Landau damping in space plasmas with generalized r, q distribution function

PHYSICS OF PLASMAS 12, 122902 2005 Landau damping in space plasmas with generalized r, q distribution function M. N. S. Qureshi Key Laboratory of Space Weather, CSSAR, Chinese Academy of Sciences, Beijing 100080, China; Department of Physics, Government College University, Lahore 54000, Pakistan; and Graduate University of Chinese Academy of Sciences, Beijing 100039, China J. K. Shi Key Laboratory of Space Weather, CSSAR, Chinese Academy of Sciences, Beijing 100080, China S. Z. Ma Graduate University of Chinese Academy of Sciences, Beijing 100039, China Received 21 July 2005; accepted 27 October 2005; published online 12 December 2005 Space plasmas possessing non-maxwellian particle distribution functions with an enhanced high-energy tail and shoulder in the profile of distribution function take an important role to the wave particle interaction. 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. 2005 American Institute of Physics. DOI: 10.1063/1.2139504 I. INTRODUCTION In space plasmas the distribution function sometimes follows an inverse power law in the high-energy tail f v such as kappa distribution function containing the spectral index. 1 4 In the limit the distribution approaches to 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=+1 reduces to kappa distribution function and approaches to Maxwellian if r=0 and q. 6 The generalized r,q distribution has been used to provide the best fits for the magnetosheath electron data 7 from AMPTE satellite and solar wind proton data 8 from CLUSTER. In both the fittings, the observed distribution functions deviated significantly from the Maxwellian in the low-energy portion shoulders as well as in the high-energy tail of the distribution function. Recently, Kiran et al. 9 calculated the power dissipated through obliquely propagating Alfven waves to heat the solar wind protons and determined the evolution of power dissipation of protons with increasing Heliocentric distance using the generalized r,q distribution function and found good agreement with the observation. Therefore, on the basis of the observations and theory, it is expected that some space plasmas can be modeled more effectively by 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, 10 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 having velocity that nearly matches the wave phase velocity /k exchange energy with the wave. If there are more particles going slower than the phase velocity, gaining energy from the wave, than those going faster, losing energy to the wave, then there is net loss of energy and the wave damps. O Neil 11 has considered the damping of large amplitude plasma waves in the limit i / r 1. In this limit particles with v r /k are trapped. The physical picture developed by O Neil 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-line orbits 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 small amplitude longitudinal plasma oscillations in collisionless Maxwellian plasma. Such damping is weak for frequencies near the electron plasma frequency since the wave speed u=/k is much larger than the electron thermal velocity. Wave attenuation becomes more severe at higher wave frequencies where u/ 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. Landau damping remains an important topic in the recent years. Many authors discussed Landau damping from different aspects; since it is impractical to mention all of 1070-664X/2005/1212/122902/6/$22.50 12, 122902-1 2005 American Institute of Physics

122902-2 Qureshi, Shi, and Ma Phys. Plasmas 12, 122902 2005 them, therefore, few shall be cited here. 1,2,12 15 The significant increase of Landau damping in -distributed plasmas was first discussed in the pioneering paper of Thorne and Summers. 1 They derived the plasma dispersion relations modeled with kappa distribution function in unmagnetized, hot plasma and computed the damping rate for different values of kappa. The work described in the Refs. 1 3 and 11 15, and in many related papers is important because of its wide range of applications in space physics. In this paper, we illustrate this phenomenon explicitly for the case of Landau damping of small amplitude electron plasma waves and ionacoustic waves modeled with the generalized r,q distribution function. In Sec. II we briefly describe the generalized r,q distribution function and its limiting forms. The general dispersion relations for electrostatic waves in a uniform, unmagnetized but fully hot plasma are derived in Sec. III. In Sec. IV we discuss the numerical results for the damping rates of electron plasma waves and ion-acoustic waves and illustrate the role of the ion temperature T + to electron temperature T ratio in determining the damping characteristics of ionacoustic wave. At last, we will give a summary of this paper. II. GENERALIZED r, q DISTRIBUTION 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. The generalized r,q distribution function can be written as where 3q 1 3/21+r q f r,q = 4 3 q 3/21+r1 + 3/21+r 1+ 1 q 1 v2 2 1+r q, 1 = T m 3q 1 1/1+r q 3/21+r3/21+r. q 5/21+r5/21+r Here v is the particle speed, is the particle thermal speed, T is the particle temperature, and is the Gamma function. We also note here that in the generalized r,q distribution function the spectral indices r and q satisfy the constraints q1 and q1+r5/2, 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. Figure 1 illustrates the log of distribution function versus v and r for q=2 and 5. From Fig. 1 we note that, for the fixed value of q, ifr increases the shoulders in the profile of distribution function tend to be broadened and the distribution function for the high-energy particle reduces. Similarly, for a fixed value of r, if q increases the distribution function for the high-energy particle reduces, too. 6 2 FIG. 1. The log of the generalized r,q distribution function vs v and spectral indices r, with q=2 upper panel and q=5 lower panel. The enhancement of shoulders and reduction of high-energy tail can be seen with the increase of r. III. DISPERSION 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 = r +i 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 where 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 k D 1 and k D 1, respectively, where

122902-3 Landau damping in space plasma Phys. Plasmas 12, 122902 2005 D = T 2 1/2 4Ne = q 11/1+r 5/21+rq 5/21+r is the electron Debye length and is the electron plasma frequency. Note that henceforth, as in Eq. 3, we shall use to mean electron plasma frequency and D as electron Debye length. The general dispersion relation comes as 2 2 1+ p, k 2 2 A + Z r,q 1 =0, 4 where A = 3q 1 1/1+r q 1/21+r1 + 1/21+r. 2q 3/21+r1 + 3/21+r 5 The summation is made over all the charge species in the plasma,, is plasma frequency, and s21+r + Z r,q 1 1 = 1 q ds 6 B s 1+ q 1 is the generalized plasma dispersion function whose general properties are described by Qureshi et al., 6 where =/k, and s=v/, and B = 3q 1 3/21+r q 4q 3/21+r1 + 3/21+r. A. 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. Therefore, we neglect the ion terms in Eq. 4 and only consider the electron terms; the resulting dispersion relation for high-frequency Langmuir oscillations is Z 1 r,q + A + Ck 2 D 2 =0, 3 7 8 where and i = 3 D 4 k 3 31+ 1 D q 1 1 r+1 q k 2 2 +3 D E, 11 q D = q 3/21+r1 + 3/21+r 5/21+rq 5/21+r, 12 3/2 E = q 11/1+r 5/21+rq 5/21+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 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 below, 1,16 i = 8 1/2 1 k 3 D 3 exp 1/2k2 D 2 + 3/2. B. Low-frequency oscillations: Ion-acoustic waves 14 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. 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 e Z 1 r,q e + A + Fk 2 D 2 + 1 2A + 1 ez 1 r,q e =0, 15 where A is given by Eq. 5, =m e /m i 1/2, =T i /T e 1/2, and F is given as where A is given by Eq. 5 and C = 3q 1 1/1+r 3/21+rq 3/21+r. 25/21+rq 5/21+r 9 Approximate solution of Eq. 8 can be found by using the appropriate limiting forms for e 1 of the generalized plasma dispersion function 6. We solve the Eq. 8 under the above approximation and obtained the real and imaginary parts of the dispersion relation as F = 2q 1 1/1+r 5/21+rq 5/21+r. 16 In Eq. 15 we assume that the spectral indices r and q are the same for ion distribution as that of the electron distribution. Approximate solutions of Eq. 15 can be obtained by assuming e 1 and e /1 as1, 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 r =1+ 3 2 k2 D 2, 10 r Gk D = 2A + G 2 k 2 2 D 1/2, 17

122902-4 Qureshi, Shi, and Ma Phys. Plasmas 12, 122902 2005 i 2 BGk D = 2A + G 2 k 2 2 D 21+ 1 31+ 1 q 1 1 2 2A + G 2 k 2 2 D 1+r q, 18 where A and B are given by Eqs. 5 and 7, respectively, and G is G = q 1 1/1+r. 5/21+rq 5/21+r1/2 19 The approximate solutions of Eq. 15 of ion-acoustic branch have already been obtained 1,16 in the above assumptions and can readily be obtained in the limiting case r=0, q of Eqs. 17 and 18 as r k D = 1+k 2 2 D 1/2, 20 i = 2 k D 8 1+k 2 2 D 2 1+ 1 3 exp 1 2 2 1+k 2 2 D. 21 IV. NUMERICAL RESULTS We have computed Eq. 11 for the damping rate of electron plasma waves in the limit k D 1, which comes from the limiting expansion e 1 of the generalized plasma dispersion function Eq. 6, for different r and q. Because of k D 1, we took the value of k D from 0.1 to 0.4 in the calculation. It is evident from Eq. 11 that the damping rate for the Langmuir waves is strongly dependent on the spectral indices r and q. Figure 2 exhibits the general behavior of the damping characteristics for the electron plasma waves modeled by the generalized r,q distribution function. Figure 2a shows the log of magnitude of damping rate versus q and the wave number k D with a fixed value of r at 1. We choose q from 2 to 6 in Fig. 2a and also in the following figures because the damping characteristics of the waves show significant changes in this range whereas beyond this range the damping is weakly dependent on q. Itcan be seen from Fig. 2a that if 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 enhancement in the number of high-energy resonant particles. It can also be seen from Fig. 2a, the damping rate increases if k D increases, and the increase in damping rate gradually sharpens when q gradually increases. Figure 2b shows the damping rate versus q and r for the fixed value of k D =0.35, where r changes from 1 to 4 and q from 2 to 6. There is a monotonic increase in damping rate as r decreases. It is because of the fact that with the decrease in the value of r the shoulders in the profile of distribution function tends to disappear and the contribution of low-energy particles to the damping decreases. For a fixed FIG. 2. The log of magnitude of damping rate for the electron plasma waves a vs q and k D when r=1 and, b vs r and q when k D =0.35. r if q decreases the damping rate increases. It is evident from Fig. 2 that the damping rate for Langmuir waves increases for the low values of r and q, and increases with the increase in the value of k D. However, damping rate changes slowly with the change in q as compared to change in r. 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 k D 1, which come from the limiting expansion e 1 of the generalized plasma dispersion function, Eq. 6. Because k D 1, we took k D 4 in the calculation. Figure 3a shows the log of magnitude of damping rate for the ion-acoustic waves versus k D and r when ion is cold, i.e., 2 =T i /T e =10 2 and q=3. Here r is varied from 0.5 to 4 because for further increase in r the damping changes a little. From Fig. 3a we can see, for a fixed r, the damping rate first increases and reaches a maximum, then decreases. For different r the maximum is different but it lies between k D =5 and k D =7. From Fig. 3a we can also note that when k D 7 the damping rate increases monotonically with increasing r, but when k D 7 the damping rate decreases monotonically with increasing r.

122902-5 Landau damping in space plasma Phys. Plasmas 12, 122902 2005 FIG. 3. The log of magnitude of damping rate for the ion-acoustic waves vs k D and r with q=3 a when ions are cool, i.e., 2 =T i /T e =10 2, and b when ions are hotter, i.e., 2 =T i /T e =10 1. FIG. 4. The log of magnitude of damping rate for the ion-acoustic waves vs k D and q with r=1 a when ions are cool, i.e., 2 =T i /T e =10 2, and b when ions are hotter, i.e., 2 =T i /T e =10 1. Figure 3b is the same as the Fig. 3b but for hotter ions, i.e., 2 =T i /T e =10 1. We can see from Fig. 3b, for the fixed k D, the damping rate for the ion-acoustic waves decreases monotonically if r increases. For a fixed r, the damping rate also decreases monotonically if k D increases. From Fig. 3, we can note that the change in damping rate is not the same for the hot and cold ions. Moreover, wave damps more rapidly when the ion temperature is higher. Figure 4a shows the log of magnitude of damping rate for the ion-acoustic waves versus k D and q when ion is cold, i.e., 2 =T i /T e =10 2. Here r is fixed at 1 and q is varied from 2 to 6. From Fig. 4a we can see, for a fixed q, the damping rate first increases and reaches a maximum, then decreases. For different q the maximum is different but it lies between k D =5 and k D =6.5. From Fig. 4a we can also note that when k D 6.5 the damping rate increases monotonically with increasing q, but when k D 6.5 the damping rate decreases monotonically with increasing q. Figure 4b is the same as the Fig. 4a but for hotter ions, i.e., 2 =T i /T e =10 1. We can see from Fig. 4b, for the fixed k D, the damping rate for the ion-acoustic waves decreases monotonically if q increases. For a fixed q, the damping rate also decreases monotonically if k D increases. From Fig. 4, we can say that the change in damping rate is not the same for the hot and cold ions. Moreover, wave damps more rapidly when the ion temperature is higher. So, from Fig. 2, we can say 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 q and r that correspond to the high-energy tail component and narrow shoulders in the profile of distribution function, respectively. From Figs. 3 and 4, for the ionacoustic waves, the damping rate increases for k D 7 when r or q increasing, but for the values k D 7 it increases for the decrease in r or q. When ions are hotter, damping rate monotonically increases for all the values of k D when r or q decreases. We note that there is a substantial enhancement in the damping rate for the ion-acoustic waves when ions are hot and for temperature T i T e waves become overdamped. Moreover, the damping rate is comparatively less sensitive to the values of r and q for ion-acoustic waves k D 1 than that for electron plasma waves k D 1. V. SUMMARY We have studied the Landau damping for Langmuir waves under the condition k D 1 and ion-acoustic waves under the condition of k D 1 in a hot, isotropic, and unmagnetized plasma modeled with the generalized r,q distribution function. We have derived the dispersion relations

122902-6 Qureshi, Shi, and Ma Phys. Plasmas 12, 122902 2005 and calculated the damping rates for the electron plasma Langmuir waves and the ion-acoustic waves. The solutions for the Langmuir waves are obtained by using the appropriate limiting expansion e 1 of the generalized plasma dispersion function, and the solutions for the ion-acoustic waves are obtained by using the limiting expansion e 1. For the Langmuir waves the damping rate is strongly dependent on the spectral indices r and q, and increases if the value of either q or r decreases. In the case of ion-acoustic waves, the rate of Landau damping first increases in the range k D 7 for the increase in the value of r or q, but for k D 7 the damping rate increases for the decrease in the value of r or q. As compared to the electron plasma waves, the dependence of the ion-acoustic wave s damping rate on the spectral indices r and q is weak. The Landau damping for such waves is strongly dependent on the ion temperature and experience 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. ACKNOWLEDGMENTS This research was supported by National Natural Science Foundation of China under Grant No. 40390150 and the International Collaboration Research Team Program of the Chinese Academy of Sciences. 1 R. M. Thorne and D. Summers, Phys. Fluids B 3, 21171991. 2 D. Summers and R. M. Thorne, Phys. Fluids B 3, 1835 1991. 3 D. Summers and R. M. Thorne, Phys. Plasmas 1, 2012 1994. 4 R. L. Mace and M. A. Hellberg, Phys. Plasmas 2, 2098 1995. 5 V. M. Vasyliunas, J. Geophys. Res. 73, 2839 1968. 6 M. N. S. Qureshi, H. A. Shah, G. Murtaza, S. J. Schwartz, and F. Mahmood, Phys. Plasmas 11, 3819 2004. 7 M. N. S. Qureshi, M. Phil. thesis, University of the Punjab, 2001. 8 M. N. S. Qureshi, G. Pallocchia, R. Bruno, M. B. Cattaneo, V. Formisano, H. Réme, J. M. Bosqued, I. Dandouras, J. A. Sauvaud, L. M. Kistler, E. Möbius, B. Klecker, C. W. Carlson, J. P. McFadden, G. K. Parks, M. McCarthy, A. Korth, R. Lundin, A. Balogh, and H. A. Shah, in Solar Wind Ten AIP Conf. Proc. No. 679, edited by M. Velli, R. Bruno, and F. Malara AIP, New York, 2003. 9 Z. Kiran, H. A. Shah, and G. Murtaza, Parallel proton heating in solar wind using the generalized r,q distribution function, Solar Physics submitted. 10 L. Landau, J. Phys. Moscow 10, 251946. 11 T. O Neil, Phys. Fluids 8, 2255 1965. 12 J. J. Podesta, Phys. Plasmas 12, 52101 2005. 13 W. E. Drummond, Phys. Plasmas 11, 552 2004. 14 D. V. Rose, J. Guillory, and J. H. Beall, Phys. Plasmas 12, 14501 2005. 15 R. W. Short and A. Simon, Phys. Plasmas 5, 4124 1998. 16 D. G. Swanson, Plasma Waves Academia, San Diego, 1989.