The inversion of incoherent scatter spectra with a non-Maxwellian electron distribution

ARTICLE IN PRESS Journal of Atmospheric and Solar-Terrestrial Physics 72 (2010) 492–497

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The inversion of incoherent scatter spectra with a non-Maxwellian electron distribution Bin Xu a,b,n, Zhange Wang b, Kun Xue b, Jian Wu b, Zhensen Wu a, Jun Wu b, Yubo Yan b a b

School of Science, Xidian University, Xi’an 710071, China National Key Laboratory of Electromagnetic Environment, China Research Institute of Radiowave Propagation, Qingdao 266107, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 16 April 2008 Received in revised form 17 January 2010 Accepted 31 January 2010 Available online 10 February 2010

With the action of powerful, high-frequency (HF) radio waves, the ionosphere plasma will depart from the equilibrium state and the non-Maxwellian distribution function can be produced. An artificial fieldaligned irregularities (AFAI) model is introduced to describe the distortion from the normal shape, and the measured data are analyzed with this model during ionosphere heating at a 186-km height on August 15th, 2006. The electron temperature and density deduced from the AFAI model are compared with the results obtained from a standard procedure. The inversion of the electron temperature is evidently affected, and the overestimation is up to 22.9%. Owing to the introduction of the AFAI model, the new irregularities’ parameters can be obtained, which implies that incoherent scatter radar is feasible as a ground-based instrument to diagnose information on irregularities. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Incoherent scatter spectra Non-Maxwellian distribution HF heating Electron temperature

1. Introduction Particles have a Maxwellian velocity distribution under the condition of thermal equilibrium in the ionosphere plasma. However, non-Maxwellian characteristics can be induced by convective electric fields in the high-latitude ionosphere, energetic particle precipitation in the auroral region, and artificial ionospheric heating. Until recently, most analyses of distorted spectra have been performed with non-Maxwellian ion velocity distributions. Raman et al. (1981) presented the theoretical interpretation of incoherent scatter spectra with large electric fields for the first time. They used the empirical ion velocity distribution function given by St-Maurice et al. (1976). Using Raman’s model, studies of the auroral F region have been performed by Lockwood and Winser (1988) and Suvanto et al. (1989). Hubert (1983, 1984) provided an analytical expression for the non-Maxwellian ion velocity distribution function that depends on the electric field and is valid for all angular directions with respect to the magnetic field. Barakat et al. (1983) and Winkler et al. (1992) used numerical simulations with a MonteCarlo technique to evaluate the ion velocity distribution function. Owing to the interaction between electromagnetic waves and

n Corresponding author at: School of Science, Xidian University, Xi’an 710071, China. E-mail addresses: [email protected] (B. Xu), [email protected] (Z. Wang), [email protected] (K. Xue), [email protected] (J. Wu), [email protected] (Z. Wu), [email protected] (J. Wu), [email protected] (Y. Yan).

1364-6826/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jastp.2010.01.006

plasma, the incoherent scatter spectra can be disturbed by heating. Rietveld et al. (1993a; 1993b) presented the enhancement of the ion line and plasma line during ionospheric heating in their reviews. Expanding Boltzmann’s equation by a spherical function, Gurevich (1978) deduced the electron velocity distribution function with the action of the HF wave in the low ionosphere. The tail of the electron velocity distribution function was enhanced by the HF electric field, that is to say, the number of electrons with higher energy increased. The airglow observations validated the deviation of the electron distribution function from a Maxwellian distribution in the high energy range (Gurevich and Milikh, 1997; Gustavsson, 2005; Mantas, 1994; Mishin et al., 2000). Incoherent scatter spectra with non-Maxwellian electron velocity distributions (super-Gaussian and kappa) were studied by Zheng et al. (1997) and Saito et al. (2000), and the features of the ion line and plasma line were discussed. The fit of the measured spectra with the assumption of a Maxwellian distribution may have resulted in some errors in the plasma parameters, and the spectra line feature can be used to deduce the form of the non-Maxwellian electron distribution. The incoherent scatter spectra from the ionospheric heating campaign with the AFAI model are inverted in this paper. In the next section, the theoretical formulae of incoherent scatter spectra are given, and the effects of a non-Maxwellian index and the ionosphere parameters on the power spectra are discussed. On this basis, an inversion of the measurement spectra is performed, and the effects of non-Maxwellian factors on the electron temperature and density are argued. Finally, the major conclusion is given.

ARTICLE IN PRESS B. Xu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 72 (2010) 492–497

2. Power spectra of the non-Maxwellian electron velocity distribution function The interaction between waves and particles during ionospheric modification can give rise to the temperature enhancement of the local electrons. The enhanced temperature diffuses quickly along the magnetic line; thus, the field-aligned inhomogeneous structure of the temperature is formed. Because of the locally high temperatures, electron movement from the high temperature zone to the low temperature zone can be caused simultaneously. The injection of the heating wave, the thermal conduction along the magnetic direction, and the thermal diffusion of the electron density will achieve a dynamical equilibrium, and the artificial field-aligned striations are produced (Gurevich et al., 1995; 1998; Kelley et al., 1995). In light of the formation mechanism, the electron temperature in the irregularities is higher than that of the background electron. It is assumed that the striation electrons and the background electrons obey the Maxwellian distribution, and the non-Maxwellian electron distribution can be described by the superposition of a high temperature distribution and a background distribution. The electron velocity distribution function can be written as fe ¼

1R

p1=2 a

expðv2 =a2 Þ þ

R

p1=2 vTh

expðv2 =v2Th Þ

ð1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ 2kb Te =me is the thermal velocity of the background electrons, R is the ratio between the irregularities electrons and the total electrons, Te is the background electron temperature, electron me is the p ffiffiffiffiffiffiffiffiffiffiffiffiffi mass, kb is the Boltzmann constant, vTh = ma, and m ¼ Tm =Te is the irregularities’ temperature coefficient. With the increase of the irregularities electrons temperature and content, the number of electrons with a higher velocity increases, which is represented as the non-Maxwellian high-tail. In fact, the super-Gaussian and kappa distributions can also be used to describe the electrons’ non-Maxwellian characteristics (Zheng et al., 1997; Saito et al., 2000). However, compared to the Maxwellian distribution, the zero velocity electrons and the highenergy electrons of the super-Gaussian distribution decrease, the electrons with moderate energy increase, and the characteristic of the kappa distribution is contrary. As only the ‘‘low-energy’’ electrons can play an important role in the ion line, the electrons with a velocity larger than 2.5-times the thermal velocity contribute almost nothing to the ion line. Consequently, the appearance of the kappa distribution for the ion line calculation is ‘‘the electrons near a zero velocity increase, the higher energy electrons mainly decrease’’, and the super-Gaussian distribution cannot produce a high-tail. Therefore, these distributions are inappropriate to describe the non-Maxwellian characteristics induced by heating, and the AFAI model is adopted. According to the theoretical results of incoherent scatter spectra deduced by Sheffield (1975), and neglecting the magnetic field and collision, the power spectra are given by     2p  Ge 2 o 2pZ Ge 2 o 1 Þ þ ð2Þ f ð fð Þ Sðk; oÞ ¼ e k  e k k e i k where k is the wave vector, Z is the charge number of ions, e =1+ Ge + Gi is the dielectric function, and Ge and Gi are defined as Z þ1 4pe2 ne0 k@fe =@v ð3Þ dv Ge ðk; oÞ ¼ 2 1 me kb okv Gi ðk; oÞ ¼

Z

þ1

1

dv

4pe2 ni0 k@fi =@v 2 mi kb okv

ð4Þ

where e is the electronic charge, ne0 and ni0 are the number densities of the electrons and ions, respectively, fe and fi are the velocity distribution functions of the electrons and ions, respec-

493

tively, the ion velocity distribution is supposed to be Maxwellian, and fe is given by Eq. (1). The effects of the ordinary ionosphere parameters and the non-Maxwellian indexes on the incoherent scatter spectra are presented (Fig. 1). With the increase of the electron temperature, the ion resonance frequency increases, the damp attenuates, and the peak-to-valley ratio enhances. With the increase of the ion temperature, the damp gains, the peak-to-valley ratio declines, and the spectra width broadens. With the increase of the electron density, the ion resonance frequency and the area increase and the spectra peak becomes more keen-edged. The ion resonance frequency shifts with the ion drift velocity. With the increase of the irregularities’ temperature and content, the amplitudes of the ion line and the spectra area become reduced and the peak-tovalley ratio increases. According to the discussion above, the effects of the non-Maxwellian factors and the plasma parameters on the spectra characteristics are overlapped, which means that the one-to-one relationship between the control factors and the spectra characteristics is broken, and many-to-one or many-tomany relationships are set up. As a result, it can be anticipated that a larger error will be found in the plasma parameters. However, the control mechanism of these parameters on the spectra is not the same, and the non-Maxwellian indexes are independent from the parameters of other ionosphere. Therefore, the two indexes can be inversed from the incoherent scatter data.

3. Inversion results of the incoherent scatter spectra The ion line spectra are presented during the Chinese campaign with the EISCAT (European Incoherent Scatter) UHF radar, observed vertically in a mono-station way, on August 15th, 2006. The heater was switched on/off every 4 min, and the antenna of transmission was directed along the geomagnetic field. The measurement results from 9:38 to 11:14 near 186 km are shown in Fig. 2, and the integral time is 240 s. At the time of 10:54, there is an abnormal ion line spectrum in the figure, which is a broken point and will not be considered in future analyses. The incoherent scatter spectra descend during heating, and the peak-to-valley ratio of the spectra line increases. Either the increase of the electron temperature and non-Maxwellian indexes or the decrease of the electron density and ion temperature can result in these changes. Consequently, the plasma parameters obtained by fitting the experimental spectra with the Maxwellian distribution may be affected. Using the standard analysis procedure, the electron temperature and electron density with the HF modification experiments off and on are presented in Figs. 3 and 4. The increase of electron temperature is evident, and the maximum percentage is up to 27% (627 K). However, the change of electron density is very faint. In general, the increase of the inversion parameters will cause the non-uniqueness of the inversion results. There are two reasons for the non-uniqueness; one is that the inversion parameters are dependent on each other. The figures about the independence of the non-Maxwellian indexes and the ionosphere parameters are given in the last section. In fact, the independence can also be directly seen in the spectra formula, as these parameters play different roles in the distribution function. As a result, there is no distinguishing between the ordinary ionosphere and the irregularities’ parameters for the inversion process in theory, and we can get these new inversion parameters. The other reason is that there are errors between the measured results and the theoretical calculations. The inversion problem can be presented as a mathematical process. The object function is E ¼ normðyi yi 0 Þ, where yi 0 is the measured result and yi is the

ARTICLE IN PRESS B. Xu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 72 (2010) 492–497

200

150

150

100 50 Te=1000K Te=2000K Te=4000K

0

-10 0 10 Frequency (kHz)

100 50 0 -50 -20

20

200

200

150

150

Power (K/kHz)

Power (K/kHz)

-50 -20

100 50 10

ne=2*10 m

0

-3

ne=2*1011m-3 ne=2*1012m-3

-50 -20

-10 0 10 Frequency (kHz)

120 Power (K/kHz)

Power (K/kHz)

200

vTh=2a vTh=4a

80 60 40

20

50 0

vi=0 vi=100m/s vi=400m/s -10 0 10 Frequency (kHz)

120

vTh=0

100

Ti=500K Ti=1000K Ti=2000K -10 0 10 Frequency (kHz)

100

-50 -20

20

Power (K/kHz)

Power (K/kHz)

494

20

R=0 R=0.2 R=0.4

100 80 60 40 20

20 0 -20

-10 0 10 Frequency (kHz)

0 -20

20

-10 0 10 Frequency (kHz)

20

0

heater off heater on

0 20 Frequency (kHz)

0.1 0.05 0 -20

-20

heater off heater on

0 20 Frequency (kHz)

0 20 Frequency (kHz)

0.1 0.05 0 -20

heater off heater on

0.05

-20

heater off heater on

0 20 Frequency (kHz)

heater off heater on

0 20 Frequency(kHz)

0.1 0.05 0 -20

heater off heater on

0.05

-20

heater off heater on

0 20 Frequency (kHz)

Fig. 2. Measurement results of incoherent scatter spectra.

0.1 0.05 0 -20

0 20 Frequency (kHz)

0.1

0

Power (K/MHz)

0 -20

0 20 Frequency (kHz)

0.1

0

Power (K/MHz)

heater off heater on

0.05

Power (K/MHz)

0.05

0

0.1

heater off heater on

0 20 Frequency (kHz)

0.1 0.05 0 -20

Power (K/MHz)

0.1

-20 Power (K/MHz)

0 20 Frequency (kHz)

0.05

Power (K/MHz)

Power (K/MHz)

-20

heater off heater on

Power (K/MHz)

0

0.1

Power (K/MHz)

0.05

Power (K/MHz)

0.1

Power (K/MHz)

Power (K/MHz)

Fig. 1. Variations of power spectra with non-Maxwellian indexes.

heater off heater on

0 20 Frequency (kHz)

0.1 0.05 0 -20

heater off heater on

0 20 Frequency (kHz)

ARTICLE IN PRESS B. Xu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 72 (2010) 492–497

6000 5000

Te (K)

4000 3000 2000 1000 0

9:38

9:54

10:10 10:26 10:42 10:58 Time

Fig. 3. Electron temperature with the HF modification experiments either off (white) or on (black: Maxwellian; gray: the irregularities model).

2.5

x 1011

ne (m-3)

2 1.5 1 0.5 0

9:38

9:54

10:10

10:26 Time

10:42

10:58

Fig. 4. Electron density with the HF modification experiments either off (white) or on (black: Maxwellian; gray: the irregularities model).

theoretical value 8 y1 ¼ Sðo1 ; p1 . . .pj Þ > > > > < y2 ¼ Sðo2 ; p1 . . .pj Þ ^ > > > > : yi ¼ Sðoi ; p1 . . .pj Þ

ð5Þ

Here, S is the physical model that is defined in Eq. (2), oi is the Doppler frequency, and pj is the inversion parameter in which i and j are the sample and parameter numbers, respectively. A set of initialization physical parameters are used to compute the theoretical model, and the difference between the measured and theoretical results, i.e., the object function, is calculated further. Some mathematical methods are used to adjust the physical parameters, which make the theoretical values continuously approach the measured results. When the steady-state is achieved, the adjusted parameters are the inversion results. In order to ensure that the inversion has a unique solution, it is necessary to satisfy iZj. Theoretically, if there are no errors in the measured results, all of the undetermined parameters can be obtained with i=j. However, due to the measured errors, the redundant data are needed to reduce the effects of the errors. In the data analysis of the incoherent scatter radar, Te, Ti, ne, and vi

495

are the direct inversion variables. Due to the large mass, the impact of the heating on the ion temperature is trivial. Thus, only the electron temperature, the electron density, and the two nonMaxwellian indexes are used as the independent variables. The ion temperature and drift velocity are given by the average value at the unheated time. In addition, the F-region measurement has a high precision in the EISCAT experiment, and there are 121 frequency samples, which means that the samples are large enough. Sometimes the real optimization solutions may not be achieved, and the inversion results are the local minimum. For example, the L-M method may give a local extremum, and the global method can improve this problem. The Pattern Search Method is used in this paper to solve the local extremum problem. In order to confirm the convergence, the variations of the objection function with the iterative number are shown in Fig. 5. At the end of the iteration, there is no change in the objection function. Therefore, the iteration is steady and the determination of the non-Maxwellian indexes is rational. A comparison of the electron temperature and density from the two inversion methods is presented (Figs. 3 and 4, respectively). When the heater is on, the electron temperature still obviously increases. However, the incremental amplitude is lower than that from the standard analysis procedure. The mean value is 161 K, and at the time of 10:22, the increase reaches the maximum of 375 K. The standard analysis procedure will overestimate the electron density, and the maximum is up to 8.1%, which will result in a completely different conclusion. In most cases, the reduction of the electron density can be seen in the inversion results from the irregularities model, which is consistent with the prediction of the theoretical model, whereas the phenomenon does not appear in the Maxwellian analysis results. The overestimation of the electron temperature can also be given a theoretical explanation. The enhancement of the electron temperature can be induced by two reasons. For one thing, the electron is easily accelerated by the artificial electric field due to its small mass. Then, the accelerated electrons collide with other electrons or ions and neutral particles, and the direction of the electron velocity is changed. The kinetic energy of the electron obtained from the HF electric field is converted into the electron thermal energy, which is represented as the increase of electron temperature. In light of the large neutral particle density in the low altitude region, this is the prime reason for the increase of the electron temperature. In contrast, in the high ionosphere region, the non-linear interaction between electromagnetic waves and the ionospheric plasma plays a more important role. With the injection of the powerful high-frequency pump waves on the ionosphere plasma, Langmuir waves and ion acoustic waves are excited. These waves are then weakened by the non-linear Landau damping. The wave energy is changed into electron kinetic energy, as Landau damping is proportional to the gradient of the velocity distribution of particles moving with nearly the phase velocity of the waves, and the slope of the electron distribution is smaller in the case of the non-Maxwellian distribution. In addition, the phase velocity of the ion acoustic waves is faster, and thus the waves propagate farther into the tail of the ion distribution, and the ions available to damp the wave are fewer than those available for the Maxwellian distribution. As a result, both the electron and ion damping decrease with the increase of the non-Maxwellian factors, and the enhancement of the electron temperature is weakened. The electron density and background electron temperature can be inversed from the Maxwellian model, whereas the introduction of the AFAI model adds two independent inversion parameters: the irregularities ratio and temperature coefficient. The two irregularities indexes of the 12 heating cases are shown in Figs. 6 and 7, respectively. The irregularities temperature

ARTICLE IN PRESS B. Xu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 72 (2010) 492–497

0

20 40 60 Iterative number

0.01

0

0

0.005

0

20 40 60 Iterative number

0

20 40 60 Iterative number

0.01 0.005 0 0

20 40 Iterative number

0.4 0.3 0.2 0.1 0

0

20 40 60 Iterative number

Object function

0

20 40 60 Iterative number

0.01

0

2

0.015

Object function

Object function

0.005

4

x 10-3

6 4 2 0

8

0

20 40 60 Iterative number

x 10-3 Object function

0

6

8

6 4 2 0

0

0.015 0.01 0.005 0

0

20 40 60 Iterative number

x 10-3

3 2 1 0

0

10 20 30 Iterative number

0

20 40 60 Iterative number

0.01 0.005

20 40 Iterative number

0.02

4

0.015

Object function

2

x 10-3

Object function

4

8

Object function

Object function

x 10-3

Object function

Object function

Object function

6

Object function

496

0

8

x 10-3

6 4 2 0

0

20 40 60 Iterative number

Fig. 5. Variations of the object function with iterative number.

10

0.25

9

0.2

8 m

R

0.15

7 0.1

6 5

9:38

9:54 10:10 10:26 10:42 10:58 Time

0.05

9:38

9:54 10:10 10:26 10:42 10:58 Time

Fig. 6. Variations of the irregularities’ temperature coefficient with time, T.

Fig. 7. Variations of the irregularities ratio with time, T.

coefficient ranges from 5.07 to 9.66, and its average value is 7.12. The irregularities ratio ranges from 5% to 22%, and its average value is 11%. These parameters denote the electrons’ energy state with artificial field-aligned irregularities.

parameters are determined by the spectra shape, the precision of the incoherent scatter radar will be affected by the nonMaxwellian factors. Therefore, using the Maxwellian distribution, the electron temperature and density will be overestimated. However, the effect on the electron temperature is more obvious. The percentage of overestimation of the electron temperature is up to 22.9% during this campaign, and that of the electron density is 8.1%. According to the computation, we can prove that the introduction of the non-Maxwellian index can bring an important influence on the determination of the ionospheric parameters, and it can be anticipated that a larger error of temperature will be obtained with a higher heating power. Thus, the tail enhancement of the electron velocity distribution should be considered in the analysis of HF ionosphere heating data.

4. Conclusion A non-Maxwellian distribution will be produced in plasma heated by sufficiently strong electromagnetic radiation. The lowenergy electrons are preferentially heated by the driving electric field, which shows that the tail of the electron velocity distribution function is enhanced. This effect has important consequences: reductions of the absorption rate, the electron heat flux, the threshold of ion acoustic drift instability, and the excitation and ionization rates. Using the distribution above, the incoherent scatter radar data are analyzed in this paper. The effects of the non-Maxwellian indexes on the spectra shape are discussed first. With the increase of m and R, the peak-to-valley ratio increases and the area is reduced. Since the plasma

Acknowledgments The authors appreciate the EISCAT Scientific Association. EISCAT is an international association supported by China (CRIRP),

ARTICLE IN PRESS B. Xu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 72 (2010) 492–497

Finland (SA), France (CNRS), the Federal Republic of Germany (MPG), Japan (NIPR), Norway (NFR), Sweden (NFR), and the United Kingdom (STFC). This work is supported by the NSFC with Grant no. 40310223 and by the National key Laboratory of Electromagnetic Environment (LEME). References Barakat, A.R., Schunk, R.W., St-Maurice, J.P., 1983. Monte Carlo calculation of the O+ velocity distribution in the auroral ionosphere. J. Geophys. Res. 88, 3237–3241. Gurevich, A.V., 1978. Nonlinear Phenomena in the Ionosphere. Springer, Berlin. Gurevich, A.V., Lukyanov, A.V., Zybin, K.P., 1995. Stationary state of isolated striations developed during ionospheric modification. Phys. Lett. A 206, 247–259. Gurevich, A.V., Milikh, G.M., 1997. Artificial airglow due to modifications of the ionosphere by powerful radio waves. J. Geophys. Res. 102 (A1), 389–394. Gurevich, A.V., Hagfors, T., Carlson, H.C., et al., 1998. Electron temperature measurements by incoherent scattering in the presence of strong small scale temperature irregularities. Phy. Lett. A 246, 335–340. Gustavsson, B., 2005. The electron energy distribution during HF pumping a picture painted with all colors. Ann. Geophys. 23 (5), 1747–1754. Hubert, D., 1983. Auroral ion velocity distribution function: generalized polynomial solution of Boltzmann’s equation. Planet. Space Sci. 31 (1), 119–127. Hubert, D., 1984. Non-Maxwellian distribution function and incoherent scattering of radar waves in the auroral ionosphere. J. Atmos. Terr. Phys. 46, 601–611. Kelley, M.C., Arce, T.L., Salowey, J., et al., 1995. Density depletion at the 10 m scale induced by Arecibo heater. J. Geophys. Res. 100, 17367.

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Lockwood, M., Winser, K.J., 1988. On the determination of ion temperature in the auroral F-region ionosphere. Planet. Space Sci. 36, 1295–1304. Mantas, G.P., 1994. Large 6300 A airglow intensity enhancements observed in ionosphere heating experiments are excited by thermal electrons. J. Geophys. Res. 99 (A5), 8993–9002. Mishin, E., Carlson, H.C., Hagfors, T., 2000. On the electron distribution function in the F region and airglow enhancements during HF modification experiments. Geophys. Res. Lett. 27 (18), 2857–2860. Raman, R.S., St-Maurice, J.P., Ong, R.S.B., 1981. Incoherent scattering of radar waves in the auroral ionosphere. J. Geophys. Res. 86 (A6), 4751–4762. Rietveld, M.T., Kohl, H., Kopka, H., Stubbe, P., 1993a. Introduction to ionospheric heating at Tromsø-I. Experimental overview. J. Atmos. Terr. Phys. 55 (4), 577–599. Rietveld, M.T., Kohl, H., Kopka, H., Stubbe, P., 1993b. Introduction to ionospheric heating at Tromsø-II. Scientific problems. J. Atmos. Terr. Phys. 55 (4), 601–613. Saito, S., Forme, F.R.E., Buchert, S.C., Nozawa, S., Fujii, R., 2000. Effect of a kappa distribution function of electrons on incoherent scatter spectra. Ann. Geophys. 18 (9), 1216–1223. Sheffield, J., 1975. Plasma Scattering of Electromagnetic Radiation. Academic Press, New York, pp. 113–128. St-Maurice, J.P., Hanson, W.B., Walker, J.C.G., 1976. Retarding potential analyzer and measurements of the effect of ion–neural collision on ion velocity distribution in the auroral ionosphere. J. Geophys. Res. 81, 5436–5446. Suvanto, E., St-Maurice, J.P., Barakat, A.R., 1989. Analysis of incoherent scatter radar data from non-thermal F-region plasma. J. Atmos. Terr. Phys. 51, 483–495. Winkler, K.J., St-Maurice, J.P., Barakat, A.R., 1992. Results from Monte Carlo calculations of auroral ion velocity distributions. J. Geophys. Res. 97, 8399–8423. Zheng, J., Yu, C.X., Zheng, Z.J., 1997. Effects of non-Maxwellian (super-Gaussian) electron velocity distribution on spectrum of Thomson scattering. Phys. Plasmas 4 (7), 2736–2740.