3. Results

[8]  1. Below we consider the main morphological characteristics of the giant pulsations detected at MOL. The giant pulsations are clearly seen in the MOL magnetograms due to the characteristic drop-like shape of the envelope with smoothly increasing amplitude and quasi-sinusoidal oscillations. There is an extreme in the diurnal distribution of Pg within the time interval 0400-0700 MLT. The oscillation amplitude in the majority of cases is 5-15 nT, reaching in some cases 25 nT. The duration of the Pg pulsations varied from 20 to 100 min. One maximum is observed in the Pg spectral structure usually falling into the period range 100-129 s. A tendency of the decrease of the frequency ( f ) of Pg pulsations depending on the local time (MLT) of observation of the pulsation amplitude maximum is detected. Within the time interval 0000-0800 MLT as a rule Pg with higher carrier frequency of wave packets are observed, whereas after 0800 MLT the Pg frequency is usually much lower. This dependence of f on MLT was earlier noted by Green [1979] and Thompson and Kivelson [2001].

[9]  The maximum probability of Pg observations at MOL falls on the spring equinox season (60% of cases). In the autumnal equinox and summer solstice 23% and 17% of analyzed events, respectively, were registered. All the Pg cases fall on the declining phase of the 21d cycle and the following minimum of solar activity. The Pg generation was observed at weakly disturbed magnetic field ( Kp<2 ) and quiet solar wind characterized by low values of the velocity ( V 350-400 km s^-1 ), density ( n 6-8 cm^-3 ), and the magnitude of the interplanetary magnetic field ( B 2-4 nT). Almost all cases of Pg were observed at MOL on 4-5 days after the beginning of a small magnetic storm, i.e., at the recovery phase. Thus the morphological properties of the giant pulsations detected at MOL coincide with the characteristics of Pg pulsations studied earlier by other authors on the basis of the auroral observatories in the northern and southern hemispheres of the magnetosphere [Annexstad and Wilson, 1968; Brekke et al., 1987; Chisham and Orr, 1991; Green, 1979].

Figure 1

[10]  2. Figure 1 shows examples of the Pg pulsations observations in the D component at MOL on 3 October 1985 within the time interval 0503-0528 UT (Figure 1a) and 6 January 1987 within the time interval 0517-0624 UT (Figure 1b). On can see that the duration of the Pg wave packet on 3 October 1985 is shorter than for the Pg event on 6 January 1987. At the same time the Pg amplitude on 3 October 1985 is higher than on 6 January 1987. As a rule, in all the analyzed cases the higher Pg amplitude, the less the width of its wave packet. In other words, the amplitude of giant pulsations is not an independent parameter as it is typical for linear wave packets.

[11]  The solid curve in Figure 1 shows the envelopes of the wave packet amplitudes of Pg pulsations. Figure 1 fairly visually illustrates similarity of the wave packets amplitude envelope of Pg pulsations to the shape of the envelope of solitary waves (solitons) [Dodd et al., 1988; Kadomtsev, 1988]. However the visual similarity of the envelope of the wave packet amplitudes of Pg pulsations to the shape of the soliton envelope is a necessary but not sufficient fact in order to identify Pg to solitons. Guglielmi [1979] described a rather simple criterion of soliton identification in the near-Earth plasma based on the analysis of the nonlinear parabolic Schrödinger equation with a derivative (NESCD). The solution of this equation is an envelope soliton. This equation was for the first time obtained by Mio et al. [1976a] and Mjolhus [1976]. Then Mio et al. [1976b] showed that in the long-wave approximation the NESCD equation is transformed into the nonlinear Schrödinger (NESC) equation. The so called soliton relation for the Alfvén waves propagating along the external magnetic field was obtained from NESC by Guglielmi [1979] and has the form

where A, t, and f are the amplitude, duration, and carrier frequency of a soliton, respectively.

[12]  The linear dependence of the Pg wave packet amplitude on could have served as a proof that giant pulsations are the envelope solitons. It is worth noting that this criterion is most acceptable for isolated wave packets of geomagnetic pulsations. Because we considered Pg observed at MOL as separated wave packets, this criterion may be applied in our case.

[13]  Now we consider the results of realization of the above described criterion of soliton identification in wave packets of Pg pulsations. To do that we estimate the maximum amplitude, carrier frequency, and duration of every Pg wave packet. Initially we drew the wave packets amplitude envelopes of Pg for the radial ( H ) and azimuth ( D ) components. Using the obtained Pg pulsations wave packets envelope, we have found instantaneous amplitudes H (t) and D (t) on H and D components accordingly. The amplitude of Pg pulsations A (t) was calculated using the expression

For the analysis the maximum A (t) of each Pg wave packet was used. It is necessary to note that the average dispersion at determination of amplitude was less than 0.05 of its maximum value. Determining the duration of wave packets of Pg pulsations, we used the results of Bondarenko et al. [1979] and Guglielmi [1979], where the duration of solitons was estimated at the 0.5 level of the maximum amplitude ( A ). For such a method of determination of the Pg wave packet duration the error t did not exceed 30 s. The carrier frequency of separated wave packets of Pg was determined using the Gilbert transformation [Vainshtein and Vakman, 1983]. It should be noted that the carrier frequency of wave packets of Pg pulsations in individual events corresponded to the interval 5.7 < f < 11.9 mHz. The value of f averaged over all analyzed cases was equal to 8.60 1.51 mHz. The dispersion of the filling in frequency of separated Pg wave packets did not exceed 0.17 of the mean value of carrier frequency.

Figure 2

[14]  Figure 2 shows the relation between the amplitude, carrier frequency, and width of a wave packet of Pg pulsations for 30 events. Each point in the graph corresponds to an individual case of Pg observation. One can see in Figure 2 that the dependence obtained is fairly well approximated by the linear function. The correlation coefficient, showing how closely the calculated line fits the experimental data, is 0.88. The regression equation between A and 1/t f^1/2 have the form

It should be noted that at the approximation of the dependence between A and 1/t f^1/2 , there appears to be a shift of the amplitude equal to 1.7. It is obvious that the shift is coupled with the uncertainties of determination of the amplitude and carrier frequency of the Pg pulsations wave packets. However, the value of the shift is considerably less than the amplitude maximum value and practically does not influence the qualitative agreement between the experimental data and results of the theory. It follows from the result presented in Figure 2 that there exists a linear dependence of A on 1/t f^1/2. This corresponds qualitatively to theoretical dependence A 1/t f^1/2 and makes it possible to assume that the analyzed cases of Pg pulsations at MOL observed as separated wave packets are envelope solitons.