Pramana – J. Phys. (2017) 88: 23 c Indian Academy of Sciences DOI 10.1007/s12043-016-1332-5

Nonlinear in –positron–ion plasmas including charge separation

A MUGEMANA1, S MOOLLA1,∗ and I J LAZARUS2

1School of Chemistry and Physics, University of KwaZulu-Natal, Durban 4000, South Africa 2Department of Mathematics, Statistics and Physics, Durban University of Technology, Durban 4000, South Africa ∗Corresponding author. E-mail: [email protected]

MS received 11 September 2015; revised 10 June 2016; accepted 20 July 2016; published online 4 January 2017

Abstract. Nonlinear low- electrostatic waves in a magnetized, three-component consisting of hot , hot positrons and warm ions have been investigated. The electrons and positrons are assumed to have Boltzmann density distributions while the motion of the ions are governed by fluid equations. The system is closed with the Poisson equation. This set of equations is numerically solved for the electric field. The effects of the driving electric field, ion temperature, positron density, ion drift, Mach number and propagation angle are investigated. It is shown that depending on the driving electric field, ion temperature, positron density, ion drift, Mach number and propagation angle, the numerical solutions exhibit waveforms that are sinusoidal, sawtooth and spiky. The introduction of the Poisson equation increased the Mach number required to generate the waveforms but the driving electric field E0 was reduced. The results are compared with satellite observations.

Keywords. Nonlinear waves; low frequency; ion-acoustic waves.

PACS Nos 52.35.Qz; 52.35.Fp; 52.35.Mw

1. Introduction these waves is the burstiness, i.e. rapid changes in the amplitudes or of the order of a few The existence of electron–positron–ion (e–p–i) triplets milliseconds to a few hundreds of milliseconds [8]. in most astrophysical environments attracts the atten- Further, the observations show that the electrostatic tion of many researchers because of their potential solitary waves (ESW) identified are parallel propaga- relevance to space plasmas [1–4]. The study of both lin- ting waves relative to the ambient magnetic field and ear and nonlinear propagation in e–p–i plasmas exhibit large amplitude, spiky behaviour. It was shown plays a vital role in understanding different types of that the nonlinear coupling between the ion cyclotron collective processes in space plasmas. The properties and ion acoustic modes lead to the generation of par- of wave motion in the presence of heavy ions are signif- allel electric fields with the periods of the waves vary- icantly different from those in electron–positron (e–p) ing from the ion cyclotron range to the ion acoustic plasmas [5,6]. The presence of ions leads to the exis- tence of several low-frequency waves which otherwise range [13,14]. Ion-acoustic solitons in e–p–i plasmas do not propagate in e–p plasmas [7]. under different regimes and models have been studied Recently, it has been suggested that the nonlinear by different researchers. Kourakis et al [15,16] have study of wave propagation can be helpful to under- shown the existence of envelope structures of solitons stand nonlinear structures similar to the broadband and holes in e–p–i plasmas. Dubinov and Sazonkin electrostatic noise (BEN) observed in the Earth’s mag- [17] investigated the nonlinear theory of ion-acoustic netosphere by numerous satellites (spacecrafts) such as waves in plasmas with cold ions and inertialess isother- Geotail [8], Polar [9],Viking [10], FAST [11] and Clus- mal electrons and positrons. By using a gas dynamic ter [12]. The waveform observations with the high time approach and the Bernouilli pseudopotential technique, resolution confirm that one of the common features of they reported that the propagation velocity of a solitary

1 23 Page 2 of 8 Pramana – J. Phys. (2017) 88: 23 wave is always higher than the linear ion sound veloc- structures in the auroral plasma in the presence of an ity. Mahmood et al [18,19] studied arbitrary-amplitude oxygen beam including charge separation. The inclu- solitons propagating obliquely with respect to an exter- sion of charge separation effect tends to, in most cases, nal magnetic field in a homogeneous magnetized e–p–i increase the frequency of oscillation of the nonlinear plasma and found that the amplitude of solitary struc- structures. It was shown that for a weakly magnetized tures increases with increase in positrons. Nejoh [20] plasma, the amplitude of the oscillations are found investigated the effect of ion temperature on the large- to be constant while they are modulated for strongly amplitude ion-acoustic waves in e–p–i plasmas and magnetized plasmas. observed that the ion temperature decreased the ampli- Later, Moolla et al [24] studied nonlinear low- tude and increased the maximum Mach number of the frequency structures in an e–p–i plasma. The nonlinear ion-acoustic waves. electric field structures found were based on the Using fluid theory, Reddy et al [14] and Bharuthram quasineutrality approximation. Bharuthram et al [25] et al [21] studied nonlinear low-frequency waves in an studied the evolution of nonlinear waves in different e–i plasma. Moolla et al [22] studied high-frequency plasmas and showed that the nonlinear waves evolve in nonlinear waves in the Earth’s magnetosphere and a consistent fashion irrespective of the plasma compo- showed that a nonlinear coupling between large-ampli- sition. tude electron cyclotron and electron-acoustic waves In this paper, we extend the work of Moolla et al can account for the high-frequency component of the [24] by including the Poisson equation, thereby allow- field-aligned bipolar electric field pulses observed ing for the charge separation effect and we numerically within the broadband electrostatic noise in the auroral, solve the resulting set of coupled nonlinear equations. polar and magnetotail regions of the Earth’s magneto- The organization of the paper is as follows: In §2, we sphere. The sawtooth and spiky structures found are in present the model and the basic equations, while the agreement with the observations of Ergun et al [11]. numerical solutions are discussed in §3. Finally, in §4, Moolla et al [23] studied nonlinear low-frequency the results are briefly summarized.

0.015 (a) 0.01 0.005

Ε 0 -0.005 -0.01 -0.015 0.15 (b) 0.1 0.05

Ε 0 -0.05 -0.1 -0.15 0.25 0.2 (c) 0.15 0.1 0.05

Ε 0 -0.05 -0.1 -0.15 -0.2 -0.25 0.4 0.3 (d) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0 10 20 30 40 50 60 S

Figure 1. Numerical solution of the normalized parallel electric field for M = 2.5, δi = 0.0, (ni0/ne0) = 0.5, (Ti/Th) = ◦ 0.0, R = 3.0, θ = 2 and E0 = 0.01 (a), 0.1(b), 0.2(c)and0.3(d). Pramana – J. Phys. (2017) 88: 23 Page 3 of 8 23

2. Model and basic equations ∂vix ∂vix 1 ∂pi εie ∂ϕ +vix + =− +εiiviysin θ, ∂t ∂x nimi ∂x mi ∂x We consider a collisionless, magnetized three-compo- (2) nent plasma consisting of warm ions (i), hot positron ∂v ∂v iy + v iy = ε  v θ − ε  v θ, (ph) and hot electron species (eh). The ion species are ∂t ix ∂x i i izcos i i ixsin (3) drifting along the magnetic field with speed v and the 0 ∂v ∂v wave propagation is taken to be in the x direction at an iz + v iz =−ε  v θ, ∂t ix ∂x i i iycos (4) angle θ to the magnetic field B0, which is assumed to be in the x–z plane, as shown here. ∂p ∂p ∂v i + v i + p ix = . ∂t ix ∂x 3 i ∂x 0 (5) The electrons and positrons are assumed to follow Boltzmann distribution with equal temperature Th and equilibrium densities ne0 and np0, respectively, as   eϕ neh = ne0 exp , (6) Th   eϕ nph = np0 exp − , (7) Th The continuity and momentum equations for the Equations (1)–(7) are closed with the Poisson equa- positive ions are expressed as follows: tion 2 ∂ni ∂nivix ∂ ϕ + = 0, (1) ε =−e(nph − neh + ni). (8) ∂t ∂x 0 ∂x2

0.4 0.3 (a) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (b) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (c) 0.2 0.1 0 Ε -0.1 -0.2 -0.3 -0.4 -0.5 0.4 0.3 (d) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0 10 20 30 40 50 60 S

Figure 2. Numerical solution of the normalized parallel electric field for E0 = 0.3, M = 2.5, R = 3.0, δi = 0.0, ◦ (ni0/ne0) = 0.5, θ = 2 and (Ti/Th) = 0.0(a), 0.05 (b), 0.15 (c) and 0.2 (d). 23 Page 4 of 8 Pramana – J. Phys. (2017) 88: 23

In eqs (1)–(8), i = eB /mi is the ion cyclotron fre- In the limit, 0   quency, εi =+1(−1) for i = c(h), ni is the ion 2 2 2 2 2 2 i + A  4A i cos θ, (10) density, vix, viy and viz are the components of the ion we obtain two modes from eq. (9). The first mode is velocity along the x,y and z directions, respectively, the ion cyclotron mode given by pi is the ion pressure, ϕ is the electrostatic potential A22 2 θ and mi is the ion mass. 2 2 2 i cos ω+ = i + A − (11) 2 2 i + A 2.1 Linear analysis and the second mode is the ion-acoustic mode given by A22 2 θ 2 i cos First, we investigate the linear modes of our system. ω− = . (12) 2 + A2 Equations (1)–(8) are linearized and yield the following i dispersion relation: The two modes obtained for this e–p–i plasma have similar characteristics as the electron cyclotron and 2 1 2 2 ω = (i + A ) electron acoustic waves investigated by Lazarus et al 2 [26] for a four-component, two-temperature electron– 1 2 2 2 2 2 2 1/2 ± [(i + A ) − 4A i cos θ] , (9) positron plasma. 2 2 2 In the long wavelength limit (k λDh  1),the where A2 parameter can be reduced to k2v2 ne 2 2 2 ia A2 = k2v2 + 0 k2v2 A = 3k vti + , 3 ti ia (13) 2 2 np + ne k λDh + (np0/ne0) + 1 0 0 and the dispersion relation (9) is consistent with the v = (n m/n m )1/2v ia i0 e0 i th is the ion acoustic speed, findings of Moolla et al [24], where they studied λ = (ε T /n e2)1/2 Dh √0 h e0 is the electron Debye length a three-component plasma by using quasineutrality and vti = Ti/mi is the ion thermal speed. approximation.

0.4 0.3 (a) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (b) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (c) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (d) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0 10 20 30 40 50 60 S ◦ Figure 3. Numerical solution of the normalized parallel electric field for E0 = 0.3, M = 2.5, R = 3.0, δi = 0.0, θ = 2 , (Ti/Th) = 0.0and(np0/ne0) = 0.3(a), 0.35 (b), 0.4 (c) and 0.5 (d). Pramana – J. Phys. (2017) 88: 23 Page 5 of 8 23   2.2 Nonlinear analysis ∂E n n = M2R2 e0 p0 −ψ − ψ + n , ∂s n n e e in (15) In the nonlinear regime, we transform eqs (1)–(5) and i0 e0 3 (8) to a stationary frame through a variable s = (x − ∂nin nin[−E − Mviyn sin θ] = , (16) vt)/(v/i). We replace ∂/∂t by −i(∂/∂s) and ∂/∂x ∂s 2 2 (ni0/ne0) (M − δi) − 3ninpin by (i/v)(∂/∂s) in eqs (1)–(5) and (8), and we define 2 ∂pin 3pinnin[−E − Mviyn sin θ] the electric potential ψ = eϕ/Th and electric field E = = , (17) ∂s (ni /ne )2 (M − δi)2 − 3ninpin −∂ψ/∂s. In addition, we assume that point quasineu-  0  0 n + n = n s = ∂v n n M trality ( i0 p0 e0) is applicable only at 0. iyn = e0 in ∂s n (M − δ ) Before proceeding with the analysis, all parameters are i0  i    normalized as follows: the velocities√ with respect to ni0 (M−δi) × sin θ M− −vizncos θ , (18) Cs = Th/mi the ion thermal velocity , the densities ne0 nin with respect to the total unperturbed electron density ∂vizn ninMviyn cos θ ne0, pressures with respect to ne0Th and potential with = , (19) ∂s (ni /ne )(M − δi) Th/e. This results in the following set of differential 0 0 equations: where δi = v0/Cs is the ion drift, M = v/Cs is the Mach number and R = ωpi/i. The additional ∂ψ n =−E, (14) subscript introduced in eqs (15)–(19) indicates nor- ∂s malized quantities.

0.4 0.3 (a) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (b) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (c) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (d) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (e) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0 10 20 30 40 50 60 S ◦ Figure 4. Numerical solution of the normalized parallel electric field for E0 = 0.3, M = 2.5, R = 3.0, θ = 2 , (Ti/Th) = 0.0, (ni0/ne0) = 0.5andδi =−0.20 (a), −0.10 (b), 0.0 (c), 0.10 (d) and 0.20 (e). 23 Page 6 of 8 Pramana – J. Phys. (2017) 88: 23

3. The numerical results number to generate waveforms, but for a slightly lower driving amplitude, to obtain spiky structures. Using the Runge–Kutta method, the set of nonlinear differential equations (14)–(19) is solved numerically. 3.2 Effect of ion temperature The following initial conditions are used: At s = 0, ψ = 0, E = E0, nin = ni0/ne0, pin The effect of ion–electron temperature ratio Ti/Th on = ni0Ti/ne0Th,butviyn0 and vizn0 are calculated the parallel electric field for the following fixed param- self-consistently. The results are discussed next. eters M = 2.5, δi = 0.0, (ni0/ne0) = 0.5, R = 3.0 and θ = 2◦ is represented in figure 2. It is found that increasing Ti/Th from 0.0 to 0.2, results in the . τci . τci 3.1 Effect of the driving amplitude, E0 period of the waves increasing from 3 44 to 3 62 . The graphs (a)–(d) in figure 2 show that the variation For fixed parameters M = 2.5, δi = 0.0, (ni0/ne0) = of ion–electron temperature ratio does not affect the ◦ 0.5, (Ti/Th) = 0.0, R = 3.0andθ = 2 ,we nonlinearity of the waves. vary the driving electric field. The results are shown in figure 1. By increasing the driver strengths, the period 3.3 Effect of positron density of oscillations increases from 1.06τci to 3.44τci,where τci = 2π/i is the ion cyclotron period. We observe We next investigate the effect of positron density on the transition from ion-cyclotron waves to ion-acoustic the electric field. It is seen from figure 3 that as the waves. Bharuthram et al [21] and Moolla et al [24] positron density increases from 0.3 to 0.5, the wave- found that the driving field strength for the onset of forms become more nonlinear and the period of these spiky structures were 1.1 and 0.3, respectively and waveforms increases from 1.94τci to 2.83τci. Similar in our study this value is reduced to 0.2. Thus, the results have been found by Moolla et al [24]. Thus, for introduction of Poisson equation increases the Mach both quasineutral and charge separated plasma of this

0.4 0.3 (a) 0.2 0.1 0 Ε -0.1 -0.2 -0.3 -0.4 -0.5 0.4 0.3 (b) 0.2 0.1 0 Ε -0.1 -0.2 -0.3 -0.4 -0.5 0.4 0.3 (c) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (d) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0 10 20 30 40 50 60 S ◦ Figure 5. Numerical solution of the normalized parallel electric field for E0 = 0.3, R = 3.0, δi = 0.0, θ = 2 , (Ti/Th) = 0.0, (ni0/ne0) = 0.5andM = 2.2(a), 2.3 (b), 2.4 (c) and 2.5 (d). Pramana – J. Phys. (2017) 88: 23 Page 7 of 8 23 type, the increase in positron density enhances nonlin- 3.5 Effect of Mach number earity making spiky structures easier to generate. This M behaviour can be due to the fact that the positrons are Figure 5 shows the effect of Mach number on the electrostatic waves when E0 = 0.3, R = 3.0, δi = 0.0, much lighter than the ions, resulting in a plasma of θ = ◦ (T /T ) = . (n /n ) = . less heavier particles. Hence, a weaker driving elec- 2 , i h 0 0and i0 e0 0 5. With a variation of M from 2.2 to 2.5, the period of oscil- tric field is required to accelerate lighter particles to the lations slightly decreases from 3.45τci to 3.36τci and nonlinear regime. the nonlinearity of the waves is not affected. This trend can be explained by the fact that increasing the wave 3.4 Effect of ion drift speed causes an increase in the wave frequency and consequently a decrease in the wave period. The effect of ion drift velocity on the electric field structures is shown in figure 4. It is seen that when 3.6 Effect of propagation angle δi < 0 (antiparallel ion drift), the periods of the waves δ > are significantly higher than when i 0 (parallel For E0 = 0.3, R = 3.0, δi = 0.0, M = 2.5, drift). However, the nonlinearity is unaffected and all (Ti/Th) = 0.0and(ni0/ne0) = 0.5, the curves rep- waveforms are spiky in nature. These results are sim- resenting different values of propagation angle θ are ilar to those of Moolla et al [24] and agree with represented in figure 6. This figure shows that the the satellite observations. These observations show period of oscillations slightly decreases from 3.45τci to that the periods of the BEN structures change signif- 3.34τci with increasing θ. The observed increase in fre- icantly, implying that the ion drift changes rapidly. quency with increasing θ is due to the transition from This means that they are accelerated in bursts lead- the ion-acoustic to the higher frequency ion-cyclotron ing to rapid changes in their periods (from parallel to mode. Here, we notice that the variation of propagation antiparallel drift), consistent with the features of BEN angle from 2◦ to 80◦ does not affect the nonlinearity of [22,27]. the wave structures.

0.4 0.3 (a) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (b) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (c) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0.4 0.3 (d) 0.2 0.1

Ε 0 -0.1 -0.2 -0.3 -0.4 0 10 20 30 40 50 60 S

Figure 6. Numerical solution of the normalized parallel electric field for E0 = 0.3, M = 2.5, R = 3.0, (Ti/Th) = 0.0, ◦ ◦ ◦ ◦ (ni0/ne0) = 0.5, δi = 0.0andθ = 2 (a), 25 (b), 50 (c) and 80 (d). 23 Page 8 of 8 Pramana – J. Phys. (2017) 88: 23

4. Summary [11] R E Ergun, C W Carlson, J P McFadden, F S Mozer, G T Delroy, W Peria, C C Chaston, M Temerin, R Elphic, R We have studied nonlinear low-frequency waves in Strangeway, R C Pfaff, A Cattell and A Cattell, Geophys. Res. Lett. 25, 2025 (1998) e–p–i plasma including charge separation. By using [12] S Pickett, J D Menietti, D A Gurnett, B T Tsurutani, P fluid equations for the warm ions with Poisson’s equa- Kintner, E Klatt and A Balogh, Nonlinear Process. Geophys. tion, nonlinear electrostatic waves have been investi- 10, 3 (2003) gated in a plasma consisting of Boltzmann electrons [13] M Temerin, K Cerny, W Lotko and F S Mozer, Phys. Rev. and positrons. The effects of driving electric field, ion Lett. 48, 1175 (1982) temperature, positron density, ion drift velocity, Mach [14] R V Reddy, G S Lakhina, N Singh and R Bharuthram, Nonlinear Process. Geophys. 9, 25 (2002) number and propagation angle were studied. The elec- [15] I Kourakis, F Verheest and N Cramer, Phys. Plasmas 14(22), tric fields of the nonlinear waves were investigated and 1 (2007) we have shown that for high positron density, the spiky [16] I Kourakis, W M Moslem, U S Abdelsalam, M R Sabry and structures are easier to generate. Our model also shows P K Shukla, Plasma and Fusion Research 4, 1 (2009) that the Mach number and the angle of propagation do [17] A E Dubinov and M A Sazonkin, Plasma Phys. Rep. 35(1), 14 (2009) not affect the nonlinearity of wave structures. [18] S Mahmood, H Mushtaq and H Saleem, New. J. Phys. 5, 289 (2003) [19] S Mahmood and N Akhtar, Eur. Phys. J. D 49, 217 (2008) References [20] Y N Nejoh, Phys. Plasmas 3, 1447 (1996) [21] R Bharuthram, R V Reddy, G S Lakhina and N Singh, Phys. [1] B Kozlovsky, R J Murphy and G H Share, Astrophys. J. 604, Scr. 98, 137 (2002) 892 (2004) [22] S Moolla, R Bharuthram, S V Singh and G S Lakhina, [2] H R Miller and P J Witta, Active galactic nuclei (Springer- Pramana – J. Phys. 61(6), 1209 (2003) Verlag, Berlin, 1987) [23] S Moolla, R Bharuthram, S V Singh, G S Lakhina and R V [3] F C Michel, Rev. Mod. Phys. 54, 1 (1982) Reddy, Phys. Plasmas 17, 022903 (2010) [4] W H Lee and E R Ruiz, Astrophys. J. 632, 421 (2005) [24] S Moolla, I J Lazarus and R Bharuthram, J. Plasma Phys. [5] F B Rizzato, J. Plasma Phys. 40(22), 289 (1988) 78(5), 545 (2012) [6] S I Popel, S V Vladimirov and P K Shukla, Phys. Plasmas 2, [25] R Bharuthram, S V Singh, S K Maharaj, S Moolla, I J 716 (1995) Lazarus, R V Reddy and G S Lakhina, J. Plasma Phys. 80, [7] R P Hamid, Phys. Lett. A 373, 847 (2009) 825 (2014) [8] H H Matsumoto, T Kojima, Y Miyatake, M Omura, I Okada, [26] I J Lazarus, R Bharuthram, S V Singh, S R Pillay and G S Nagano and M Tsutsui, Geophys. Res. Lett. 21, 2915 (1994) Lakhina, J. Plasma Phys. 78(6), 621 (2012) [9] F S Mozer, R Ergun, M Temerin, C Cattell, J Dombeck [27] H Kojima, H Matsumoto, T Miyatake, I Nagano, A and J Wygant, Phys. Rev. Lett. 79, 1281 (1997) Fujita, L A Frank, T Mukai, W R Paterson, Y Saito, S [10] R G Boström, B Gustafsson, G Holback, H Holmgren and P Machida and R R Anderson, Geophys. Res. Lett. 21, 2919 Koskinen, Phys. Rev. Lett. 61, 82 (1988) (1994)