Influence of Generalized (r, q) Distribution Function on Electrostatic Waves

Transcription

1 Commun. Theor. Phys. (Beijing, China) 45 (2006) pp c International Academic Publishers Vol. 45, No. 3, March 15, 2006 Influence of Generalized (r, q) istribution Function on Electrostatic Waves M.N.S. Qureshi, 1,2,3 SHI Jian-Kui, 1 and MA Shi-Zhuang 3 1 Key Laboratory of Space Weather, the Chinese Academy of Sciences, Beijing , China 2 epartment of Physics, Government College University, Lahore 54000, Pakistan 3 Graduate University of the Chinese Academy of Sciences, Beijing , China (Received July 11, 2005) Abstract Non-Maxwellian particle distribution functions possessing high energy tail and shoulder in the profile of distribution function considerably change the damping characteristics of the waves. In the present paper Landau damping of electron plasma (Langmuir) waves and ion-acoustic waves in a hot, isotropic, unmagnetized plasma is studied with the generalized (r, q) distribution function. The results show that for the Langmuir oscillations Landau damping becomes severe as the spectral index r or q reduces. However, for the ion-acoustic waves Landau damping is more sensitive to the ion temperature than the spectral indices. PACS numbers: Fp, g Key words: generalized (r, q) distribution function, electrostatic waves, damping 1 Introduction In space plasmas the distribution function often follows an inverse power law in the high-energy tail f v α such as kappa distribution function containing the spectral index κ. 1 4] In the limit κ the distribution approaches the Maxwellian. Vasyliunas 5] appears to have been the first to employ the general form of the kappa distribution function and to note its relation with the Maxwellian. A more generalized distribution function is the generalized (r, q) distribution function, which in the limiting case r 0 and q κ+1 reduces to kappa distribution function and approaches Maxwellian if r 0 and q. 6] On the basis of the theory and observations, it is noted that many space plasmas can be modeled more effectively by kappa or generalized (r, q) distribution function. Plasma waves can experience substantial attenuation during propagation through the tenuous, collisionless plasmas in space. As initially formulated by Landau in his elegant paper, 7] this collisionless damping is caused by resonant interactions with the particles moving with a velocity comparable to wave phase speed. When plasma is magnetized, additional resonant interactions occur associated with multiples of the cyclotron frequency of each species. Treatment of the wave growth or damping under such conditions is more complex, but it remains analogous to what is now termed Landau damping. 1] In this process, the particles, which have velocity nearly matching the wave phase velocity ω/k, exchange energy with the wave. If there are more particles going slower than the phase velocity and thus gaining energy from the wave, than those going faster and losing energy to the wave, then there is net loss of energy and the wave damps. The physical picture developed by O Neil 8] in the limit ω i /ω r 1 is that the bulk of the energy loss of the wave is due to the net increase in energy of these particles as their phase space orbits in the wave frame change from straight lines to rotational orbits, and then phase mix. In this process particles initially going slower than the wave are speeded up and the particles going faster than the wave are slowed down. In the original paper, Landau predicted the damping of longitudinal plasma oscillations in collisionless Maxwellian plasma. Such damping is weak for frequencies near the electron plasma frequency since the wave speed u ω/k is much larger than the electron thermal velocity Ψ. Waves attenuation becomes more severe at higher wave frequencies u/ψ is reduced. Such Landau damping limits electron plasma oscillations to a narrow frequency range just above the electron plasma frequency. However, the presence of high-energy tail component and shoulders in the profile of distribution function can considerably change the rate of resonant energy transfer with plasma waves. In this paper, we illustrate this phenomenon explicitly for the case of Landau damping of both electron plasma waves and ion acoustic wave. We present the solutions for the direct comparison with the previous work 9] in which the solutions are expressed in terms of the plasma dispersion function 10] corresponding to the Maxwellian distribution. In Sec. 2 we briefly describe the generalized (r, q) distribution function and its limiting forms. In Sec. 3, we derive the general dispersion relations for electrostatic waves in a uniform, unmagnetized but fully hot plasma. In Sec. 4 we discuss the numerical results for the damping rates of electron plasma waves and ion acoustic waves and illustrate the role of the ration of the ion temperature (T + ) The project supported by National Natural Science Foundation of China under Grant No and the International Collaboration Research Team Program of the Chinese Academy of Sciences

2 No. 3 Influence of Generalized (r, q) istribution Function on Electrostatic Waves 551 to electron temperature (T ) in determining the damping characteristics of ion-acoustic wave. 2 Generalized (r, q) istribution Function In the present paper we consider the generalized (r, q) distribution function, which is of a more general form of the kappa distribution function having two spectral indices r and q, hence better suited to model the plasmas exhibiting characteristics which cannot be described by the kappa distribution function, and is of the form f (r,q) 3(q 1) 3/2(1+r) Γ(q) 4πΨ ( v 2 ) 1+r] q, (1) Γ (q 3/2(1 + r)) Γ (1 + 3/2(1 + r)) q 1 Ψ 2 Ψ T 3(q 1) 1/(1+r) Γ (q 3/2(1 + r)) Γ (3/2(1 + r)) m Γ (q 5/2(1 + r)) Γ (5/2(1 + r)) is the particle thermal speed, T is the particle temperature, and Γ is the Gamma function. We also note here that in the distribution (1) the spectral indices r and q satisfy the constraints q > 1 and q(1 + r) > 5/2, which arise from the normalization and the definition of temperature for the above distribution, and has been normalized f (r,q) d 3 v 1. In general, if we increase the value of r and fix the value of q, the contribution of the high energy particles reduces but the shoulders in the profile of distribution increases. Similarly, if we fix the value of r and increase the value of q, the result will be the same. 6] 3 ispersion Relation for Electrostatic Waves We follow the standard techniques to derive the dispersion relation for the electrostatic oscillations in the plasma which is initially uniform and unmagnetized with complex wave frequency ω ω r + iω i and real wave number k. We also assume that both electrons and ions may be described by the generalized (r, q) distribution as the Eq. (1) the spectral indices r and q and the thermal speed Eq. (2) can be adjusted to provide a best fit to observations. This procedure yields the propagation and damping characteristics of both electron plasma waves and ion-acoustic waves in the limiting cases when kλ 1 and kλ 1 respectively, λ ( T 4πNe 2 ) 1/2 ( Ψ ω p ) (q 1) 1/(1+r) Γ(5/2(1 + r))γ(q 5/2(1 + r)) 3Γ(3/2(1 + r))γ(q 3/2(1 + r)) is the electron ebye length and ω p is the electron plasma frequency. The general dispersion relation comes as 1 + 2ωp,α 2 A + ξα k 2 Ψ 2 Z (r,q) 1 (ξ α ) ] 0, (4) α α A 3(q 1) 1/(1+r) Γ (q 1/2(1 + r)) Γ (1 + 1/2(1 + r)) 2Γ (q 3/2(1 + r)) Γ (1 + 3/2(1 + r)) (2) (3). (5) The summation is made over all the charge species in the plasma, ω p,α is plasma frequency, and + Z (r,q) 1 1 (ξ α ) B s ξ α q 1 s2(1+r)] q ds (6) is the generalized dispersion function, whose general properties are described by Qureshi et al., 6] ξ α ω/kψ α, s v/ψ α, and 3(q 1) 3/2(1+r) Γ(q) B 4Γ (q 3/2(1 + r)) Γ (1 + 3/2(1 + r)). (7) 3.1 High Frequency Electron Plasma Waves For the longitudinal (Langmuir) waves propagating in unmagnetized, collisionless plasma it is justified to consider the ions as immobile uniform background that simply maintains the charge neutrality (unless T e T i, which will be considered in Subsec. 3.2). Therefore, we omit the ion terms in Eq. (4) and only consider the electron terms. The resulting dispersion relation for high frequency Langmuir oscillations is ξ e Z (r,q) 1 (ξ) + A + Ck 2 λ 2 0, (8) A is given by Eq. (5) and C 3(q 1) 1/(1+r) Γ (3/2(1 + r)) Γ (q 3/2(1 + r)). (9) 2Γ (5/2(1 + r)) Γ (q 5/2(1 + r))

3 552 M.N.S. Qureshi, SHI Jian-Kui, and MA Shi-Zhuang Vol. 45 Approximate solution of Eq. (8) can be found by using the appropriate limiting forms (for ξ e 1) of the generalized plasma dispersion function (6). 10] We solve Eq. (9) under the above approximation and obtain the real and imaginary parts of the dispersion relation as ω r ω p 2 k2 λ 2, (10) ω i ω p ( 3π 4 ) k 3 λ {( 1 q 1 k 2 λ 2 ) r+1 ] q + 3 E}, (11) Γ(q) ( ) ( ) Γ 5/2(1 + r) Γ q 5/2(1 + r) ] 3/2 Γ ( q 3/2(1 + r) ) Γ ( 1 + 3/2(1 + r) ) 3Γ ( 3/2(1 + r) ) Γ ( q 3/2(1 + r) ), (12) E (q 1)1/(1+r) Γ (5/2(1 + r)) Γ (q 5/2(1 + r)). (13) We here note that the real part of the dispersion relation does not contain r or q and is identical to the Maxwellian solution in the long wavelength limit. But the Landau damping rate ω i is strongly dependent on the spectral indices r and q. In the limiting case r 0, q, the damping rate Eq. (11) for the electron plasma waves reduces to the Maxwellian solution as follows: ω i π ) 1/2 1 ( ω p 8 k 3 λ 3 e (1/2k2 λ 2 )+3/2]. (14) 3.2 Low Frequency Oscillations: Ion-Acoustic Waves If the electrons are hot (T e T i ), an electrostatic wave in which ions do play a major role is found at lower frequencies. These waves are characterized by phase velocities lying between the thermal velocities of ions and that of the electrons and propagate only when ω ω p (Note that henceforth, as in Eq. (3), we shall use ω p to mean electron plasma frequency). To include the low frequency oscillations in plasma we include the ion terms in Eq. (4) and obtain the dispersion relations for a single ion as ξ e Z (r,q) 1 (ξ e ) + A + F ]k 2 λ τ 2 A + 1 ( ετ ξ ez (r,q) ξe )] 1 0, (15) ετ A is given by Eq. (5), ε (m e /m i ) 1/2, τ (T i /T e ) 1/2, and F is given as F 2(q 1) 1/(1+r) Γ (5/2(1 + r)) Γ (q 5/2(1 + r)). (16) In Eq. (15) we assume that the spectral indices r and q are the same for ion distribution as those of the electron distribution. Approximate solutions of Eq. (15) can be obtained by assuming ξ e 1 and ξ e /ετ 1 as τ 1, and then to use the power series and asymptotic series expansions for generalized plasma dispersion function (6), respectively. Thus, the real part of the frequency and the damping rate become as ω r εgkλ, (17) ω p (2A + G 2 k 2 λ 2 1/2 ω i πε2 BGkλ ω p (2A + G 2 k 2 λ { ετ ( 1 ) 1+r } q ] q 1 τ 2 (2A + G 2 k 2 λ 2, (18) A and B are given by Eqs. (5) and (7), respectively, and G is ] 1/2 G. (19) (q 1) 1/(1+r) Γ (5/2(1 + r)) Γ (q 5/2(1 + r)) The approximate solutions of Eq. (15) of ion-acoustic branch have already been obtained 9] in the above assumptions and can readily be obtained in the limiting case r 0, q of Eqs. (17) and (18), ω r εkλ, (20) ω p (1 + k 2 λ 2 1/2 ω i π ε 2 kλ { ω p 8 (1 + k 2 λ ετ 3 exp 1 ]} 2τ 2 (1 + k 2 λ 2. (21) 4 Numerical Results We have computed Eq. (11) for the damping of electron plasma waves in the limit kλ 1, which comes from the limiting expansion (ξ e 1) of the generalized plasma dispersion function Eq. (6) for different values of r and q,

4 No. 3 Influence of Generalized (r, q) istribution Function on Electrostatic Waves 553 and have considered the range of kλ from 0.15 to 0.4. It is evident from Eq. (11) that the damping rate for the Langmuir waves is strongly dependent on the spectral indices r and q. Figures 1 and 2 exhibit the general behavior of the damping characteristics for the electron plasma waves modeled by the generalized (r, q) distribution function. In Fig. 1 we have taken r as a fixed value of 1 and changed the value of q as 2, 3, 5, and 10 in the computation. It can be seen that as the value of q decreases the Landau damping increases. It is due to the fact that for the low values of q the high-energy tail component in the distribution function tends to increase and hence, increase the number of high-energy resonant particles. In Fig. 2, q is fixed at 3 and we have changed the values of r as 0.5, 1, and 2. Here we note that as we increase the value of r the damping rate decreases. This is because as we increase the value of r the shoulders in the distribution function become more prominent and the contribution of high-energy particles reduces. Therefore, there is a decrease in the number of high-energy resonant electrons and rate of damping reduces. Furthermore, it can be noted that the rate of Landau damping is more sensitive to the spectral index r than the spectral index q. From Figs. 1 and 2, we notice that the damping rate for Langmuir waves increases as the values of r or q decrease and becomes maximum for the lowest values of r and q. Fig. 1 The damping rate of electron plasma waves for different values of q with r 1. Fig. 2 The damping rate of electron plasma waves for different values of r with q 3. The damping of ion-acoustic waves is represented by Eq. (18). We have computed Eq. (18) for different values of r and q in the limit kλ 1, which comes from the limiting expansion (ξ e 1) of the generalized plasma dispersion function Eq. (6). Figure 3 shows the damping rate of ion-acoustic waves versus kλ in the range kλ 4 to 16, the value of q is fixed at 3 and the value of r is varied as 0.5, 1, and 2. In Fig. 3(a), the graphs are plotted in the limit when ions are cool, i.e., τ 2 T i /T e The rate of damping increases as the value of r increases when kλ 7, but as the value of kλ increases approximately from 7.5 the damping also increases for the decrease in the value of r. Figure 3(b) shows the damping characteristics when ions are hot, i.e., τ 2 T i /T e We can see that as the value of r increases the rate of damping decreases. The comparison of Figs. 3(a) and 3(b) shows that there is a marked enhancement in the damping rate when the ions temperature is higher. Figure 4 shows the damping of ion-acoustic waves versus kλ in the range kλ 4 to 16, in which the value of r is fixed at 1 and the value of q is changed as 2, 3, and 5. In Figs. 4(a) and 4(b) we consider τ 2 T i /T e 10 2 and τ 2 T i /T e 10 1, respectively. The damping rate increases for the high values of q when kλ 7, however, as the value of kλ increases from 7 it decreases as the value of q increases. In Fig. 4(b) we get the same result, i.e., the damping rate increases as the value of q reduces. Again we can see that the damping rate is more sensitive to the ratio of ion to electron temperatures than the values of spectral indices r and q. Figures 3 and 4 indicate that for the ion-acoustic waves, the damping rate increases if we reduce the value of one of the spectral index and keep the other fixed; but the damping becomes more severe if the ion temperature increases. Figures 1 and 2 show that the rate of Landau damping is strongly dependent on the spectral indices r and q for the electron plasma waves and it increases for the low values of r and q, which corresponds to the high-energy tail component and narrow shoulders in the profile of distribution function. The comparison of Figs. 3 and 4 indicates that the presence of a hot ion component, i.e. T i /T e > 10 1, substantially enhances the damping rate of ion-acoustic waves, and that for T i T e waves become overdamped. However, the results are comparatively less sensitive to the values of r and q in the limit kλ 1 than to Figs. 1 and 2 in the limit kλ 1.

5 554 M.N.S. Qureshi, SHI Jian-Kui, and MA Shi-Zhuang Vol. 45 Fig. 3 (a) The damping rate of ion-acoustic waves for q 3 for different values of r when ions are cool, τ 0.1; (b) The damping rate of ion-acoustic waves for q 3 for different values of r when ions are hot, τ 0.3. Fig. 4 (a) The damping rate of ion-acoustic waves for r 1 for different values of q when ions are cool, τ 0.1; (b) The damping rate of ion-acoustic waves for r 1 for different values of q when ions are hot, τ Summary We have studied the Landau damping for Langmuir waves under the condition kλ 1 and ion-acoustic waves under the condition of kλ 1 in a hot, isotropic, and unmagnetized plasma modeled with the generalized (r, q) distribution function. We have derived the dispersion relations for the electron plasma (Langmuir) waves and the ion-acoustic waves and the damping rates are calculated. The solutions for the Langmuir waves are obtained by using the appropriate limiting expansion (ξ e 1) of the generalized plasma dispersion function, and the solutions for the ion-acoustic waves are obtained by using the limiting expansion (ξ e 1). For the Langmuir waves the damping rate is strongly dependent on the spectral indices r and q, and increases if we decrease the value of either q or r. In the case of ion-acoustic waves, the rate of Landau damping first decreases in the range kλ 7 for the decrease in the value of r or q, but as the value of kλ increases from 7 the damping rate increases as the value of either r or q reduces. As compared to the electron plasma waves the damping rate for the ion-acoustic waves is weakly dependent on the spectral indices r and q. The Landau damping for such waves is strongly dependent on the ion temperature and experiences only moderate Landau damping as long as T i T e. The rate of Landau damping increases greatly when T i /T e > 10 1, and the waves become overdamped when T i T e. References 1] R.M. Thorne and S. Summers, Phys. Fluids B 3 (1991) ]. Summers and R.M. Thorne, Phys. Fluids B 3 (1991) ]. Summers and R.M. Thorne, Phys. Plasmas 1 (1994) ] R.L. Mace and M.A. Hellberg, Phys. Plasmas 2 (1995) ] V.M. Vasyliunas, J. Geophys. Res. 73 (1968) ] M.N.S. Qureshi, et al., Phys. Plasmas 11 (2004) ] L.. Landau, J. Phys. (U.S.S.R) 10 (1946) 25. 8] T. O Neil, Phys. Fluids 8 (1965) ].G. Swanson, Plasma Waves, Academic, San iego (1989). 10] B.. Fried and S.. Conte, The Plasma ispersion Function, Academic, New York (1961).