for different waves and instabilities but no one has yet analyzed it for FHI.Our purpose is to investigate how these beams affect the firehose instability that could develop in the earth's magnetosphere's auroral acceleration area, where there are significant horizontal gradients in plasma density that might be exacerbated by precipitating electrons during substorm moments.Our investigation is justified by the possibility that the parallel potential drop along the auroral field lines in the aurora region may cause down-flowing electron and up-flowing ion beams [21].
There is much observational evidence that gives reason for the present study.The S3-3 satellite and Polar satellite data have reported the ion beams moving upward [32,33].Ions are frequently shown to have been propelled transverse to the ambient magnetic field in the auroral area [34,35].According to conventional wisdom, a field-aligned potential accelerates both upward and downward electron beams, resulting in "inverted V" electron distributions.The ion beam's energy is nearly proportional to the size of the potential drop as determined by the improvement of the electron loss cones.To check up-going ions, data from near perigee, which was observed over the southern pole zone at an altitude of about 0.8 RE and close to the auroral acceleration region, were employed [36].Within this paper, we have extended the research work done for the firehose instability by Lazor et al. [37].They have studied firehose instability in solar flares and solar wind with Kappa distribution function.Further, for the study of firehose instability with GLCD function, we have taken the guidelines from the study done by Ahirwar [38].They have studied EMIC waves by applying the GLCD function in the auroral acceleration region.Then the concept of the beam effect on firehose instability is framed through the work done by Ahirwar et al. [12].They have studied the effect of beam velocity on EMIC waves in the auroral region with particle aspect approach.Instead of the particle approach, we have used the kinetic approach as charged particle species in low collisional plasma more accurately described by the non-Maxwellian velocity distribution function.Such plasma systems generate micro instability of short wavelength order of ion gyro frequency, therefore in the present paper kinetic theory is appropriate for mathematical description.Given the ample information available on particle aspect analysis, our inquiry will only focus on the kinetic approach.Since in this study, plasma is homogenous, tranquil, and without any background turbulence quasi-linear theory is applied and how the non-Maxwellian distributed instability is affected by the presence of beams, is investigated.

Mathematical Model
The general dispersion relation from Tautz and Schlickeiser [39] in the non-relativistic limit is provided as follows for the transverse waves travelling by the side of the ambient magnetic field.
Where c is the speed of light in a vacuum, ω stands for the frequency and K denotes the wave vector parallel of the different plasma states respectively., 0 / ( ) is the (non-relativistic) gyrofrequency, is the plasma frequency for different species of particles, where s denotes the ions/ electrons and ± signs are used to designate the right-and left-handed circularly polarised electromagnetic waves, respectively.

Anisotropic distribution function
To evaluate the dispersion relation, growth rate, and growth length, deal with a bi-Maxwellian plasma with density distribution and presume that the particle distribution function is divisible in perpendicular and parallel velocities [11,38,40] i.e. can be represented in the following equation , when we take into account the LCD function for the density of particles moving perpendicularly [41] ( ) ) exp{ ( ) / ( ) .
The loss-cone feature's steepness is gauged by the distribution index J.This is a bi-Maxwellian distribution in the case of J=0 and reduces to the Dirac Delta function at J  [40].The GLCD function is defined by Summers et al. [11]. , are the squares of the effective thermal velocities of the different species (electron, ion) about the outer magnetic field.

Dispersion Relation
Inserting the value of equation ( 6) in ( 5) and the obtained value in equation ( 1) we get the required As the frequency of the firehose instability is within the range of the ion gyro frequency, it may be assumed that , im r i e

    
So equation ( 7) takes the form

Growth Rate (𝛾)
Now adding the influence of beam velocity on the growth rate of FHI Where , ./D i e v represents the beam velocity for ions/electrons, the effect of the beam appears through this term in the above equation.Modified growth rate () term derived as Where is the frequency of the instability when influenced by the beams.The waves receive energy from particles moving at speeds that are near their phase velocity.It can be shown that the ion and electron beam velocities have an impact on the growth rate and growth length of FHI, which is moving and growing parallel to the external magnetic field with the GLCD function.

Result and Discussions
In the auroral acceleration region using auroral plasma parameter growth rate and growth length of firehose instability with GLCD function is studied by taking the effect of beam velocity of electron/ions.The traits of FHI are illustrated by using the following parameters [15,13,24,27 ].
Figure 1 shows that for each value of ion beam velocity, a peak is seen it may be because of wave-particle interaction at this particular parallel wave number.After reaching the maximum growth at this wave number the growth has started reducing for each value of ion beam velocity.The growth rate is reported to be larger at lower ion beam velocities, which may be the result of Doppler shifting effects.For the small range of K∥, the growth rate increases, and at larger K∥ values, damping takes place.This might be a result of the Larmor radius effects.Figure 2 shows that as the wave number increases the growth length of FHI also increases for each value of ion beam velocity.It is seen that the growth length is greater at higher values of ion beam velocity.Figure 3 shows that for each value of electron beam velocity, a peak is seen it may be due to the interaction of wave-particle interaction at this particular paralleled wave number.After reaching the maximum growth at this particular wave number the growth has started reducing for each value of electron beam velocity.It is observed that the growth rate is lower at lower values of the value of electron beam velocity.Figure 4 shows that as the wave number increases the growth length of FHI also increases for each value of electron beam velocity.Also, higher electron beam velocity lowers the growth length this can be because a larger mirror ratio has a stronger mirroring effect.The FHI in the inertial limit is a crucial aspect of the altitude range between 1000 km and 2.5RE, albeit it might be a problem that needs more research.

Conclusions and Recommendations
In the present study using the kinetic approach and GLCD function, the beam effects of ions and electrons on the dynamics of FHI are studied.The significance of this study is here as the mirroring structure of magnetic field lines is taken into account which provides evidence of magnetosphereionosphere coupling.The following are some key observations from this study: 1. Ion beams, amplify the growth rate of FHI at lower values of the velocity of the ion beam, whereas electron beam velocity amplifies the growth at higher values of electron beam velocity.The direction of the ion beam is opposite and the direction of the electron beam is the same as the direction of the instability is the possible reason for such activity.2. The research shows that the beam velocities in opposition to wave propagation greatly influence the dynamics of FHI and may be the reason for FHI dissipation in the auroral acceleration region.3. It is evident that FHI serves as a means of transferring energy from hot plasma to cold plasma by slowing the growth rate that may be caused by a change in the resonance condition.The cold plasma is heated by the free energy set up by the hot plasma for stimulating the instability and subsequently dampening the instability.4. Ion and electron beams are discovered to be more effective at lower parallel wave numbers (K∥) in the presence of a gradient in a magnetic field.According to the analysis, FHI is produced as a result of the gradient in the magnetic field's effect on the diamagnetic drift, which also causes a gradient in plasma density.In this study, the background plasma is considered homogenous, calm and without any turbulence.More research may be done on the dynamics of FHI while it is growing at an angle to the surrounding magnetic field and turbulence is present in the background plasma environment.The threshold criteria of FHI depend upon temperature anisotropy and beta (β) parameter (ratio of plasma pressure to magnetic pressure).The presence of background turbulence greatly alters the FHI borderline stability threshold by enhancing the limit of temperature anisotropy and by lowering the limit of the beta (β) parameter.When FHI is evolving in such a plasma system having background turbulence, fluctuation and in-homogeneities the moment-based quasi-linear theory is a more applicable tool to describe it rather than the quasi-linear theory which is used to describe homogenous plasma [42].The present study may be extended by considering the effect of background turbulence on the kinetics of FHI.
The firehose instability may be triggered by the varying magnetic field and density inhomogeneities in the far magnetosphere during substorm times, and it then spreads towards the ionosphere, causing acceleration and current pattern, therefore, the investigation about FHI is significant in the auroral acceleration region (lies in the magnetosphere) as it directly affects our home planate.
,(10), and(11) are solved by applying Mathcad software to obtain the value of growth rate and growth length.The effects of the ion/electron beam velocity, on FHI, are shown in Figures 1, 2, 3, and 4 followed by the analysis.

Figure 1 .
Figure 1.The growth rate versus wave number Figure 2. The growth length Lg versus wave number parallel K∥ for different values of ion beam parallel K∥ for different values of ion beam velocity V D, i = -1×10