Electron velocity distribution function in a plasma with temperature gradient and in the presence of suprathermal electrons: application to incoherent-scatter plasma lines

. The plasma dispersion function and the reduced velocity distribution function are calculated numerically for any arbitrary velocity distribution function with cylindrical symmetry along the magnetic ﬁeld. The electron velocity distribution is separated into two distributions representing the distribution of the ambient electrons and the suprathermal electrons. The velocity distribution function of the ambient electrons is modelled by a near-Maxwellian distribution function in presence of a temperature gradient and a potential electric ﬁeld. The velocity distribution function of the suprathermal electrons is derived from a numerical model of the angular energy ﬂux spectrum obtained by solving the transport equation of electrons. The numerical method used to calculate the plasma dispersion function and the reduced velocity distribution is described. The numerical code is used with simulated data to evaluate the Doppler frequency asymmetry between the up-and downshifted plasma lines of the incoherent-scatter plasma lines at di(cid:128)erent wave vectors. It is shown that the observed Doppler asymmetry is more dependent on deviation from the Maxwellian through the thermal part for high-frequency radars, while for low-frequency radars the Doppler asymmetry depends more on the presence of a suprathermal population. It is also seen that the full evaluation of the plasma dispersion function gives larger Doppler asymmetry than the heat ﬂow approximation for Langmuir waves with phase velocity about three to six times the mean thermal velocity. For such waves the moment expansion of the dispersion function is not fully valid and the full calculation of the dispersion function is needed


Introduction
We want to estimate the ®eld-aligned electron mean drift velocity e from incoherent scatter Doppler measurement of the plasma lines (Vidal-Madjar et al., 1975;Bauer et al., 1976;Showen, 1979). In order to do this we need to solve accurately the plasma dispersion relation for electrostatic waves at high frequencies and thus to have an accurate model of the electron velocity distribution function.
A common way of representing the whole electron velocity distribution function is to separate it into two populations: the ambient or bulk population f v and the suprathermal or tail population f s v, and special care needs to be taken for the treatment of the transition region between the suprathermal and ambient electrons. At ionospheric heights about the p 2 region, the bulk population of the electrons is collision-dominated and thus the velocity-space distribution is expected to be very close to a Maxwellian. In this case, the parameters describing the state of the thermal population are: the electron density n e , the electron temperature e and the potential source of inhomogeneity such as the spatial gradients of electron temperature $ e and pressure $p e , as well as possibly an electric ®eld E. These parameters are provided by the analysis of the measurement of the ion line incoherent scattering. On the other hand, the suprathermal component f s v is taken from a complete kinetic electron transport code which takes into account the ionization and heating resulting from both solar insolation and particle precipitations.
In the ®rst part, we describe and review the original theory developed to calculate the velocity distribution function of the ambient electrons in the presence of a temperature gradient and/or an electric ®eld (Spitzer and H arm, 1953). Thereafter we present and discuss the calculations we use to represent the suprathermal part of the distribution function. We then describe a numerical method to calculate the full two-dimensional dispersion relation. We test our numerical code and discuss the results on simulated Doppler asymmetry data for radars with dierent wave vector and compare the results given by the heat¯ow approximation of Kofman et al. (1993).

The ambient velocity distribution
For low energy and for a fully ionized plasma consisting of electrons and one ion species, the distribution function of the electrons in a highly collisional regime, i.e. in a regime where the velocity-space distribution of the electrons is close to a Maxwellian (Gombosi and Rasmussen, 1991), can be approximated by the Spitzer-HaÈ rm distribution function of Cohen et al. (1950) and Spitzer and H arm (1953).
This time-independent distribution function is the result of the presence of a weak electric ®eld and a temperature gradient. The distribution function is expanded as a power series in the Knudsen number which represents the ratio of the microscopic length scale to the macroscopic length scale. In this theory only the ®rst order in is kept, which is known as the principle of local action (Woods, 1993). This restriction to small values of implies that the electron mean free path k e is much smaller than the dierent scale lengths considered r log e , r log p e and eiau e (Ljepojevic and MacNeice, 1989 where i is the electric ®eld, e the electron temperature, p e the electron pressure and r represents the derivative along the line of sight. For small Knudsen numbers, i.e. i ( 1 and ( 1, perturbation methods apply and the ambient electron velocity distribution function f is expanded about a local Maxwellian f 0 v n e a 2p 3a2 av 3 e expÀvav e 2 a2 with thermal velocity v e u e am e 1a2 and takes the following form where l is the cosine of the pitch angle measured from an axis parallel to the direction of the temperature gradient and electric ®eld, is the charge number of the ion species and x is the ratio vav e . The functions i and are the solutions of two second-order dierential equations [Eq. (40) of Spitzer and H arm (1953) and Eqs.
(6)±(13) of Cohen et al. (1950)] derived from the Boltzmann's equation where only the long-range electron-electron and the electron-ion interactions have been taken into account through two Fokker-Planck collision operators. This approximation is valid for low energy only, so that the upper boundary of integration of these functions should not be too large compared to the mean thermal velocity v e . We have recalculated the solutions to these equations for dierent values of the upper boundary. Figure 1 shows the two functions i and for those dierent values of the upper boundary of integration x max .
By taking the ®rst-and third-order velocity moments of the perturbation functions i and one de®nes four transport coecients c i , d i , c and d . These are the normalized transport coecients relative to a Lorentzian gas (Spitzer and H arm, 1953;Shkarofsky, 1961). Equations 4±7 show the relations between these coecients, the velocity moments of the distribution function and the transport coecients r e , s e , l e and j e . where r e is the electrical conductivity, s e is the current ow conductivity due to a temperature gradient at constant electron density, l e is the heat¯ow conductivity due to an electric ®eld at constant electron temperature and j e is the thermal conductivity. Table 1 presents the values of the normalized transport coecients we have recalculated and the original values of Spitzer and HaÈ rm (1953). With the exception of the values for x max 2X8, the values of the transport coecients are in good agreement (under 17) with the values calculated by Spitzer and HaÈ rm (x max 3X2).
In the work of Spitzer and HaÈ rm, the electron mean free path k e is taken to be the mean free path due to electron-electron collisions and electron-ion collisions. We shall correct the electron mean free path to take into account the electron-neutral collision term (Banks, 1966). We de®ne the electron mean free path as 1 or as a function of the electron-charged particle free path k e : k e k e 1 k e ak en X 10 The electron-neutral collisions tend to reduce the electron mean free path, and in the limit of low neutral particle densities we recover the electron mean free path value of a fully ionized plasma (Banks, 1966). It is important to note that the dierential equations for the perturbation functions i and have not been modi®ed, thus the departure of the velocity distribution function from the Maxwellian state is still caused by Coulomb interactions through the two Fokker-Planck collision operators for distant interactions.
In the ionosphere, a so-called polarization electric ®eld E builds up such that the ions and electrons are constrained to drift as a single gas, which maintains bulk charge neutrality. E is determined by the current J and it exists whenever there is a gradient in the electron density or in the temperature (Min et al., 1993). It is given by E J r e $p e en e À s e r e $ e X 11 If the ®eld-aligned current is attributed to the¯ow of the suprathermal electrons only then the Jar e term is small compared with the gradient terms and we get the following relation between the electric ®eld E and the gradient of temperature $ e E $p e en e À 3c u 2c i e $ e X 12 Using Eqs.
(1) to (7), this leads to the following relationship between the two Knudsen numbers i and In the rest of this paper we always consider the presence of such a polarization electric ®eld. The two Knudsen numbers for the Spitzer-HaÈ rm distribution then always satisfy Eq. (13).

The suprathermal velocity distribution
The suprathermal velocity distribution f s we use is derived from the angular energy¯ux / calculated by the electron transport model code along the Earth magnetic ®eld described in Lilensten et al. (1989) and Lummerzheim and Lilensten (1994).
In the ionosphere, primary photoelectrons or precipitating electrons move along the magnetic ®eld, produce heat and provoke processes such as excitation and ionization. In an ionization process, the incident electron mostly scattered forward is called the primary electron, while the extracted electron may be scattered in any direction and is called the secondary electron. This code calculates the energy¯ux of the electrons by solving the vertical kinetic transport equation. This equation simply expresses the fact that the variation of the steady-state electron¯ux with the scattering depth for a given altitude, energy and pitch angle, is the dierence between whatever leaves that energy, altitude or angle slab and whatever enters it. The variations in energy or angle due to collisions are described through dierential cross-sections. An additional energy loss arises from the heating of the ambient thermal electron gas due to hot electrons to thermal electrons interactions. This loss process is assumed to be a continuous energy loss of the hot electrons to the thermal electrons, without any de¯ection during the process.
We are using the angular energy¯ux calculated by this code as our input to calculate the velocity distribution. The electron velocity distribution is simply related to the angular energy¯ux by where i 1 2 m e v 2 and X is the solid angle. With the assumption that the angular energy¯ux is symmetric around the magnetic ®eld, f s is a two-dimensional function of the energy i or the velocity v and of the pitch angle h or the cosine of the pitch angle l cos h to the magnetic ®eld at a given altitude. The angular energy¯ux / is calculated over an energy grid of 215 points ranging from i min 0X3 eV to i max 350 eV and over a l-grid corresponding to the points of the double-Gauss quadrature integration rule (Stamnes et al., 1988). The number of points in the l-grid is often referred to as the number of streams. The double-Gauss quadrature refers to two Gauss quadratures applied separately on the upper and lower hemispheres. The main advantage of this double-Gauss scheme is that the quadrature points (in even orders) are distributed symmetrically around jlj 0X5 and clustered both towards jlj 1 and l 0, whereas in the single Gauss scheme they are clustered towards jlj 1. This clustering towards l 0 will give superior results near the boundaries where the functions to integrate vary rapidly or can even be discontinuous, i.e. around l 0.
The angular¯ux calculations we are using were obtained by running the code for 25 June 1994 at 14:00 UT over Tromsù assuming an A p index of 3 and a F10.7 index of 75. The ionospheric parameters used as input to the code have been computed by the IRI 90 model (Bilitza, 1990). Figures 2 and 3 show two examples of calculation of the distribution function for an eight-point angular quadrature. Figure 2 shows only the¯ux for one angle, the¯ux at this height is nearly isotropic and one could not separate the¯ux. From a height of about 200km and above, the velocity distribution starts to develop an anisotropy mostly in the direction of the magnetic ®eld, i.e. for jlj 9 1. This feature is clearly seen in Fig. 3: the two angular distributions in the lowest plate are for nearly parallel and anti-parallel directions to the magnetic ®eld and they clearly present dierences in intensity, while in the highest plate (angular distributions for the directions nearly perpendicular to the magnetic ®eld), the two curves cannot be separated.
An interesting function which illustrates the regions in phase space where the heat¯ux is predominantly carried is the ratio of the integrated heat¯ux up to velocity v xv e and normalized to the total net heat¯ux q s (Gray and Kilkenny, 1980). We de®ne in this way the where u s is the mean drift velocity of the suprathermal velocity distribution. Note that with the symmetry around the magnetic ®eld both the mean drift velocity u s and q s are vectors parallel to the magnetic ®eld of component u s and q s , respectively. Figure 4 shows the values of the parameter a at dierent altitudes for a standard set of suprathermal distribution function calculated by the transport code for an eight-stream run. At high altitudes (see Fig. 4 at 246 km for example), the local skewness is more than the net skewness for velocity v $ 30v e , which means that locally the distribution can have skewness of opposite sign compared to the total skewness of the distribution.
We now have a representation for the ambient and the suprathermal distributions, the next operation consists in the treatment of the transition region between the suprathermal and the ambient electrons. Sophisticated methods such as the numerical resolution of the nonlinear Boltzmann equation (Ashihara and Takayanagi, 1974;Jasperse, 1976), as well as full analytical treatment such as the one proposed by Krinberg (1973) have been studied to solve this problem. However, it has been shown later that a good approximation for the complete distribution function can be obtained by joining the two distribution functions at the energy for which the two distributions have equal intensities (Krinberg and Akatova, 1978;Stamnes and Rees, 1983). For simplicity we choose this method and in the rest of this paper the terminology truncated distribution refers to a distribution cut at the velocity where the ambient population equals the suprathermal population.

Numerical two-dimensional plasma dispersion
In linear theory the dierential scattering cross-section d 2 radX dx per angular frequency and per solid angle for a multi-component, uniform, stationary, along the magnetic ®eld and non-relativistic plasma with the collisions eects included through a BGK model is given by (Sheeld, 1975;BjùrnaÊ and Trulsen, 1986;Ichimaru, 1992) d 2 r dXdx 1 p p n e r 2 0 jn Â n Â pj 2 kY xY 16 f aYk f aYk an aYk denotes the velocity probability distribution function for the k th component of the particle species a (e for the electrons and j for the ions). m a is the collision frequency of the particle species aY r 2 0 e 2 a4p 0 m e 2 is the electron radius, n is the unit vector pointing from the scattering volume towards the receiver and p is the unit polarization vector of the incident radiation; x is the frequency shift between the transmitted radio wave x 0 and the received frequency x r , k is the wave vector shift de®ned as the dierence between the returned wave vector and the transmitted radio-wave vector k 0 .
h and a are respectively the dielectric function and the opposite of the susceptibility function for the particle species a.
In order to calculate the dispersion relation, we need to calculate integrals of the and types de®ned by for velocity probability distribution f de®ned in a cylindrical coordinate system along the magnetic ®eld (which is the same direction as the temperature gradient), and when the scattered wave vector k is aligned to the local magnetic ®eld line. When m 0, one can note by applying the Plemelj formula that the imaginary part of is proportional to  the reduced velocity distribution function p k along the direction of k.
When the collision frequencies are very small, we found that can be expressed in the form kY x 9 1 kv e n x kv e Y 29 with n y 2p where w i and l i are respectively the weights and points of a n-points double-Gauss quadrature. In the same way, can be formulated with n y À2p , where Z is the plasma dispersion function (Fried and Conte, 1961). The normalized Doppler shift of the Maxwellian distribution is where n=k/k and When collisions are not negligible, the n and n functions are modi®ed to the following expressions The integral over the normalized velocity is either of Cauchy principal values type or integral of rational functions. Two dierent quadratures are used to calculate these integrals.

Test of n and n on a Maxwellian
We performed tests on the numerical evaluation of the n and n functions for a Doppler-shifted two-dimensional Maxwellian distribution. The result for the n Fig. 6. On the left, the real and imaginary parts of the n function for same complex argument as in Fig. 5. On the right, their relative error with the real and imaginary parts of the function (Ichimaru, 1992). The normalized Doppler shift of the Maxwellian distribution is x d 0X5 function is compared with the function of a reduced Doppler-shifted Maxwellian (Ichimaru, 1992). The result for the n function is compared with Zxa 2 p a 2 p where Z is the plasma dispersion function de®ned by Fried and Conte (1961).
The input for the code consists of a two-dimensional array ®lled with sampled data in both pitch angle and velocity. The velocity points are normalized to the mean drift velocity v e . The parameters used for our test (Figs. 5 and 6) are, for the velocity space: 250 points ranging from 0 to 20v e . It is much more than required and it is seen that the accuracy is not improved by increasing the sampling rate, nor by taking more points in the tail of the distribution function. On the other hand, the test shows that the precision is highly dependent on the number of points in the pitch angle quadrature for the calculation in the near thermal region, i.e. for jvj 4ve, but not too much for velocities jvj b 4ve.
In the thermal region, the accuracy is drastically improved by going from an eight-point double-Gauss quadrature (the relative error is about 10 À1 ), to a 32point quadrature where the relative error is better than 10 À4 . For larger velocities the accuracy is quite stable and is better than 10 À7 .

Test of n and n on the Spitzer-H rm distribution
We also performed tests on the Spitzer-H arm distribution function. We looked at the in¯uence of the upper boundary of integration x max of the i and functions when evaluating n and n . The values of x max we used are the ones listed in Table 1.
For our test we used 5 Á 10 À2 , although the linear theory of heat conduction breaks down for such large values of , that is these values give negative Fig. 7. On the left, the real and imaginary parts of n for real argument (g 0) and for Knudsen number 5 Á 10 À2 and i À3 c a4c i . On the right the dierence between n and Zxa 2 p a 2 p for the four dierent values of x max of Table 1 values of the velocity distribution function (Forslund, 1970). We used the same velocity grid as for the Maxwellian distribution while we increased the number of points in the pitch angle grid to 256 points. The results are shown in Figs. 7 and 8. One can see in the real part of the dierence between n and in Fig.8, the artifact of the discontinuity of the distribution function at x max . This eect is larger for the lowest value x max 2X8 of the boundary i.e. xakv e AE2X8 2 p . For larger values of x max the discontinuity of the thermal distribution is pushed down at higher velocities and is attenuated due to the Maxwellian behaviour at large velocities.

Test of n and n on the suprathermal distribution
We used a 32-stream suprathermal calculation at an altitude of 202 km as input. The transport code calculation of the distribution function was then interpolated over a 1024 double-Gauss points. The n and n functions were then computed using the distribution function evaluated on this denser l-grid. The suprathermal velocity distribution used are very much identical to the one presented in Fig. 3. When comparing with the n and n functions of a Maxwellian or a Spitzer-H arm distribution, it is interesting to see how the characteristics of the distribution function are mapped on the n and n shape. In order to integrate correctly the irregularities or`spikes' corresponding to the discrete solar emission lines, we have to increase the order of the pitch angle quadrature up to 512 or even 1024 points. Increasing further the number of points in the l-grid space does not improve the results for large values of xakv e , i.e. above jxakv e j b 5. On the other hand, for jxakv e j`5 the code is probably not so robust Fig. 8. On the left, the real and imaginary parts of n for real argument and for 5 Á 10 À2 and i À3 c a4c i . On the right the dierence between n and for the four dierent values of x max of Table 1 to the spikes, as can be seen in the upper left plate in Fig. 9, and further developments need to be made.
There are several remarks to be made about the n and n functions. First about the imaginary part of the n function (lower left plate in Fig. 9) which is proportional to the reduced distribution function as is seen in Eq. (28). If the distribution were isotropic the¯at part around zero should be equal on both sides of zero up to the value corresponding to the minimum energy of the suprathermal distribution. The eect of the anisotropy on the reduced velocity distribution function is to create a discontinuity at zero velocity and thus introduce a zero-order skewness. Secondly, on both the real and imaginary parts of the n functions (right plates in Fig.  9.), one can observe the signature of the distribution function itself. In particular, the typical N 2 dip above 2 eV which corresponds to excitation of the vibrational levels in N 2 (see Fig. 2) can clearly be identi®ed around jxakv e j 6X5.

Results
We have used the two-dimensional code of the n and n functions to calculate the frequency of the up-and downshifted Langmuir waves which are the highfrequency solutions of the plasma dispersion equation with the function hkY x given in Eq. (18). We have performed these calculations for two dierent distributions, one that takes into account the deviation from the Maxwellian on the ambient part with the Spitzer-Harm distribution and the other one on the suprathermal part with the distribution calculated from the electron transport code. Fig. 9. On the left, the real and imaginary parts of n for real argument and on the right, the real and imaginary parts of n for real argument of a suprathermal distribution at the altitude of 202 km. These calculations were performed using a 32-stream calculation of the transport code and the distribution function was then recalculated over 1024 double-Gauss points in order to perform the calculations of n and n over this l-grid We used the simulated data for 25 June 1994 at 14:00 UT over Tromsù assuming again an A p index of 3 and an F10.7 index of 75. The ionospheric parameters of the thermal part are shown in Fig. 10 and the velocity moments of the suprathermal distribution, as well as the moments of the Spitzer-HaÈ rm distribution, are shown in Fig. 11.
The lowest right plate in Fig. 10 shows the Knudsen number and i . The largest value is about 4X5 10 À3 . Such values are reasonable and allow the use of the linear theory of Spitzer-H arm. The corresponding polarization electric ®eld i of i is also of the order of the expected value i.e. under 10 À2 lV m À1 . Figure 11 shows the calculated suprathermal centred velocity moments up to the third order, i.e the heat¯ow, for both the raw distribution as calculated by the transport code and the truncated distribution we use in our calculations and which have been processed according to the strategy described at the end of Sect. 3. The lower right plate in Fig. 11 also shows the heat¯ow q of the ambient Spitzer-H arm distribution function calculated numerically and the heat¯ow used by Kofman et al. (1993) which was originally given by Banks (1966) q B À7X710 5 5 2 e r e eV cm À2 s À1 Y 36 assuming a Coulomb logarithm log K 15 and d calculated by Spitzer and H arm (see Table 1). We note that the heat¯ow given by Eq. (36) has larger values by a factor up to 1.5 than the heat¯ow q we calculated. The reason for this is that the approximation given by Eq. (36) is valid for a fully ionized gas only. We have taken into account the electron-neutron collisions in the mean free path (Eq. 10) and the eect is to decrease the two Knudsen numbers and thus the net heat¯ow (Banks, 1966). Figures 12 and 13 show the frequencies of the upshifted Langmuir waves of the plasma lines and the frequency dierence for the three EISCAT radars: VHF (224 MHz), ESR (500 MHz) and UHF (931 MHz). Figure 12 shows the calculation for a deviation on the ambient part, i.e. the Spitzer-H arm distribution. The frequency asymmetry calculated is compared with the heat¯ow approximation of Eq. (9) of Kofman et al. (1993), Figure 13 shows the calculation in the presence of a suprathermal part and assuming that the ambient part is Maxwellian. The frequency asymmetry calculated is also compared with the results given by the heat¯ow approximation, assuming that the total distribution does not deviate dramatically from Maxwellian.
The best agreement between the full dispersion estimation and the heat¯ow approximation for the Spitzer-HaÈ rm distribution is for low-frequency radars like VHF radars. For these radars the phase velocity v / is between 12v e and 25v e as shown in Fig. 14. At such high velocities the moment approximation can be safely used, i.e. the classic expansion 1Àx À1 1xx 2 Á Á Áx n is to be valid at the third order. For the UHF radar the phase velocity v / is between 3v e and 6v e (see Fig. 14) and the approximation breaks and we note a large deviation between the two calculations. This deviation can be observed on the real part of the dierence between n and (upper right plate in Fig. 8) and has to be compared with the asymptotic behaviour in xakv e À5 that we would get by subtracting to the heat¯ow approximation of Eq. (9) in Fig. 11. The parameters of the suprathermal part of the distribution function and the two odd moments of the ambient (Spitzer-HaÈ rm) distribution. In all four plates, the moments of the raw suprathermal distribution function are represented by circles while the moments of the distribution we use for further calculations are represented by solid lines. In the upper right plate (mean drift velocity), the calculated mean Doppler velocity of the ambient distribution v is represented by the dash-dot line and as expected is equal to zero (see Eqs. 11±13). On the lower right plate (heat¯ow), the calculated heat¯ow of the ambient distribution q is represented by the dash-dot line and the dashed line corresponds to the heat¯ow q B given by Eq. (36) Kofman et al. (1993), especially for values of xakv e smaller than 5.
Another remark is about the very large asymmetry observed around 250 km, which is over 10 kHz for the full dispersion calculation. We can see that due to the behaviour of the dispersion function at 4`xakv e`5 , we do not need large heat¯ow values to observe large asymmetry between the up-and downshifted plasma line frequencies. This is very satisfying in that we do not need to invoke larger heat¯ow values through processes such as the electron thermal runaway (Mishin and Hagfors, 1994;Nilsson et al., 1996) to explain the large deviation which were reported by Kofman et al. (1993), especially during 12 May 1992. On the contrary, our smaller heat¯ow values corrected for partially ionized plasma are in good agreement with the theory of Schunk and Walker (1970) and Banks (1966) and are able to create frequency asymmetry of the order of that observed by Kofman et al. (1993).
In the presence of a suprathermal distribution we can make the following remarks. For UHF radars, i.e. at phase velocity v / between 3v e and 6v e , we note that the full dispersion calculation gives similar results as the Maxwellian approximation while the heat¯ow approximation gives larger deviation. In order to understand the small eect of the suprathermal distribution for highfrequency radars, we note that the real part of n of the thermal distribution (Fig. 6) has much larger amplitude than the one of the suprathermal distribution ( Fig. 9) at the considered phase velocity. At large phase velocities v / , i.e. for VHF radars, the thermal n is very small, whereas the one of the suprathermal is still not negligible. This is seen clearly when comparing the mean width of the real part of n in Fig. 6 and the real part of n in Fig. 9. Thus the eect of the suprathermal is important and should be taken into account. Another remark to be made is that if all the ®ne structures observed on the suprathermal n in Fig. 9 in the region jxakj`6v e are real and not artifacts of our calculations, they should map on the frequency asymmetry as it appears in Fig. 13.

Conclusion
We developed and tested a computer code to calculate the plasma dispersion function and the reduced distribution function for any arbitrary distribution function given in two dimensions: velocity and pitch angle. This code has been applied for two types of electron velocity distribution deviating from the Maxwellian distribution, one in the ambient part through a temperature gradient and the other one assuming the presence of a suprathermal electron population.
We used the code to estimate the frequency asymmetry between the up-and downshifted plasma lines which can be observed by incoherent-scatter radar technique. For high-frequency radars such as UHF radars we showed that the frequency asymmetry between the plasma lines is mostly due to a deviation from the Maxwellian in the ambient part of the electron Fig. 12. The upper plate presents the calculated upshifted plasma frequency for the Spitzer-HaÈ rm distribution for the three dierent EISCAT radars. In the lower plate we present the frequency dierence between up-and downshifted lines for the three radars. The Maxwellian approximation is shown with circles, the full twodimensional dispersion estimation is the solid line and the heat¯ow approximation (Kofman et al., 1993) is shown with the dashed line Fig. 13. Same plates as in Fig. 12. The compared distribution functions are a Maxwellian and a Maxwellian superposed with a suprathermal. The line codes are identical to the codes used in Fig. 12 distribution. On the other hand, for low-frequency radars such as VHF radars the Doppler frequency of the plasma lines is more in¯uenced by the presence of a suprathermal electron population.
We also pointed out a discrepancy between the full estimation of the plasma dispersion function and the heat ow approximation for waves with phase velocity such that the moment expansion is not valid. The discrepancy is in the right direction and allows to explain large Doppler asymmetry of the plasma lines without need to increase the value of the heat¯ow. An analytic model of a distribution deviating from the Maxwellian distribution would be a very useful tool to study the dierence between the exact calculation and the moment approximation of the plasma dispersion function. In the left plates, the ratio v / av e where v / is the phase velocity of the Langmuir wave for the three dierent radars (from top to bottom UHF, ESR and VHF). In the right plates, the integrated heat¯ux up to the phase velocity v /