Ion-acoustic waves with non-planar wavefronts

Shujaht BUKHARI ,Muhammad SHAHID and Ata-ur RAHMAN

1 Department of Physics,The University of Azad Jummu and Kashmir,Muzaffarabad 13100,Azad Kashmir

2 Department of Physics,Women University of Azad Jammu &Kashmir,Bagh 12500,Pakistan

3 Department of Physics,Islamia College Peshawer,Peshawar 25120,Pakistan

Abstract The ion-acoustic (IA) mode exhibiting various orbital angular momentum (OAM) states is examined in a plasma with drifting electrons.The constituent plasma species are modeled with a non-gyrotropic Maxwellian distribution and discussion of dispersion relation and growth rate of twisted IA waves under various conditions is presented.In the domain of kinetic model,the twisted IA waves are characterized by Laguerre-Gaussian (LG) solutions,where plasma distribution function and electric field are decomposed into axial and azimuthal components.The plasma response function is obtained under paraxial approximations and investigated for threshold condition of instability growth rate with helical electric field structures.The impact of an extra electron specie on the instability is demonstrated through a comparison of twisted waves for single and double electron species.

Keywords: instability,twist parameter,wave excitation,orbital angular momentum,threshold limit

1.Introduction

Because ion-acoustic (IA) waves propagate through positively charged ions,they frequently interact with their electromagnetic fields.Basically,these are longitudinal perturbations of plasma density leading to the propagation of electrons and dynamical ions in a phase space [1].The coupling of ions and electrons is provided by an electric field.The theoretical prediction of IA waves was carried out in 1929 by Tonks and Langmuir [2] whereas their experimental verification was done in 1933 by Rewans [3].Subsequently,other scientists [4-6] performed many experiments and confirmed the existence of these waves.Their discovery has motivated the researchers towards investigation of their propagation and excitation in different plasma environments,e.g.,see references [7-12].Ion-acoustic waves exhibit many applications in various regimes of the plasmas.Many authors[13-15] showed the significant damping of IA waves in collisionless plasmas for same temperatures of both ions and electrons.Later,Bhadra and Varma [16] examined the properties of IA waves by including the collisional effects of ionion interactions.In further studies,Buti [17] also incorporated the effects of Coulomb collisions on the propagation of IA waves and found that such collisions increase damping.The prediction of the existence of super nonlinear waves was initially made in a two-component electron-ion plasma featuring Maxwellian-distributed electrons [18].Recently,the nonlinear characteristics of ion acoustic waves that arise from the interaction between the ionospheric plasma of Mars and the solar wind are investigated [19].

Ocean circulation,quantum vortices in liquid nitrogen,spiral galaxies in the Milky Way,solar winds,typhoon vortices,optical and electromagnetic fields,and other examples of vortex-like formations can all be found in nature.The vortex beams are characterized by the Laguerre-Gaussian(LG) modes through Hilbert factor exp(ilϕ),where “l” represents the azimuthal mode index and carries finite amount of orbital angular momentum (OAM) [20].The vortex beam is a classical tool for investigating the characteristics of optical vortices because it can be easily generated in the laboratory[21].The presence of angular momentum and co-rotating solar wind associated with the plasma species has been found in the study [22] of comet Halley.It is interesting to note that when rotating solar wind interacts with a cometary plasma,it induces the generation of angular momentum in the plasma particles.This interaction gives rise to helical field structures,which are commonly observed in various astrophysical plasmas such as interplanetary clouds,coronal plasmas,solar jets,and other similar phenomena.

The exchange of energy and momentum occurs when electromagnetic waves (laser) interact with the matter [23].It is now confirmed that both spin and orbital parts contribute in the total angular momentum of electromagnetic radiations.The OAM is an extrinsic characteristic of electromagnetic(EM) radiations,exhibiting the helical or non-uniform wavefronts.In contrast,spin angular momentum (SAM) is associated with planar wavefronts and is considered an inherent property of EM radiations.Beth [24] was the first to perform an experiment of transferring angular momentum to a half wave plate.Allenet al[25] theoretically proposed the vortex beams with finite OAM states and identified the propagation of optical vortices in paraxial beams.Yao and Padgett [26]investigated the mechanism of generating non-planar structures and corresponding OAM states.Mendonça [27] identified the excitation of twisted modes due to finite OAM states.Later,various environments of space and astrophysical plasmas are identified where distinct and finite OAM states might contribute to plasma excitation.The examples of these environments include the co-rotating solar winds and spinning regions of the Jovian planets.The presence of helical phase structures in space plasmas results in notable effects of OAM on the propagation of helicon waves [28].Recent studies indicated an increasing interest on the inclusion of OAM states in the study of plasma modes [29-33].These studies include the investigation of the characteristics of the nonlinear twisted waves [30],plasmons with twisted effects [31],magnetic field generation by LG light beams [32],phonons with finite OAM states [33],excitation of twisted waves [34],twisted DA waves with self-gravitational effects[35],instability of ion-acoustic twisted waves in permeating and superthermal plasmas [36,37].

In present work,a kinetic theory of the plasma physics is employed for the study of unstable ion-acoustic twisted waves in a plasma with (a) single and (b) double species of electrons.A linearized set of Vlasov and Poisson equations is used for the investigation of twisted waves.Analytical expressions for plasma response function are derived for the examination of twisted IA waves.The instability conditions for two distinct cases are also obtained.We particularly concentrate on the influence of finite OAM states on the instability conditions and growth rate of waves in a plasma which is in thermal equilibrium.Such thermal equilibrium may be attained after many successive collisions of the plasma species.The impact of an extra specie of electrons on the growth rate is highlighted.

2.Twisted ion-acoustic waves and instability threshold

2.1.With single species of electrons

We assume an unmagnetized and collisionless plasma composed of drifting electrons and non-drifting positive ions.Any type of perturbation in a plasma from equilibrium corresponds to perturb the distribution function and plasma densities,fα=fα0+fα1andnα=nα0+nα1withα=e,i refers to electrons and ions.The digits 0 and 1 in the subscripts represent the quantities in equilibrium (unperturbed) and nonequilibrium (perturbed) states,respectively.The physical model employed in the present study is based on the collisionless Vlasov-Poisson equations in which the particle distribution functionfα(r,v,t) evolves according to the Vlasov equation,

whereqα(mα) is the charge (mass) ofαspecies,respectively.After many successive interparticle collisions,a thermal equilibrium is attained in a plasma and particle distribution function in such a situation will always exhibit the Maxwellian velocity distribution as

wherenα,vαandTαare the density,velocity and temperature of theα-species,respectively.The Vlasov equation can be more conveniently solved by expressing the electric field in the form of electrostatic potential as,E1=-∇ϕ1and is examined by the Poisson’s equation

withfα1(r,v,t) ≪[fα0(v)] being the perturbed distribution function (DF) [Maxwellian equilibrium DF] andϕ1(r,t)refers to the perturbed shielding potential.Throughout the present analysis,a field free plasma is assumed accounting for zero external electric (E0) and magnetic (B0) fields and the plasma equilibrium state is characterized by charge neutrality condition ∑nαqα=0.

To analyze the effect of finite OAM states [27] on the electrostatic IA waves,the electric field can be expressed asE1=-ikeffϕ1(r,t)in cylindrical coordinates system due to helical trajectories via new effective wave vectorkeff=withbeing the unit vectors.The perturbed electrostatic potential ϕ1is now proportional to exp[i(lφ+kz-ωt)] with an extra effect of azimuthal angle “ φ” due to phase structure which is always absent in plane wave solutions.

The perturbed distribution function [27] can be written in terms of generalized LG beam solution in this form

where the LG mode functions and normalization constant are defined as ψpl(r,z)=andCpl=(1/2)π-1/2[(l+p)!/p!]1/2,respectively.Moreover,fpl(v) is mode amplitude whereasindicates the Laguerre polynomials in terms of Rodrigues formula with ζ=[r/w(z)]2andw(z) the beam waist.Likewise,the twisted electrostatic potential ϕ1can be expressed by taking into account the helical phase structures,as

here,p(=0,1,2,...) is the radial mode index andl(=0,±1,±2,...) represents the azimuthal mode index or quantum number which is proportional to the OAM.It can be seen that equations (4) and (5) may reduce to the Gaussian plane beam solution by assumingp=l=0,which is the lowest order mode of the LG beams.

To proceed further,we adopt the standard procedure[38,39] for obtaining the generalized dielectric response function with contribution of finite OAM in an electrostatic collisionless electron-ion plasma,as

where,the axial and azimuthal dispersion functions are respectively,defined asandwith arguments µze≡ (ω-kve0)/kvtetaking into account drifting electrons and µzi=ω/kvtifor nonstreaming positive ions while µφα=ω/lqφvtαrepresents the azimuthal argument.The third term in both square brackets comes from the contribution of OAM states associated with nonplanar wavefronts and by ignoring it,we can obtain the expression for planar waves.For twisted IA waves,the resonance condition is modified as ω=kvz-lqφvα.It is interesting that,in the limitl→0,the dielectric constant (6) is reduced to the planar case which shows its generality.

The main aim of the present study is to investigate the twisted IA modes and the effects of finite OAM states on instability growth rate.For this,we employ constraints on the plasma waves of the formvte,withvte(vti) is the thermal speed of electrons(ions).In present case,the axial and azimuthal arguments for plasma species obey the following conditions;µze≪1,µzi≫ 1 and µφα≫1.For further proceedings of the mathematical analysis,asymptotic expansions are used for electrons and ions.As a result,the dielectric constantD(k,lqφ,ω) takes the form

where (sucessepti)bilities of the electrons χe(k,lqφ,ω) and ions χik,lqφ,ω are respectively defined as

and

Here,λDeandare the Debye lengths of electrons and ions,respectively.The twist parameteris the ratio of longitudinal to azimuthal wave number.The real part,i.e.Re[D(k,lqφ,ω)]=0 of the dielectric constant leads to the wave frequency,as

In plasmas,the constituent particles may have their speeds either slower or faster than the wave speed.In case of larger number of faster particles,the resonant particles loose net amount of energy by transferring it to the wave which exhibits growth rate of instability.Otherwise,resonant particles gain energy from the wave resulting in the damping of wave.The instability growth rate of IA waves carrying OAM states is obtained,as

where Γ is defined as

Equation (11) shows that the growth rate is modified significantly by the streaming speedve0and parameterβ.The characteristic behavior of the wave,i.e.either it is growing or is damped depends on the sign of (1 -ve0/vph) in equation(11).Hence,the cutoff value forve0,above which instability arises is calculated as

Equation (13) is the threshold limit for instability of IA waves with helical structures above which instability steps in invoking γia＞0.

2.2.With two species of electrons

Here,we examine the twisted IA waves in a magnetic field free plasma,whose constituent particles are the ions,cold electrons and hot electrons.The phase speed of the wave is restricted asvti≪ω/k＜vtc,vth.The dispersion functionsZ(µzα) andZ(µφα) in equation (6) can be asymptotically expanded according to the constraints µzi≫ 1,(µzc,µzh) ≪ 1 and µφα≫1 for ions,cold and hot electrons,respectively.Now,following the same procedure as in previous case,we eventually arrive at

where

and χihas the same value as given in equation (9).Here,is the effective Debye length with.In present model,only hot electron streaming speed (vh0=v0) is taken into account while drift speed of cold electrons is zero,i.e.,vc0=0.Equation (14) represents the generalized form of dielectric constant of twisted IA waves in two-temperature electron plasma exhibiting finite OAM states.After the decomposition of equation (14) into real and imaginary parts and assumingDr(k,lqφ,ω)=0,we come up with the following dispersion relation

Note that the dispersive characteristics of twisted IA waves are now significantly affected by the modified IA speed in a two-temperature electron plasma.The modified IA speed is expressed as

whereσ(=Th/Tc) andf(=nh0/nc0) are the hot-to-cold electron temperature and density ratios,respectively.The assumptionnh0=0 simply implies thatf=0 while forTh=0,one may obtainλDh=0 andλDef=λDc.Thus,replacing the subscript c with e,equation (16) simply reduces to the case for IA waves in a plasma with single species of electron.

The growth rate with two-temperature electrons retaining hot electron streaming effects can be written,as

with

The growth rate is now significantly modified withβ,the ratio of hot-to-cold electrons densityfand temperature ratioσ.In a limiting case,whennh0=v0=0 and subscript c is simply replaced by e,the expression [40] for decay rate of twisted IA waves is recovered.Equation (18) satisfies the following instability condition

which is the onset of the instability.The threshold limit for instability of twisted IA waves is modified withnh0andTh.The ignorance of the contribution of OAM states assumesl→ 0,which corresponds toβ→ ∞.In this situation,the factorsthat appear as a multiple of twist parameter in equation (19)become zero more quickly as compared toβ→ ∞.This leads to vanish the products in equation (19) and at condition (l→0),the analytical result is valid for planar IA waves.

3.Results and discussion

In this section,the numerical calculations in the kinetic framework are used to examine the results of equations (10),(11),(16) and (18).With non-zero streaming speed of electronsve0and finite OAM states,the wave frequency,its growth rate and cutoff limit for instability are drastically altered.Special attention and care are needed for the examination of the influence of these parameters on wave dynamics.It is interesting to mention that,the smaller values of azimuthal wave numberlqφ,i.e.,lqφ≪kleads to the larger values ofβ,i.e.β≫ 1.In the limit of largerβvalues (i.e.β→ ∞),the solution for straight propagating IA waves is retrieved.For the validation of obtained results,plasma parameters corresponding to the physical situations [41,42]are used.

The pictorial illustration of the wave frequency of IA waves exhibiting finite OAM via twist parameter [while ignoring spin angular momentum] is shown in figure 1,in which the impact of β on wave frequency is depicted.It is interesting to mention that parameter β (=k/lqφ) describes the helical phase structures and is ultimately controlled by azimuthal number l (=0,1,2,…).

The important point is that the handedness depends on the sign of l (i.e.either positive or negative) and is normally taken by convention.For larger l,i.e.,l → ±∞,the twist parameter β → 0 and consequently at β=0,a phase singularity arises.An increasing amplitude of the wave frequency is noticed in higher momentum states as smaller β values indicate the enhanced OAM effects.Therefore,it is easier to conclude that the real frequency of twisted IA waves is more as compared to the planar waves.Furthermore,for smaller wavelengths,i.e.at longer wave numbers,all curves admit large number of waves through a certain point in unit time.

Figure 2 presents the contour plots of growth rate against wave number “k” and twist parameter “β” showing the impact of finite OAM.It has already been described in above discussion that for larger values of azimuthal wave number,β reduces which shows an enhanced twisted effect.Figure 2 shows the threshold at β=10,above which IA wave is Landau damped while below this value it suffers the instability.Furthermore,the amplitude of the unstable wave increases dramatically at smaller values of β,i.e.β → 0.For a smaller value of β (higher value of azimuthal number “l”),a large number of resonant particles contribute their energy to the wave.It is also well-known that larger values of “β”correspond to the planar waves.

Figure 1.Real frequency of twisted IA waves “ωia” versus normalized wave number “k” plots at different values of twist parameter“β”,i.e.β=0.7 (black dashed curve),β=1.0 (red solid curve),β=1.5 (blue dotted curve) and β=50 (green dashed curve) at Δ=100.

Figure 2.Contour plots of instability growth rate against wave number and azimuthal quantum number (l) via twist parameter β for ve0/Cia=10.8 and Δ=95.

Therefore,it can be expected that twisted waves exhibit more instability than planar waves.In our present problem,the source of free energy is the streaming speed of inertial electrons.In the absence of streaming speed,the transmission of energy from resonant species to the wave becomes almost zero and twisted ion-acoustic waves show damping for the whole range of β.The contour plots also highlighted another interesting fact that for shorter wavelengths (longer wave numbers),the damping rate vanishes and only growth rate of instability appears.This implies that the threshold limit,i.e.equation (13) for unstable waves is modified significantly with the twist parameter and wave number k.

Figure 3 depicts the growth rate “γia” plotted against the twist parameter “β” and wave number k illustrating the changes in instability for three distinct values of the streaming speed “ve0”.The amplitude of the unstable waves first increases with wave number,reaches at the maximum value and then starts to decrease [all curves show almost same trend].It can be observed that twisted IA waves gain more energy from the resonant plasma species for larger values of electron’s streaming speed.

In plasmas,kinetic effects associated with the streaming particles,such as Landau damping and wave-particle resonances,become more significant at higher streaming speeds.These effects can modify the dispersion properties of the acoustic waves,potentially leading to increased instability.This observation confirms that the free energy source available for the excitation of twisted ion-acoustic waves is streaming speed.Figure 3 also reveals that the maxima of instability growth rate is shifted towards right with streaming speed (see 3D curve with green grid lines).

Figure 3.Three dimensional plots of instability growth rate against twist parameter and normalized wave number at ve0=20.8Cia(upper curve with green grid lines),ve0=15.8Cia (middle curve with blue grid lines) and ve0=10.8Cia (bottom curve with red grid lines) at Δ=95.

Figure 4 indicates the growth rate of twisted IA waves against normalized wave number at distinct values of streaming speed for the case where k ＜ lqφ,so that β=k/lqφ=0.1.Here,we have noticed that twisted IA waves are Landau damped in longer wavelength limit while they suffer growth rate of instability for shorter wavelengths.

Thus,a cutoff exists at certain value of wave number above which waves become unstable instead of exhibiting damping.It can be seen that the threshold limit for unstable waves reduces by increasing the streaming speed.This shows that waves are more unstable when larger number of electrons with high streaming speeds are available.It is important to note that the increase in streaming speed of particles can influence the instability of acoustic waves through various mechanisms,including Doppler shifts,resonant interactions,specific instability mechanisms and kinetic effects.The specific nature and magnitude of the instability depend on the characteristics of the system and the interplay between the streaming particles and the acoustic waves.

Figure 4.Growth rate of instability “γia” versus wave number “k”plots for distinct values of streaming speed ve0,i.e.ve0/Cia=7.0(black dashed curve),ve0/Cia=7.5 (red dotted curve),ve0/Cia=8.0(green solid curve) and ve0/Cia=8.5 (blue dashed curve) at fixed β=0.1 and Δ=95.

A comparison of the plots of growth rate of twisted IA waves for (a) single species of electrons and (b) two species of electrons is presented in figure 5.The significant impact of an extra species of electrons on the dynamics of twisted IA waves is observed.An enhancement in the magnitude of unstable wave is identified in the presence of two species electrons.It demonstrates that twisted IA waves are more unstable in a two-species electron plasma than in a singlespecies electron plasma.

4.Conclusion

To conclude,we have investigated the instability of IA waves with finite OAM states in a plasma which is collisionless and magnetic-field free.The solutions of the model equations of kinetic theory are obtained with Laguerre-Gaussian (LG) profiles of perturbed quantities.The frequency and growth rate of IA plasma waves with non-zero OAM states were obtained and numerically analyzed.It is highlighted that the wave characteristics and instability of low frequency IA twisted wave are significantly modified with finite OAM states via twist parameter β(=k/lqφ).The analysis illustrates a clear threshold at β=10,signifying that when the value of β surpasses this threshold,the IA wave experiences Landau damping.Conversely,when β is below this critical value,the IA wave exhibits instability,characterized by significant growth in its amplitude.

Moreover,the enhanced instability of twisted IA waves is attributed to the increase in electron drift speed,which serves as a recognized source of instability.The results indicate that when the electron drift speed ve0is at 20.8 times the ion-acoustic phase velocity Cia,the twisted ion-acoustic waves exhibit a higher degree of instability.Furthermore,a comparison of instability growth rate of twisted IA waves for(a) single species of electrons and (b) two species of electrons is presented to showing the significant modification in the magnitude of growth rate in the presence of an extra electron’s species.The presence of angular momentum and co-rotating solar wind associated with plasma species,as identified in the study of comet Halley [22],suggests that there are dynamic plasma phenomena occurring in the vicinity of the comet.These findings open the possibility for further investigations to explore and expand our understanding of the plasma environment surrounding comet Halley.Our results may have relevancy in that environment.Furthermore,the study also has its relevance to the plasma waves propagating in the spinning regions of space plasmas such as those found in the Earth’s magnetosphere,planetary magnetospheres and interplanetary medium.

Figure 5.Contours of instability growth rate “γia” against wave number and azimuthal quantum number (l) via twist parameter β for (a)single species of electron and (b) two species of electrons.