Geophysical Institute, Acad. Sci. Czech Republic, Prague
F. Z. Feygin and D. S. Fligel
Institute of Physics of the Earth, Moscow
In recent years, the idea of an Alfvén maser developed by Belyaev et al. [1984, 1985, 1987, 1990a] and Belyakov et al.  has gained wide recognition, contributing considerably to the theory of the generation of type Pc 1 ( f 0.2-5 Hz) geomagnetic pulsations ("pearls"). The mechanism is based on solving the self-consistent problem of the interaction of Alfvén waves with hot anisotropic protons on field lines with footpoints in conjugate ionospheres. Using the "realistic" properties of the medium, the aforesaid authors were able to explain the formation of the structured emission, a long-standing mystery in the theory of generation of the "pearls". These authors considered the quasi-linear interaction of Alfvén waves with hot anisotropic protons and precipitation of the latter into the ionosphere (ensuring the auxiliary ionization), which led them to interpret the fine effects of the dynamical spectrum of the "pearls" differently from the mechanism earlier proposed by Feygin and Yakimenko  and Gendrin et al. .
One of the components of the Alfvén maser is the ionospheric resonant cavity [ Lysak, 1988] or the ionospheric Alfvén resonator [ Polyakov and Rapoport, 1981; Prikner and Vagner, 1990], numerical simulation of which is reported by Ostapenko and Polyakov , Prikner and Vagner , and Rudenko . Resonance properties of the ionospheric cavity result in the fact that the reflection coefficient R is a function of frequency and has clearly pronounced peaks. Wave generation takes place in a narrow frequency band near one of the maxima of the reflection coefficient in multiple passage of a signal through the region of amplification. These ideas led Feygin et al.  and Nekrasov et al.  to attempt a search for simultaneous series of "pearls" propagating along the flux tube. These authors suggested that the central frequencies of the "pearls" observed simultaneously on the given station correspond to maxima of the reflection coefficient, and the difference between the central frequencies of the series of one event (consisting of several series) is similar to the frequency difference between the reflection coefficient peaks. In addition, Feygin et al.  and Nekrasov et al.  also considered other possible mechanisms for the generation of simultaneous (parallel) Pc 1 series with similar frequencies, e.g., the possibility of Pc 1 generation in different regions of the magnetosphere and their propagation over different L shells, and also the effect of the heavy ions He+ and O+ on the propagation of the ion-cyclotron waves (Pc 1 pulsations) near gyrofrequencies of these ions. In both cases, the picture of the wave process would differ significantly from that actually observed.
The idea of reconstructing the ionospheric electron density profile over the maximum of the F2 layer was first put forward and substantiated by observations of spectral resonance structures of atmospheric electromagnetic noise in the frequency range 0.5-10 Hz by Belyaev et al. [1990b].
In this paper, we employ the method of ionospheric modeling [ Prikner and Fligel, 1991, 1992, 1993] for determination of the electron density profile Ne(Z) of the ionospheric Alfvén resonator from the spectral structure of the series of Pc 1 geomagnetic pulsations on the assumption that the characteristics of reflection at conjugate points are similar during the occurrence of these pulsations.
In our work we used dynamical spectra of ten Pc 1 events observed at the Kerguelen (49o 21'S, 70o 12'E, geographic; L = 3.7 ) and Sogra ( 62o 48'N, 46o 15'E, geographic; L = 3.6 ) observatories in 1964-1966, and at the Nurmijarvi ( 60o 30'N, 24o 42'E, geographic; L = 3.5 ) observatory in 1977 [ Nekrasov et al., 1991]. All the events observed featured two to four simultaneous Pc 1 series having relatively similar time-independent mean frequencies f0i and different widths D fi of their spectra obtained from sonograms, but approximately the same times ti of group delay (except for events 4 and 5). These parameters for all ten events are listed in Table 1.
Each individual series in the sonogram consists of a number of structural elements, which are substantially the Pc 1 wave packets. The analysis of the dynamical spectra of ten Pc 1 events in the sonograms have shown that the difference between the mean frequencies of a series in each event is 0.2-0.5 Hz, and the spectral width of the individual series D fi is 0.05-0.2 Hz. The lifetimes of individual series differ.
The solution of the inverse problem of ionospheric modeling is described by Prikner and Fligel [1991, 1992, 1993] based on calculated characteristics of the ionospheric filter: the coefficient of reflection R and transmission j on both boundaries of the ionosphere. The dipole magnetic field and variations of the height characteristics of the base models of ionosphere were taken into account. This method is outlined by Prikner and Vagner . It was assumed that incident on the irregular, anisotropic, and absorbing thin-layer ionosphere are left-polarized waves with a frequency f and a wave vector oriented in the magnetic meridian plane at an angle b to the ionospheric boundary. Calculations performed by the method described by Prikner and Vagner  yield R and j:
|R(b, f) = Arefl / Ainc , j(b, f) = Aprop / Atot||(1)|
where Arefl is the amplitude of the wave reflected from the ionosphere, Ainc is the amplitude of the incident wave, Aprop is the amplitude of the wave that has propagated to the Earth's surface, and Atot is the sum of the incident wave and the wave reflected from the ionosphere.
As is shown by Belyakov et al. , Ostapenko and Polyakov , Prikner and Vagner [1983, 1990], and Rudenko , the frequency dependencies R(f) and j(f) at a prescribed incidence angle b have an oscillatory structure with several peaks arranged more or less regularly and characterize the resonator at its two boundaries. The frequency arrangement of the peaks depends on the main characteristic of the resonator: the electron density vertical profile Ne(Z), which also determines the attenuation of waves in the resonator. With regard to the height profiles ne(Z), ni(Z), and mi(Z), where ne and ni are the effective collision frequencies of the charged particles and mi is the effective ion mass, these do affect the values of R(f) and j(f) , but not the frequency arrangement of their resonance peaks.
In this work we are only interested in the frequency arrangement of the resonance peaks of R(f) and j(f), and not in their absolute values; therefore the choice of the height profiles ne, ni, and mi will not be discussed.
In modeling the resonator characteristics (1), the choice of the ionospheric model is essential. For auroral geomagnetic latitudes, models for the cases of low and high solar activity were used. Vertical profiles of Ne, ne, and ni for some models are shown in Figure 1 for heights of 200-3000 km. Models DL and NL are for daytime and nighttime conditions, respectively, and characterize the ionosphere under low solar activity ( F10.7 = 70 ), whereas models DH and NH describe the daytime and nighttime ionosphere under high solar activity ( F10.7 = 200 ). We will call these the base models. The distribution of Ne(Z) in the lowest ionosphere at Z 200 km is the same as reported by Prikner . We also used an auxiliary model DM constructed for the mean physical parameters of the ionosphere, i.e., for average (over solar activity) distributions of the temperature Te, Ti, and Tn, densities Ne and Nn, and ion masses mi according to Prikner .
To determine the Ne with an arrangement of peaks of the R(f) coefficients which is closest to the experiment, we used three types of variations of the vertical profiles of Ne at altitudes Z> 200 km with the base models DL, DH, NL, NH and DM [ Prikner and Fligel, 1993].
In type I the distributions of Ne(Z) in the base models DL and NL varied as follows. It was assumed that the maximum increase in Ne(%) relative to the base model, i.e., the value of pNe(%) occurs at an altitude of Z 300 km, and then pNe decreases with height and disappears at Z 1500 km, i.e., the changed profile of Ne(Z) at Z 1500 km coincides with the base profile.
In type II the decrease in density at altitudes Z 300-500 km ( pNe < 0 ) and its slight increase at altitudes Z 1000-1200 km was considered. This variation may be determined by "heating" of the neutral component of plasma because of friction with the accelerated ions [ Sellek et al., 1991]. The effect disappears at altitudes Z 1500 km.
For type III both positive and negative variations of Ne with a constant value of pNe(%) along the entire profile were considered. The increase in Ne in models DL and NL was assumed up to the values of Ne in models DH and NH.
The frequency distribution of the peaks of R(f) also depends on the choice of the angle of incidence of waves onto the ionosphere [ Prikner and Fligel, 1992; Prikner and Vagner, 1983, 1988]. For the Kerguelen, Sogra, and Nurmijarvi observatories, values b = 68, 70, and 73o were selected. Values b = 68 and 70o suited the experimental conditions best of all. Since the Pc 1 wave envelopes propagate in the form of left-handed waves along the field lines, their angle ( b ) of incidence onto the ionosphere is determined by the angle of entry of the field lines into the ionosphere.
Figure 2a shows the results of modeling event 4 (April 22, 1965, 0220 UT, Kerguelen observatory). The most suitable was the ionospheric model DL using type I variations with pNe = 100%, b = 70%. The peaks of R(f) are seen to agree well with the experiment (vertical lines). The third series of this event with a mean frequency f 1.1 Hz is found in the center of the twin maximum of R(f). Another example of good agreement of the modeling results with the experiment is shown in Figure 2b for event 9 (Kerguelen, March 30, 1966, 2230 UT), where the nighttime model NL with type III variations, pNe = -30% and b = 68%, is used. The Pc 1 series with central frequencies of 0.58 and 1.0 Hz agrees well with the R(f) peaks. Figure 2c shows the calculation results for event 1 (Kerguelen, April 6, 1964, 0800 UT). For this event, good agreement was obtained using a model with type III variations and pNe = 40%. The calculations of the transmission function are shown in Figure 2d with the clearly seen sharp peaks of the resonance variation of the j(f) coefficient with a very weak attenuation of the wave amplitude.
Figure 3 shows the height profiles Ne(Z) calculated for the same events 1, 4, and 9. These profiles are used in the calculations shown in Figure 2. Figure 3 compares these with the base models of Ne(Z) shown as solid lines. Fitting of the obtained profiles to the Ne(Z) profiles at Z < 300 km was carried out in the same way as Belyaev et al. [1990b] did for models D5L and N5L.
The modeling results for all the events are listed in Table 2, where f0i are the values of mean frequencies for the R(f) peaks. For all the events, except event 10, the models for low solar activity proved to be the most suitable for the experimental conditions. This is not just accidental, since the low geomagnetic activity in all the events considered is confirmed by the low values of the indices Kp | 20 | and Ap 6. For event 10, model DM proved to be the most suitable, but with somewhat decreased values of Ne over the entire profile.
For comparatively long series of the "pearls" with a constant central frequency corresponding to a maximum signal amplitude to exist, there must be relatively stable properties of both the source of amplification and the conditions of propagation in the flux tube and reflection of signals in the conjugate hemispheres during the entire time of recording. One indirect evidence of such stable conditions may be the low values of the Kp and Ap indices in all the events in question. Besides, a condition for the appearance and persistence of such a series of "pearls" is the practical coincidence of the reflection characteristics in the conjugate hemispheres (at least in the frequency intervals of the Pc 1 series). The assumption of similar reflection characteristics of waves in the conjugate hemispheres during the duration of the series of "pearls" with a time-constant central frequency is evidenced by the very fact of the existence of the discussed pulsation events. Incoherence of the reflection coefficients in frequency in the conjugate hemispheres may result in both the impossibility of a multiple amplification process (reflection in the Alfvén maser in general) and, for example, the occurrence of different shifts of the central frequency. Such events of the "pearls" do occur, but these are beyond the scope of the present paper.
Analysis of the frequency components obtained in the modeling has shown their good agreement with the observed frequencies of the series for all the Pc 1 events under study, except in events 5 and 10 (see the numerals in parentheses in Table 2). For these events, modeling resulted in occurrence of a frequency component that did not exist within the interval of the recorded series. Such a spectral component can disappear upon reflection from a conjugate ionosphere, if it is absent in the reflection spectrum.
It should be noted that the occurrence of the component f0i 0.5 Hz ( ti 152 s) in event 5 (see Table 1) is of special interest because this event is beyond the scope of the hypothesis proposed by Feygin et al.  and Nekrasov et al.  that explains the occurrence of simultaneous Pc 1 series and is corroborated by the present work. In accordance with this hypothesis, occurrence of simultaneous Pc 1 series is determined by one source generating a wide Pc 1 spectrum, in which, because of the resonance character of the reflection coefficient R(f), only those frequencies that correspond to the R(f) maxima are isolated. In this context, it is assumed that the repetition periods ti of signals in each series of a given event practically coincide. On the other hand, the values of ti in event 5 are 152, 95, and 93 s. This fact may be related to the assumption that the generation source itself occupies a wide spatial region extended in the radial direction; therefore, the generated ion-cyclotron waves come to the observation point by different paths and have different times of group delay. A similar situation may take place in event 4 as well.
Solving of the problem with the aim of determining not only frequency characteristics but also the amplitude characteristics of the observed Pc 1 series would be far harder, since this would require consideration of the ionospheric temperature characteristics Te(Z), Ti(Z), Tn(Z) and also the dependencies of ne(Z) and ni(Z). Consideration of these is especially important in the lowest ionosphere (up to the maximum of the F2 layer), where these exert strong influence on wave attenuation, thereby affecting the amplitude of the reflection and transmission coefficients.
If one assumes that the Ne(Z) vertical profiles can be obtained by the comparatively simple method described above, then, by varying the ne(Z) and ni(Z) vertical profiles, one can obtain the transmission coefficients j(b, f) . This, in turn, will make it possible to determine the ratio of pulsation amplitudes on both boundaries of the ionospheric layer. And, approximate as the obtained results may be, this still would enlarge applications of the method of numerical simulation of ionospheric filtration of the Pc 1 range signals.
Electron density vertical profiles Ne(Z) of the ionospheric Alfvén resonator are determined from ground-based observations of simultaneous series of the Pc 1 geomagnetic pulsations. The obtained Ne(Z) profiles agree quite well with the known Ne(Z) models of the ionosphere for low solar activity, which also agrees with the conditions of the experiment (low Kp indices).
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