Theoretical Study on Weibel instability in the existence of large amplitude Langmuir wave inside a Plasma

. This paper investigates how an electrostatic Langmuir wave having large amplitude affects the Weibel instability (WI) in the existence of ions and an electron beam. Two Langmuir side band waves are produced by coupling of EM perturbation to the Langmuir wave. The Langmuir wave (LW) increases the growth rate beyond its linear value. Here, we noticed that the growth rate 𝛤(𝑠𝑒𝑐 −1 ) scales linearly with the electron beam velocity 𝑣 𝑏𝑒 and 1/2 power of the electron beam density 𝑛 𝑏𝑒 . As we increase the density of ions inside the plasma, the growth rate stabilizes. Additionally, we find that the growth rate is very sensitive to the plasma frequency of ions. Therefore, our work finds an application in space, galactic cosmic rays and supernovas. Also, our work covers a range of application from the development of fusion power to understand the various astrophysical phenomena.


Introduction
Weibel instability (WI) is a type of electromagnetic instability  that can be caused by small amplitude electromagnetic fluctuations.It was first introduced by Weibel [6] in 1959 and has since been observed in several astrophysical phenomena, such as collisionless shocks, universal magnetogenesis and laser plasma interaction [1][2][3][4][5].Recently, the WI has been shown to play a vital role in Gamma ray burst (GRB) dynamics.The Weibel instability (WI) is generated by an anisotropy in the distribution function, which can be created by counter-streaming flows and a high temperature anisotropy.
This was later able to be identified in laser-plasma experiments [8].Different Weibel instability (WI) configurations have been investigated for unmagnetized plasma [9-10] and magnetized plasma [12][13][14].Weibel instability (WI) is used to describe the existence of strong magnetic fields inside of plasma.An intense magnetic field (up to 10 8 G) is produced when highly intense ultra-short laser pulses are bombarded on the solid target [15].In fast ignition target's coronal plasma, in the transportation of fast electrons, an important role has been played by the magnetic fields created due to Weibel instability (WI) [16].
Weibel instability is known to produce very intense femtosecond pulses, and Krainov [17] was the first to study this phenomenon.Siemon [20] later looked at how electron beams behave during Weibel instability, and Ghorbanalilu and Sadegzadeh [19] investigated the magnetic fields that are produced as a result of Weibel instability in relativistic shocks.Finally, Mahadavi and Khanzadeh [22] explored the effect of thermal conditions on Weibel instability and derived an expression for its growth rate.Ryutov et al. [23] explored the collisional effect on the ion Weibel instability (WI) for two counter streaming plasma flows generated with the help of high-power laser.Here, for the case of strong intra-stream collisions, the dispersion relations have been evaluated.The purpose of this paper is to call the effect of ions on the Weibel instability (WI) in the existence of Langmuir pump wave (LPW) having large amplitude.The Weibel instability (WI) is an electromagnetic (EM) perturbation (  ,  ⃗  ) that is paired with Langmuir pump wave ( 0 ,  ⃗ 0 ) to produce two Langmuir wave sidebands ( 1,2 =   ∓  0 ,  ⃗ 1,2 =  ⃗  ∓  ⃗ 0 ).These sidebands produce a nonlinear current that amplifies the original EM perturbations (  ,  ⃗  ).The responses of the beam and plasma are noted and discussed in Section II, using fluid treatment and the expression of growth rate through first-order perturbation theory.The findings of the analysis are detailed in Section III, along with a discussion of those findings.Finally, section IV provides concluding remarks for the study.

Numerical instability analysis for Weibel instability
Assume a plasma system is composed of ions and electrons ( − ).The charge, mass, temperature, and equilibrium density, respectively, of the two species ( − , ions) are given as (−,   ,   ,   ), (,   ,   ,  0 ).Along the z-direction, with a density of   and a velocity of be vz , an electron beam is travelled through this plasma system.A Langmuir wave of large amplitude is present in the plasma that has an electrostatic potential as where and   (= are the thermal velocity of electron and electron plasma frequency, respectively. The linear response of the plasma electrons, beam and ions to the electrostatic Langmuir pump wave (LPW) is attained using the fluid equation of motion and equation of continuity.On simplifying the eqn.( 2), we obtained the perturbed beam velocity as Linearization of Eq. ( 5) yields the perturbed density of an electron beam Also, on linearizing of Eq. ( 3), we get the perturbed velocity of plasma electrons Plasma electrons' perturbed density, after linearization of Eq. ( 6), is obtained as follows.
Similarly, for ion we get the perturbed velocity as and the perturbed density of ions are The system is perturbed by an electromagnetic perturbation as And the magnetic field is alongside the y-direction.The non-linear coupling between the electromagnetic perturbation and the Langmuir pump wave produced two Langmuir side band, whose electrostatic potential can be written as where  1,2 =   ∓  0 , and  ⃗ 1,2 =  ⃗   ̂∓  ⃗ 0 .
The linear response of beam electron corresponds to Langmuir potential 1  and 2  is given as and where j=1, 2.   = −

2
[  −    ] 2 is the electron beam susceptibility and is the plasma beam frequency.
Similarly, for plasma electron the linear response corresponds to Langmuir potential 1  and 2  is given as , 01 where j=1, 2.
2 is the plasma ion susceptibility.
The ponderomotive force  1,2 at ( 1,2 ,  ⃗ 1,2 ) is generated by the coupling of the magnetic field of electromagnetic perturbation (EM) with the oscillatory velocity of plasma electrons  0 provided by Langmuir pump wave (LPW).
where  0 * signifies the complex conjugates of  0 , where Also, the nonlinear plasma electron response corresponds to ponderomotive potential is given as Similarly, for ions, we obtain as where Similarly, for beam electron, we obtain nonlinear density as where  1 = − The Maxwell's wave equation for electromagnetic magnetic waves is given as Here, And the expression of Growth rate is

Results & Discussions
In the numerical calculation, we have adopted the following typical plasma parameters: plasma electron beam density   = 1.0 × 10 7  −3 , plasma ion density  0 = 2.0 × 10 10  −3 , plasma beam velocity   = 1.0 × 10 8 /.Using Eqn. ( 27) the growth rate ( −1 ) as a function of beam velocity (  ) is sketched in Figure 1.We noticed that with increasing the beam velocity, a raise in the growth rate is reported.This is because the beam velocity makes plasma system unstable.Consequently, enhanced growth rate is resulted.In Fig. 2, the growth rate versus electron beam density is plotted and we found that increasing the beam density increased the instability, so our growth rate increased.In Fig. 3, the growth rate (  −1 ) as a function of wave number   is sketched for different value of ion density.We obtained the value of growth rate  = 5.2342 × 10 5 ,  = 4.6821 × 10 5 and  = 4.2742 × 10 5 at wave number   = .33for different value of ion density  0 = 2 × 10 10  −3 ,  0 = 3 × 10 10  −3 ,  0 = 3 × 10 10  −3 and  0 = 4 × 10 10  −3 .The growth increases  = 1.3165 × 10 6 ,  = 1.1776 × 10 6 and  = 1.0750 × 10 6 at wave number   = .83.From Eqn. (27) we can notice the enhancement of growth rate nearly linearly with increasing the wave number   .Also, as we increase the value of ion density inside the plasma the plasma gets stabilizes.Using Eqn.(27), the behaviour of growth rate as a function of plasma-ion density has been portrayed in Figure 4.It has been noticed that with increasing the value of ion density, a diminishment in the growth rate is seen because addition of ions inside the plasma make plasma system stabilizes, so our growth rate decreases.

Conclusion
We have presented a comprehensive fluid explanation to investigate the effects of ion, electron beam velocity, beam density, and ion density on the Weibel instability (WI), which may develop in the laser fusion plasma as a result of an anisotropic velocity distribution (hot electron beam) in the existence of Langmuir waves (LW) with large amplitude.It is investigated how the electron beam density and beam velocity affect the growing rate of the Weibel instability.We observed that the growth rate rises with increasing electron beam density  and velocity   and stabilizes with increasing ion density.The results of the present study may be utilized to further analyze how EM fluctuations in solar plasma are affected by velocity or temperature anisotropy.Additionally, this research may shed light on how Weibel instability (WI) causes magnetic fields to form in laser-induced plasma studies.Additionally, in fusion techniques involving inertial confinement, the resultant magnetic field may affect electron energy transfer and electromagnetic wave propagation.

Figure 1 .
Figure 1.Schematic of growth rate with electron beam velocity be v .