A Model for Periodic Nonlinear Electric Field Structures in Space Plasmas

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1 Commun. Theor. Phys. (Beijing, China) 52 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 1, July 15, 2009 A Model for Periodic Nonlinear Electric Field Structures in Space Plasmas M.N.S. Qureshi, 1,2, SHI Jian-Kui, 2 and LIU Zhen-Xing 2 1 Department of Physics, GC University, Lahore 54000, Pakistan 2 State Key Laboratory of Space Weather, CSSAR, Chinese Academy of Sciences, Beijing , China (Received November 19, 2008) Abstract In this study, we present a physical model to explain the generation mechanism of nonlinear periodic waves with a large amplitude electric field structures propagating obliquely and exactly parallel to the magnetic field. The Sagdeev potential from the MHD equations is derived and the nonlinear electric field waveforms are obtained when the Mach number, direction of propagation, and the initial electric field satisfy certain plasma conditions. For the parallel propagation, the amplitude of the electric field waves with ion-acoustic mode increases with the increase of initial electric field and Mach number but its frequency decreases with the increase of Mach number. The amplitude and frequency of the electric field waves with ion-cyclotron mode decrease with the increase of Mach number and become less spiky, and its amplitude increases with the increase of initial electric field. For the oblique propagation, only periodic electric field wave with an ion-cyclotron mode obtained, its amplitude and frequency increase with the increase of Mach number and become spiky. From our model the electric field structures show periodic, spiky, and saw-tooth behaviours corresponding to different plasma conditions. PACS numbers: Mw, Cq Key words: nonlinear waves, electric field solitary structures 1 Introduction During the last decades or so there have been many satellite observations in space plasma where large amplitude nonlinear periodic electric field structures or wave packet-like electrostatic structures have been observed. [1 7] The observed waves propagate at speeds much less than the electron thermal speed, suggesting that they may be related to the nonlinear ion electrostatic waves. The measurements in the space with satellites, such as S3-3, [8] Viking, [9] Polar, [10,11] and Fast, [12] indicate that nonlinear electric field structures are often accompanied with localized nonlinear plasma waves. Figure 1 shows a measurement of large amplitude nonlinear periodic ion-cyclotron electric field structures in earth s auroral acceleration region by Polar spacecraft. [10] Indeed, the existence of solitary waves in an unmagnetized plasma has been studied extensively. [13 16] It is also of great interest to determine how a magnetic field affects the propagation of these waves. Several authors studied the nonlinear electrostatic ion-cyclotron (EIC) and ion-acoustic waves in magnetized plasma [17 19] with ideal Magneto Hydro Dynamic (MHD) equations. The results showed that nonlinear periodic density waves could exist in the space plasma. Also, some studies were restricted to the weakly nonlinear theories (not fully nonlinear study) such as BGK (Bernstein Greene Kruskal) and the KdV (Kortweg de Vries) equation. The fully nonlinear Sagdeev pseudopotential analysis has been highly successful in explaining the existence of the electrostatic density soliton in plasmas with different particle compositions. [20] In this paper, large amplitude nonlinear periodic electric field waves are studied theoretically with MHD equations and some properties of the waves are given. Fig. 1 Large amplitude nonlinear periodic electric field structures observed by Polar spacecraft in earth s auroral acceleration region [Mozer et al., 1997]. 2 Physical Model Let us consider a non-isothermal (T i T e ) plasma in which waves can be excited satisfying the following ambient conditions. Supported by National Natural Science Foundation of China under Grant Nos and , and the Specialized Research Fund for State Key Laboratories Corresponding author, nouman sarwar@yahoo.com

2 150 M.N.S. Qureshi, SHI Jian-Kui, and LIU Zhen-Xing Vol. 52 The fluid consists of electrons and ions, and plasma β 1, which implies that plasma pressure is much smaller than the magnetic pressure. The magnetic field is directed along the Z-axis, i.e., B = B 0 e Z (B 0 is the constant background field and e Z is the unit vector along the Z-axis). In consequence, we only consider electrostatic waves and the magnetic field will be passively taken into account in the gyrofrequency. The Phase velocity satisfies v Ti v p /γ v Te therefore, the Landau damping can be neglected. Here v α = (2T α /m α ) 1/2 is the particle thermal velocity, T α and m α are particle energy and mass; subscript i denotes the ions and e denotes the electrons; v p is phase velocity, θ is the angle between the wave vector K and e Z, and γ = cosθ. The wave scale λ λ D, where λ D is the Debye radius, so the charge separation effects can be neglected and the quasi-neutrality condition is may be applied, i.e., n n e n i. Here, n is the particle number density. For such low-frequency ω Ω e waves (here, Ω e is the electron gyrofrequency in the magnetic field), the electron inertia may be neglected because the electron mass is much smaller than the ion mass. The basic two-fluid equations for ions governing the dynamics of motion of such a plasma composition are then ion continuity equation and momentum equation. n + (nv) = 0, (1) t v 1 + (v )v = t mn p e m ϕ + v Ω i, (2) ( eϕ ) n = n i n e n 0 exp, (3) T e p = nt i, (T i = const.), (4) where v, and m are the ion velocity and mass, respectively, e is the elementary charge, p is the thermal pressure, φ is the electric potential, n 0 is constant background density, Ω i = (eb 0 /m i c)ẑ is the ion gyro-frequency, c is the speed of light, and B 0 is the background magnetic field. In Eq. (3), quasi-neutrality is assumed and electron inertia is neglected which is valid for waves with frequency ω Ω e. 3 Oblique Propagation Assuming all the variations in the x-z plane we introduce the following dimensionless quantities: N = n/n 0, τ = Ω i t, R = r/ρ i, Z = z/ρ i, V = v/c s, Φ = eφ/t e, M = v p /C s, and v p = ω/k (here ρ i is the ion gyro-radius). And by considering the oblique propagation with respect to the external magnetic field, we look for solutions of Eqs. (1) (4) that depend on x, z, and t through the variable S, as S = (k x x + k z z ωt) Ω i /ω = (αx + γz τm)/m, (5) where α = sin θ, γ = cosθ, k x and k z are the components of K in the direction of x and z, respectively. The Mach number M gives the velocity of the wave along the S direction. Here, we choose S as the variable, that means the wave structure is investigated in the coordinate system moving together with the wave. Normalizing Eqs. (1) (4) and then integrating using the boundary condition, when N = 1, V = 0, we get the energy integral for the classical particles in a potential well as where Ψ(N) = 1 2 ( dn ds N 4 [ 2(aN 2 /M 2 1) 2 2N 2 a M ) 2 + Ψ(N) = 0, (6) 2 (1 N + lnn) + (N 1) 2( ) 1 + γ 2 N 2 a2 M 4 2Nγ 2 a M 2 (1 N + N lnn) N 2( a ) 2E ] M (7) Here, a = 1+T i /T e and E 0 is the initial value of E. Equation (6) is analogous to the energy integral of a classical particle in a 1-D potential well and Ψ(N) is called the Sagdeev potential. According to the definition of electric field, the normalized electric field can be written as E = dφ ds = 1 dn N ds. (8) Therefore, by using Eq. (13) we can write E = ± 2 Ψ(N). (9) N2 From Eq. (6), we can analyze the nonlinear plasma density solution and from Eq. (9), we can analyze the electric field waveform. The solution for nonlinear large amplitude density waves from Eq. (6) and the corresponding electric field waveforms from Eq. (9) can be obtained numerically when the parameters lead to a Sagdeev potential Ψ(N) < 0. Our numerical solutions show that the waveform will be different for different plasma conditions and we obtain two types of ion-cyclotron density waves: (i) when a/m 2 < 1 and E 0 = 0, and (ii) when a/m 2 = 1 and E 0 = 0, which are discussed in the following sections. 3.1 Nonlinear Waves with E 0 = 0 and a/m 2 < 1 By analyzing the property of the Sagdeev potential the condition a/m 2 < 1 and E 0 = 0 there will be 1 < N 1 < N 2 and Ψ(N) has the properties: Ψ(N 1 ) = 0, Ψ(N 2 ) = 0, Ψ (N 1 ) < 0, Ψ (N 2 ) > 0, and Ψ(N) < 0 for N 1 < N < N 2. In this case Eq. (6) has a solution corresponding to density periodic waves. It follows from the properties of Ψ(N) that density periodic waves (N 2 > N 1 > 1) in the number density profile can occur when a/m 2 < 1 and E 0 = 0. These density periodic waves are identified as ion-cyclotron waves as the wave speed V p is greater than C s i.e., M > 1. For these density waves, the waveform becomes sharp and its amplitude and frequency increase with the increase of Mach number M. We can obtain the electric field waveform from Eq. (9) corresponding to

3 No. 1 A Model for Periodic Nonlinear Electric Field Structures in Space Plasmas 151 the density waves for the same plasma condition. For this condition, the electric field waveform exhibits the ioncyclotron oscillations whose amplitude increases with the Mach number M and becomes spikier. 3.2 Nonlinear Waves with E 0 = 0 and a/m 2 = 1 By analyzing the properties of the Sagdeev potential the condition a/m 2 = 1 and E 0 = 0 there will be 1 < N 1 < N 2 and Ψ (N) has the solution corresponding to periodic density waves. A typical example of corresponding density periodic waves is show in Figs. 2(a) and 2(b). Figure 2(a) depicts the numerical solution of the density periodic waves for a/m 2 = 1.0, γ = 0.4, and E 0 = 0 and Fig. 2(b) represents the numerical solution of the density periodic waves for a/m 2 = 1.0, γ = 0.7, and E 0 = 0. We can see that the density wave has a maximum value of about 2.56 and a minimum value of about 1.02 in Fig. 2(a) and a maximum value of about 10.1 and a minimum value of about 1.0 in Fig. 2(b), respectively. We note that with the increase of γ, the waveform of density waves becomes sharp and its amplitude and frequency increases. Fig. 2 The upper two panels show the density waves for a fixed value of M = 1.04 when (a) γ = 0.4 and (b) γ = 0.7. The amplitude and the frequency of the density waves increase with the decrease in angle of propagation and the structures become more spiky. The lower two panels show the saw-tooth like electric field waveform for the corresponding density waves in the upper two panels for the same parameters. We can obtain the electric field waveform from Eq. (9) when a/m 2 = 1 and E 0 = 0 corresponding to the density waves. Figure 2(c) shows the variation of the oblique electric field for a/m 2 = 1.0, γ = 0.4, and E 0 = 0 and Fig. 2(d) shows the variation of the oblique electric field for a/m 2 = 1.0, γ = 0.7, and E 0 = 0. It can be seen that the electric field structure has saw tooth form and the oscillation frequency increases with the γ. The maximum value of electric waveform also increases with γ and becomes more spiky as shown in Figs. 2(c) and 2(d). 4 Parallel Propagation When γ = cosθ = 1, then Eq. (7), becomes Ψ(N) = N 4 [ 2(aN 2 /M 2 1) 2 2N 2 a M 2 (1 N + lnn) + (N 1) 2( 1 + N 2 a2 M 4 ) 2N a M 2 (1 N + N lnn) N 2( a ) 2E ] M (10) From the above Eq. (10), we can analyze the nonlinear plasma density solution when the parameters lead to Sagdeev potential Ψ(N) < 0 and from Eq. (9), we can analyze the electric field waveform. Our numerical solutions show that the waveform will be different for different plasma conditions and we obtain two types of density waves: (i) ion-acoustic waves when a/m 2 > 1 and E 0 0, and (ii) ion-cyclotron waves when a/m 2 < 1 and E 0 0, which are discussed in the following sections. 4.1 Nonlinear Waves with E 0 0 and a/m 2 > 1 By analyzing the property of the Sagdeev potential the condition a/m 2 > 1 and E 0 0, there will be N 1 < 1 < N 2 and Ψ (N) has the solution corresponding to periodic density waves. These density periodic waves are identified as ion-acoustic waves as the wave speed V p

4 152 M.N.S. Qureshi, SHI Jian-Kui, and LIU Zhen-Xing Vol. 52 is less than C s i.e., M < 1. In this case when the Mach number M increases for a fixed E 0, the ion-acoustic density periodic wave s amplitude and frequency decrease. We can obtain the electric field waveform from Eqs. (9) and (10) under the same plasma conditions corresponding to the ion-acoustic density waves. The oscillation frequency of electric field waveform slightly decreases with the increase of M but the amplitude of the waveform increases slightly with the increase in M. Detailed characteristics of density and electric field structures can be seen in Fig Nonlinear Waves with E 0 0 and a/m 2 < 1 By analyzing the property of the Sagdeev potential the condition a/m 2 < 1 and E 0 0 there will be N 1 < 1 < N 2 and Ψ (N) has a solution corresponding to density periodic waves. These density periodic waves are identified as ion-cyclotron waves as the wave speed V p is greater than C s i.e., M > 1. An example of corresponding density periodic waves is show in Figs. 3(a) and 3(b). Figure 3(a) represents the numerical solution of the typical density periodic waves for a/m 2 = 0.6 and E 0 = 0.1, and Fig. 3(b) represents the numerical solution of the density periodic waves for a/m 2 = 0.1 and E 0 = 0.1. We can see that the density wave has a maximum value of about and a minimum value of about in Fig. 3(a) and a maximum value of about and a minimum value of about in Fig. 3(b), respectively. We note that with the increase of M for a fixed E 0, the waveform of periodic density waves becomes less sharp and its amplitude and frequency decrease. Fig. 3 The upper two panels show the density waves for the parallel propagation when (a) a/m 2 = 0.6 and (b) a/m 2 = 0.1 for the fix value of E 0 = 0.1. The peak to peak amplitude and the frequency decrease with the increase in E 0 and the waves becomes less steep. The lower two panels show the corresponding periodic electric field waveform for the same parameters which exhibit the same behavior for amplitude but frequency decreases with E 0. We can obtain the electric field waveform from Eqs. (9) and (10) when a/m 2 < 1 and E 0 0 corresponding to the ion-cyclotron periodic density waves. Figure 3(c) shows the variation of the electric field waveform for a/m 2 = 0.6 and E 0 = 0.1 and Fig. 3(d) shows the variation of the electric field for a/m 2 = 0.1 and E 0 = 0.1. It can be seen that the oscillation frequency and the amplitude of the periodic electric field waveform decrease with the increase of Mach number M. 4.3 Characteristics for Parallel Propagation In Fig. 4, the typical characteristics of the ion acoustic density periodic waves and electric field waveforms are shown. In the left panel of Fig. 4, the peak to peak amplitude of the density waves and the amplitude of the electric field waveforms are plotted against the different values of initial electric field E 0 for the fixed value of a/m 2 = We can see that the amplitude of the electric waveform increases linearly with the increase of initial electric field E 0. Amplitude of the density periodic waves also increases linearly with the E 0 but its rate of increasing is higher than the electric field waveform rate. In the right panel of Fig. 4, the peak to peak amplitude of the density waves and the amplitude of the electric field waveforms are plot-

5 No. 1 A Model for Periodic Nonlinear Electric Field Structures in Space Plasmas 153 ted against the different values of the Mach number M for the fixed value of E 0 = 0.2. We can note that the amplitude of both the electric waveform and the density periodic waves remain almost constant for low values of M till about 0.35 and then increases monotonically with the further increase in M. It can be noted from Fig. 4 that the amplitude of the electric field waveform is strongly dependent on the initial electric field for a constant Mach number but there is a slight increase in amplitude with the increase of Mach number when initial electric field remains constant. The same is the case with the amplitude of density periodic waves. Fig. 4 The left panel shows the peak to peak variation of density waves and amplitude of electric field against the different values of E 0 for a/m 2 = 12. Amplitude increases linearly for both the waveforms with E 0. The right panel shows the peak to peak variation of density waves and amplitude of electric field against the different values of M for E 0 = 0.2. Amplitude remains almost constant up to 0.35 and then increases for both the waveforms with the further increase in M. Fig. 5 The left panel shows the peak to peak variation of density waves and amplitude of electric field against the different values of E 0 for a/m 2 = 0.1. Amplitude increases monotonically for both the waveforms with E 0. The right panel shows the peak to peak variation of density waves and amplitude of electric field against the different values of M for E 0 = 0.2. Amplitude decreases for both the waveforms up to 3.5 and then becomes almost constant with the further increase in M. Typical characteristics of the ion cyclotron density periodic waves and periodic electric field waveforms are shown in Fig. 5. In the left panel of Fig. 5, the peak to peak amplitude of the density waves and the amplitude of the electric field waveform are plotted against the different values of initial electric field E 0 for the fixed value

6 154 M.N.S. Qureshi, SHI Jian-Kui, and LIU Zhen-Xing Vol. 52 of a/m 2 = 0.1. We can see that the amplitude of the periodic electric waveform as well as density periodic waves increases monotonically with the increase of initial electric field E 0. In the right panel of Fig. 5, the peak to peak amplitude of the density waves and the amplitude of the periodic electric field waveform are plotted against the different values of Mach number M for the fixed value of E 0 = 0.2. We can see that the amplitude of both the periodic electric waveform and the density periodic waves decreases monotonically for the low values of M till about 3.5 and then becomes almost constant for the further increase in M. 5 Summary and Conclusion In this paper we presented a model which could explain the generation mechanism of large amplitude nonlinear periodic electric field waves in oblique and parallel orientations related to the magnetic field. The Sagdeev potential from the MHD equations is derived to find the solution of the nonlinear periodic electric field waveform. As regarding the satellite observation we are merely looking for electrostatic solutions. The results are as follow. (i) Nonlinear periodic ion-acoustic waves and ioncyclotron density waves, and the corresponding spiky, periodic and saw-tooth electric field waveforms can be obtained when the Mach number M, E 0, and γ = cosθ satisfy certain plasma conditions in the oblique and exactly parallel propagations. (ii) For the oblique propagation when (a) a/m 2 < 1 and E 0 = 0, and (b) a/m 2 = 1 and E 0 = 0, i.e. for ion-cyclotron waves, the amplitude of the periodic density waves and electric waveform increases in both the cases with the Mach number when the angle of propagation remains constant, and with the decrease of angle of propagation when the Mach number remains constant. (iii) For the exactly parallel propagation, in case of ion-acoustic waves when a/m 2 > 1 and E 0 0, the amplitude of both the density waves and electric field waveform increases linearly with the initial electric field when the Mach number remains constant, and it remains constant for low values of Mach number and then increases slowly with the further increase in M for the fixed value of initial electric field. In case of ion-cyclotron waves for exactly parallel propagation when a/m 2 < 1 and E 0 0, the amplitude of both the periodic density waves and the periodic electric waveform increases monotonically with the initial electric field when the Mach number remains constant, but it decreases for the low values of Mach number up to 3.5 and then becomes almost constant with the further increase in M for the fixed value of initial electric field. The periodic and spiky electric field waveform from our model gives a good match with the observed electric field waveform in the different regions of space plasmas in particular with the waveform given in Fig. 1. References [1] H. Matsumoto, H. Kojima, T. Miyatake, et al., Geophys. Res. Lett. 21 (1994) [2] J.R. Franz, P.M. Kintner, and J.S. Picket, Geophys. Res. Lett. 25 (1998) [3] C.A. Cattell, J.R. Wygant, M.K. Hudson, et al., Geophys. Res. Lett. 26 (1999) 425. [4] C. Cattell, C. Neiman, J. Dombeck, et al., Nonlin. Processes Geophys. 10 (2003) 13. [5] R. Pottelette, R.E. Ergun, R.A. Treumann, et al., Geophys. Res. Lett. 26 (1999) [6] J.S. Pickett, J.D. Menietti, D.A. Gurnet, et al., Nonlin. Processes Geophys. 10 (2003) 3. [7] J.S. Pickett, L.J. Chen, S.W. Kahler, et al., Annales Geophysicae 22 (2004) [8] F.S. Mozer, M.K. Cattell, M.K. Hudson, et al., Space Sci. Rev. 27 (1980) 155. [9] M. Andre, H. Koskinen, G. Gustafsson, and R. Lundin, Geophys. Res. Lett. 14 (1987) 463. [10] F.S. Mozer, R. Ergun, M. Temerin, C. Cattell, J. Dombeck, and J. Wygant, Phys. Rev. Lett. 79 (1997) [11] J. Dombeck, C. Cattell, J. Crumley, W. Peterson, H. Collin, and C. Kletzing, J. Geophys. Res. 106 (2001) [12] R.E. Ergun, C.W. Carlson, J.P. McFadden, et al., Phys. Rev. Lett. 81 (1998) 826. [13] H. Washimi and T. Taniuti, Phys. Rev. Lett. 17 (1966) 996. [14] M. Widner, I. Alexeff, W.D. Jones, and K.E. Lonngren, Phys. Fluids 13 (1970) [15] H. Ikezi, Phys. Fluids 16 (1973) [16] B. Buti, Phys. Lett. A 76 (1980) 251. [17] M. Temerin, M. Woldorff, and F.S. Mozer, Phys. Rev. Lett. 43 (1979) [18] M.Y. Yu, P.K. Shukla, and S. Bujarbarua, Phys. Fluids 23 (1980) [19] L.C. Lee and J.R. Kan, Phys. Fluids 24 (1981) 430. [20] F. Verheest, Nonlin. Proc. Geophys. 14 (2007) 49.