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[1] Several examples of whistler wave packets accompanied by density cavities are detected in F4 experiments of Freja satellite. The density depletion seems to be a fixed structure in space from the cross-correlation of two Langmuir probes. The wavelet analysis shows the frequency drift from 1 kHz (below the low-hybrid frequency) to 200 Hz (below the proton cyclotron frequency). The wave packets are always associated with an extremely low frequency component at 20–30 Hz (around the oxygen ion cyclotron frequency). The method of the minimum variance is used to calculate the oblique propagation of the wave packets with an angle decreased from 31 to 17 degrees between the wave vector and the ambient magnetic field. The right polarization of the wave packets is shown in the direction of the ambient magnetic field. Moreover, the phase velocity of the wave packets is inversely proportional to the frequency. These features may support the formation of envelope solitary whistler waves.

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[2] Whistler waves have been studied for over a hundred years in space and laboratory plasmas as reviewed by Stenzel [1999]. Whistler waves are electromagnetic waves in magnetized plasmas at frequencies below the electron cyclotron frequency. There are also hydro-magnetic whistler waves in the range of ion cyclotron frequency. Whistlers are dispersive waves (e.g., group and phase velocity are frequency dependent), which propagate obliquely to the ambient magnetic field. The spectrum of whistler waves are readily detected in ground-based and spacecraft measurements in ionosphere and magnetosphere [Russell et al., 1971; Al'pert, 1980; Carpenter, 1988; Sazhin et al., 1992; Singh et al., 1998], and may be implied in solar observations [Mann and Baumgärtel, 1989; Chernov, 1990]. There are various of plasma instabilities in ionosphere and magnetosphere mostly due to anisotropic distribution of electrons, such as beams, loss-cones, rings, or anisotropic temperature, which lead to the emissions in the whistler branch [Gurnett and Frank, 1972; Maggs, 1976; Sazhin et al., 1993].

[3] More recently, there are different plasma waves reported with the instrument F4 of Freja satellite in auroral region near 1700 km altitude, such as the low hybrid spikelets correlated with the localized density depletions [Eriksson et al., 1994; Seyler, 1994; Dovner et al., 1997; Kjus et al., 1998], the solitary kinetic Alfvén waves with strong magnetic, electric, and density fluctuations [Louarn et al., 1994; Huang et al., 1997], the electromagnetic ion cyclotron waves at extremely low frequencies (ELF) [Erlandson et al., 1994], the envelope solitary ion-ion hybrid waves with localized density depletions [Wang and Huang, 2001, 2002], etc.

[4] A new phenomenon with evident feature of whistler modes and envelope wave packets accompanied with density cavities and ELF components are first analyzed and reported in this paper. The general feature of the wave packets is given in section 2. More detailed properties on the wave packets, e.g., the frequency drift, density cavity, ELF component, propagation angle, polarization and phase velocity are analyzed in section 3. Section 4 briefly summarizes the conclusions and discussions.

2. Observations

[5] It is possible to measure several wave fields simultaneously by a snapshot technique in the F4 experiment of Freja satellite on plasma waves. In the case of low frequency (LF) with the default mode, there are four channels, i.e., one magnetic component along the spin axis of Freja, one electric component in the plane perpendicular to the spin axis and two density fluctuations of data sampled at 4 ksamples/sec and the snapshot is 0.375 s long for every 2 s. The dynamic range to measure the magnetic, electric and density fluctuations are respectively 10^{−5}nT/ at 1 kHz, 4μ V/m-1 V/m, and 0.01–50%. The instruments of F4 experiments are described in detail in the paper of Holback et al. [1994].

[6] There are over ten of whistler wave packets are collected from the data of F4 experiment. A typical example on July 6, 1993 between 00:38:44.0 and 00:38:44.8 UT is shown in Figure 1a. The slightly enhanced magnetic (0.1 nT) and electric (10^{−3} V/m) wave packets are well coincident with the density depletions (1%). These fluctuations are much smaller than that in previous reports of the low hybrid spikelets and solitary kinetic Alfvén waves, but still reasonable signatures in the dynamic range of the F4 experiment. Some fluctuations at lower frequencies are evidently superposed on the density depletions. It is evident that the magnetic and electric waves shift from high to low frequencies, which is extensively shown in Figure 1b with the central part (00:38:44.3 and 00:38:44.45 UT) of Figure 1a. The time scale of a single pulse increases continuously from about 1 to 5 ms, which is also detected in the density fluctuations. The time scale of the envelope of the wave packets and the density cavities is about 0.2 sec.

[7] The basic plasma parameters of this event are listed in Table 1. Here, , . n_{α}, q_{α} and m_{α} refer to the density, the charge and the mass with the species α of plasma particles, respectively. e, H^{+}, O^{+} represent electron, hydrogen ion and oxygen ion, respectively. B_{0} is the ambient magnetic field strength, c is the velocity of light. The ratio between the density of the oxygen ions and the density of ambient plasma is 50% in average. The low hybrid frequency decreases from 2.3 kHz to 1.3 kHz with increasing the proportion of the density of the oxygen ions [Stasiewicz et al., 1994]. The magnetic local time is about 16.0, the magnetic latitude is about 70.0. The altitude of this event is about 1760 km.

Table 1. Plasma Parameters on 6 July 1993

α

n_{α}

f_{pα}

B_{0}

f_{cα}

e

4600cm^{−3}

600 kHz

25000 nT

700 kHz

H^{+}

2300 cm^{−3}

10 kHz

25000 nT

380 Hz

O^{+}

2300 cm−3

2.5 kHz

25000 nT

20 Hz

3. Data Analysis

3.1. Frequency Drift

[8] It is well known that the Fourier transformation (FT) spectrum is obtained by integral from t = −∞ to ∞, which is stationary or time independent. But, the observational signals f(t) are usually generated from a time-varying system, i.e., the varying properties of this system is described by the spectrum in every instant time or named time-frequency spectrum. The simplest method for FT is to integrate in a short time interval, that is, the windowed FT (FT^{w}):

Here, W_{τ} is the window function, , the bracket in the right side denote the inner product. The key problem of the time-frequency spectrum in FT^{w} is that the resolution ability or rate in both of time and frequency domains may be contradicted for a fixed width in the window function in time domain. If the width is big, the resolution rate in time domain is lost, and if it is small, the resolution rate in frequency domain is decreased.

[9] The mother function of the wavelet transformation (WT) is selected as Ψ(t) ∈ L^{2}(−∞,∞) [Charles, 1995],

to satisfy

and

where a and b are parameters of WT. Then, the WT and its inversion are defined by

[10] The main advantages of WT are compared with FT^{w} as follows.

[11] 1. The width of the wavelet in time domain depends adaptively on frequency ω, hence the resolution rate in both time and frequency domains is better than that of FT^{w}.

[12] 2. If N is the number of points in time domain, and the sample frequency be ω_{s}, then Δt = . According to the sampling theorem, discrete F_{f}(ω_{i}) in FT^{w} is a linear distribution in [0, ] divided equally by , i.e., Δω = , or Δω ċ Δt = , which is something like the uncertainty relation. If we want to know its spectrum denser in frequency, then we have to divide time domain sparer, and vice versa. However, there is no such a contrariety in WT theoretically.

[13] 3. In FT^{w}, all the discrete points are linear distribution in both the time and frequency domains. But there are a lot of nonlinearity in the practical observations, which are just the most interesting phenomena that we want to know.

[14] Therefore WT is suitable for analysis of a time series containing non-stationary power at many different frequencies as shown in Figure 1. Figure 2 gives the dynamic spectrum of magnetic and electric fields, and two density fluctuations in Figure 1. The frequency drift from 1 kHz (below the low-hybrid frequency) to 200 Hz (below the proton cyclotron frequency). It is also found an extremely low frequency (ELF) component at 20–30 Hz (close to the oxygen ion cyclotron frequency) without the frequency drift in the density fluctuations. The FT of LF component of magnetic and electric fields are shown in Figure 3, which is comparable with the wavelet analysis in Figure 2, e.g., the frequency range is from 200 Hz to 1 kHz with maximum at about 0.5 kHz in Figures 1 and 2.

3.2. Density Cavity

[15] The further analysis is performed using a digital low or high pass filter and FFT to separate the DC, ELF and LF components in the four time series of Figure 1.

Here, m and n are parameters of the filter. The DC component of the density depletions is well coincident with the LF component of electric and magnetic wave packets in the example (Figure 4), which means that the whistler waves may be trapped in the density cavities. Moreover, the ponderomotive force of the wave packets may deepen the seed density cavities [Seyler, 1994].

[16] The time delay is about 0.8 ms from the cross correlation between the time profiles (selected in Figure 1b) of two Langmuir probes (with distance of 21 m) to measure the density fluctuations (Figure 5). Hence the speed of the density cavities is about 25 km/s, the angle θ between the line of two probes and velocity of Freja is 70 degree, and the product of 25 km/s and cosθ is comparable with the speed of Freja satellite [Dovner et al., 1994]. The local density structure with time scale of 0.2 s is about fixed in space with a length of about 500 m in the direction of Freja.

3.3. ELF Component

[17] The ELF component of density fluctuations at 20–30 Hz in Figures 1 and 2 is confirmed with Fourier transformations of dn1 and dn2 in Figure 6, in which the peak frequency is below 50 Hz. The envelope of the ELF waves in Figure 5 is quite similar to that of the whistler waves in Figure 4, and well coincident with the density cavities (Figure 7). Note that the DC, LF and ELF components of the density fluctuations are comparable in magnitude and detectable simultaneously in Figures 1 and 2. In fact, the ELF component also exists in the magnetic and electric fields, but their amplitude is much smaller than that of the whistler mode (LF component).

3.4. Propagation

[18] The second example of whistler waves is selected at 04:23:21.2– 21.4 UT with three magnetic and one electric fluctuations in Figure 8 (“elmag” mode of F4 experiment). The time scale and evolution are quite similar to that in Figure 1b. The angle between two vectors of the magnetic fluctuations and the ambient (DC) magnetic field by F2 experiment is between 56–57 degree (Figure 9), which means that the propagation of the whistlers is oblique in respect to the ambient magnetic field, because the wave vector is perpendicular to the vector of magnetic fluctuations. The wave vector of the whistlers may be calculated using the data in Figure 8 and the method of the minimum variance analysis [Song and Russell, 1999]. The percent variance of three eigenvalue is 50.98, 45.84 and 3.175. The wave vector is obtained from the calculated eigenvector (177.68, −46.52, −24.32) corresponding to the third eigenvalue of the minimum variance, and the magnetic fluctuations rotate mostly in a plane constructed by the other two eigenvectors (Figure 10). The three components of the averaged ambient magnetic field by F2 experiment in the duration of the second example in Figure 8 are 24062.1, −10972.7, and 5151.57 nT, respectively. Hence the angle between the wave vector and ambient magnetic field is about 21 degree, which is consistent with the prediction of Figure 9. Moreover, if the wave vector is calculated in three time intervals of the example, the angle decreases with time from 31, to 26 and 17 degree, respectively.

3.5. Polarization

[19] The polarization sense is readily shown by the counterclockwise rotation of the magnetic fluctuations projected in the plane that perpendicular to the ambient magnetic field (Figure 11), e.g., the right circular (RC) polarization sense of the whistler waves is deduced in the direction of the ambient magnetic field. The RC polarization sense is also obtained from the analysis of the minimum variance in Figure 10, in which the counterclockwise rotation of the vector of magnetic fluctuations is detected in the direction of the wave vector (177.68, −46.52, −24.32) that is quasi-parallel to the ambient magnetic field.

3.6. Phase Velocity

[20] One component of the phase velocity may be estimated by the ratio of the electric and magnetic amplitudes [Huang et al., 1997]. The order of magnitude of the phase velocity is 10^{7} m/s (the local Alfvén velocity) by the ratio of 10^{−3} V/m (de) and 0.1 nT (dbz) in Figure 1b. Moreover, the phase velocity may be calculated at a given frequency between 300 Hz − 1 kHz in Figure 2. The calculated phase velocity is inversely proportional to the frequency, which is simulated by the curve V_{phase} ∝ 1.2/f(kHz) 10^{7}m/s (Figure 12). The frequency sample rate is 100 Hz in the wavelet analysis, the phase velocity is averaged at a given frequency with error bar as shown in Figure 12.

4. Discussion

[21] The phenomena are characterized by 1) the frequency drift from 1 kHz to 200 Hz between low hybrid and proton cyclotron frequencies, 2) the propagational angle decreases from 31, to 26, and 17 degree in respect to the ambient magnetic field, 3) the RC polarization sense, 4) the frequency dependence of the phase velocity, and 5) the wave packets at both of whistler and oxygen ion cyclotron modes are accompanied with density cavities fixed in space. These features are independent from the other envelope wave packets observed by F4 experiment of Freja, such as lower hybrid wave cavities studied intensively in many papers [Eriksson et al., 1994; Seyler, 1994; Dovner et al., 1997; Kjus et al., 1998], and the envelope soliton below the H^{+} gyrofrequency [Wang and Huang, 2001, 2002]. In comparison with the electrostatic low hybrid spikelets and associated density depletions, the electromagnetic mode and the frequency drift of whistlers are quite different from the low hybrid waves. On the other hand, both of these two waves may be trapped by the density cavities that fixed in the space. The wave mode, time scale, and associated density cavities of the electromagnetic ion cyclotron waves below proton or oxygen gyro-frequency may be comparable with the whistlers. However, the frequency drift is the most independent feature of the whistlers from the others, which may be considered as the first report in Freja data sets. On the other hand, the formation of the whistlers may be associated with that of the other wave packets. It may support the hydro-magnetic whistler mode associated with oxygen ion cyclotron mode, and the formation of the envelope solitary waves of these modes. The exact mode of the wave packets should be confirmed by the dispersion relation with parameters in Table 1. Moreover, another experiment F7 show the signature of energetic electrons or ions associated with the whistlers, which may be helpful for understanding the excitation of the wave packets. After that, the whistlers may be trapped in the density cavities, or the envelope solitary whistlers are excited together with the density cavities. It can not be excluded the other possibilities to explain the frequency drift of the wave packets, such as the nonlinear resonance or wave-wave interaction of low hybrid and proton cyclotron modes [Putterman, 1989]. The theoretical model will be studied by the authors in forthcoming papers.

Acknowledgments

[22] This study is supported by the NFSC projects 10273025, 10333030, and “973” program G2000078403. The data of the F4 and F7 experiments are collected at the Centre d'Étude des Environnement Terrestre et Planétaires in France and Alfvén Laboratory in Sweden, respectively.

[23] Arthur Richmond thanks G. P. Chernov and another reviewer for their assistance in evaluating this paper.