Waveform and packet structure of lion roars

The Equator-S magnetometer is very sensitive and has a sampling rate of normally 128 Hz. The high sampling rate allows for the first time fluxgate magnetometer measurements of ELF waves between the ion cyclotron and the lower hybrid frequencies in the equatorial dayside magnetosheath. The so-called lion roars, typically seen by the Equator-S magnetometer at the bottom of the magnetic troughs of magnetosheath mirror waves, are near-monochromatic packets of electron whistler waves lasting for a few wave cycles only, typically 0.25 s. They are right-hand circularly polarized waves with typical amplitudes of 0.5–1 nT at around one tenth of the electron gyrofrequency. The cone angle between wave vector and ambient field is usually smaller than 1.5°.


Introduction
More than 20 years ago, Smith and Tsurutani (1976) published the ®rst search-coil magnetometer observations of what they called lion roars inside magnetic troughs in the dayside magnetosheath. They found that the lion roars are narrow-banded right-hand polarized waves, basically electron cyclotron waves, that are relatively short-lived, about 2 s, and that they have typical frequencies of about 120 Hz and typical amplitudes of 0.1 nT. Some years later, Tsurutani et al. (1982) studied the same phenomenon with the plasma wave instrument on ISEE. They found these waves at somewhat lower frequencies (50±100 Hz) and also established that lion roars are intimately related to mirror mode structures and that they are caused by the perpendicular electron pressure anisotropy in the magnetic troughs of these structures.
However, due to the instrumentation employed in the earlier studies, the actual waveform of the lion roars was unknown until very recently. Zhang et al. (1998) used the waveform capture instrument onboard Geotail to study 20±300 Hz lion roars with typical amplitudes of 0.1 nT within mirror waves (their type A, similar to the lion roars analyzed in the earlier studies and the present paper) and similar waves without ambient magnetic ®eld depletion closer to the bow shock (their type B). Zhang et al. (1998) did not discuss the wave form itself very much, but rather the angle between the wave vector, k, and the ambient ®eld. They did this by applying a minimum variance analysis to the magnetic waveform and removed the AE180 -ambiguity of the k-vector using electric ®eld data. They obtained typical cone angles between the wave vector and the ambient ®eld of h kf % 10 , which is lower than the typical values found by Smith and Tsurutani (1976), but still rather oblique for electron whistlers.
In the present paper we will also study the waveform of mirror trough lion roars, but using the¯uxgate magnetometer onboard Equator-S. This will limit our study to waves below 64 Hz, but extend the frequency range down to 8 Hz. Using a dierent approach we will show for single cases as well as in statistics covering 356 cases that (1) lion roars can be found down to 8 Hz (the lower limit set in our analysis), (2) the lion roar wave forms have a clear packet structure, and (3) the average cone angle found in our frequency range, h k % 0X3 , is much smaller than previously thought and that lion roars travel essentially parallel to the background ®eld.

Instrumentation and data
The Equator-S magnetic ®eld instrument is fully described in Fornacon et al. (1999). In short, it consists of two units with a pair of three-axes¯uxgate magnetometers each. The sensors of the primary and the redundant units are mounted on two rigid booms, with the main sensor located at the end of the 1.8-m boom and the other 50 cm further inboard. The sampling rate is 128 vectors/s in normal mode, when only the outboard magnetometer is used, and 64 vectors/s for dual mode operation. The amplitude resolution is 16 bit and ranges are selected automatically in steps of 4 between 256 and 64000 nT. The data used in the present study are mainly 128 Hz data, all sampled with a resolution of about 10 pT in the 256 nT mode.
For the four months the Equator-S encountered the near-magnetopause day-and morningside magnetosheath, we searched for all occurrences of mirror modes and looked into each of their troughs or minima to ®nd lion roars. We were successful in 356 cases.

Case studies
Before proceeding to the full data set, we will ®rst present six typical examples. The upper panel of Fig. 1 shows the magnetic amplitude of a magnetosheath mirror mode observed on 24 January 1998. Note that the trough is rather deep, with Dfaf % 1, and that the ®eld in the trough's center is slightly enhanced. Actually, we found that most mirror troughs had Dfaf % 1 and many had a stronger core ®eld.
The lion roars can already be recognized in the un®ltered data (see Fig. 7 of Fornacon et al., 1999) but here we use high-pass ®ltered (f ! 4 Hz) components in mean ®eld-aligned coordinates (middle panels of Fig. 1). Three lion roar events, stretching over about 1 s each could be identi®ed in the trough. The lion roars do not appear right at the center of the trough, but rather at its edges, on both sides of the stronger core ®eld. In all three cases, the two transverse components are nearly identical but phase-shifted by 90 , indicating right-hand circular polarization of the wave, just as expected for electron whistlers.
Since the lion roar signals are rather short-lived, we use a wavelet analysis to determine spectrograms of the signals. The wavelet used is the so-called Morlet wavelet, which is best suited for this type of analysis (Lui and Najmi, 1997;Torrence and Compo, 1998). By transforming the complex signal (f rYl f x AE if y ), we calculate the right-hand and left-hand polarization spectrograms. The lower panel of Fig. 1 shows the spectrograms of the right-hand polarized signals.
The left-most diagrams show one of the strongest and longest-lived lion roar signal wave form we found in our data set, about 10 wave cycles with an amplitude of AE1 nT. The signal is nearly monochromatic with a center frequency near 15 Hz. This is less than 1/20 of the electron gyrofrequency near 350 Hz. The lion roar in the middle panels has a somewhat higher frequency of 21.5 Hz despite of the same background ®eld magnitude. Moreover, it is clearly broken into 3±4 wave packets, some of them lasting only for 2±3 wave cycles. The third lion roar (in the right-most diagrams) has again a somewhat dierent center frequency (18 Hz).
We have also determined the wave normal direction as the direction of the minimum variance, i.e., along the eigenvector having the smallest eigenvalue in a minimum variance or principal component analysis (see, for example, Smith and Tsurutani, 1976;Zhang et al., 1998). The wave normal direction, i.e., its k-vector, was clearly aligned with the background magnetic ®eld. The cone angle, h k , had values of 1X1 , 0X1 , and 0X7 during the three events.
A clear packet structure can also be seen in Fig. 2, which shows wave forms and wavelet spectrograms for two other lion roar wave packet trains inside a magnetic trough detected on 28 January 1998. Both wave trains last about 1.2 s and contain four and two clearly discernible wave packets, respectively. They occur just 4 s apart on both sides of a stronger core ®eld. While f e % 420 Hz for both events, the lion roar frequency at 40 and 25 Hz, respectively, is distinctly dierent for the two events. The cone angle between the wave vector and the ambient ®eld varied between 0.1±0.6 during the four wave packets of the ®rst lion roar event, while it attained values of 0.25 and 0.15 during the ®rst and second wave packet of the second event, respectively. Another lion roar signal is shown in Fig. 3. It occurred on 5 March 1998 and this time the mirror trough did not contain a stronger core ®eld. Again, the wave train lasts for about one second, but it shows considerable internal structure, with some wave packets lasting only for 3±4 wave cycles. In addition, one ®nds that the frequency varies during the event, from about 40 Hz to just above 20 Hz, i.e., by a factor of two. In contrast, the electron gyrofrequency drops only slightly, from 360 Hz to 310 Hz, during the event. The change in frequency proceeds in a series of glitches from packet to packet, both in upper and lower frequency cut-o. The cone angle, h k , ranged between 0.1 and 0.4 during the event.

Statistics
Including the six cases discussed above, we have analyzed the average properties of all 356 lion roar wave packets, which found in 4-months-worth of noonto-morning magnetosheath data in the frequency range between 8 and 64 Hz. In particular we will look at the frequency and its dependence on the mirror trough background ®eld, the duration of the lion roar wave packets, and the cone angle of the wave vector with the ambient ®eld. Figure 4 shows the occurrence distribution of the lion roar frequencies, normalized to the electron gyrofrequency, f e . The normalized frequencies are fairly evenly distributed in the range 0X05 f af e 0X15, with an average normalized frequency close to 0.1. Case distribution as well as average value are not too dierent to those found by Zhang et al. (1998), except that we naturally lack the comparatively few samples in their higher-frequency tail due to our lower high-frequency limit.
The packet structure of the lion roars is clearly visible in most of the cases presented in the previous section, is   Fig. 3. Mirror mode trough, embedded lion roar waveform, and right-hand polarized wavelet spectrogram also found in the bulk of the 356 cases. Figure 5 gives the occurrence rates of the duration of the lion roar wave packets in number of wave cycles. Half of all cases have a wave packet length of less than ®ve wave cycles and about 85% last less than ten wave cycles.
As for the case studies, we have also performed a minimum variance analysis for all 356 cases and compared the wave vector direction with the ambient ®eld direction. The occurrence distribution of the cone angle, h k , is given in Fig. 6. Except for ten cases with 1X5 h kf 5 , all cone angles are smaller than 1.4 . In fact, about half of the lion roar wave packets travel along the ambient magnetic ®eld with cone angles smaller than 0.2 .
These values are de®nitely much smaller than the typical cone angles of 10±30 found by Zhang et al. (1998) and Smith and Tsurutani (1976). Why? Looking into Fig. 7 of Fornacon et al. (1999), one notices that the ambient ®eld direction easily changes by 5±20 during a lion roar event lasting several seconds. We have done the minimum variance analysis on the wave packets with their average duration of 250 ms and compared the wave propagation direction with the average of the ambient ®eld over the same period. We suspect that the authors of the earlier papers have used longer stretches of data for their minimum variance analysis and/or for averaging the background ®eld direction.

Summary
The present study complements the earlier observations of lion roars, by extending the frequency range to lower values. While search coil magnetometers and plasma wave instruments are somewhat insensitive to the frequency range of some tens of Hertz, the Equator-S  Occurrence distribution between lion roar wave vector and ambient magnetic ®eld. Ten more cases (between 1.5 and 5 ) have been omitted from this diagram magnetometer, with a Nyquist frequency of 64 Hz, cannot cover the higher frequency range, but our observations show that lion roar electron whistlers can be observed down to 10 Hz, with similar characteristics as at higher frequencies, but also some new features: 1. Lion roars are near-monochromatic right-hand circularly polarized waves. 2. Lion roars have typical frequencies of 0.05±0.15 f e . 3. Lion roars have typical amplitudes of 0.2±1.0 nT. 4. Lion roars have a wavelet-like packet structure with typical durations of 3±8 wave cycles; 5. Lion roars travel nearly parallel along the ambient magnetic ®eld, more ®eld-aligned than previously thought.
Average numbers for frequency, packet duration, and cone angle h kf are given in Table 1.

Discussion
The most obvious observation is the narrow-bandedness of the signal. Since we know that it is in the right-hand parallel whistler mode and is generated inside the magnetic trap con®guration of a single mirror mode, parallel whistler instability theory applies. Under the assumption that the trapped electrons are anisotropic, dilute, and suciently energetic for justifying the application of linear theory, the upper frequency limit of the emission is expected at f max f e e e e e 1 1 The observed values for the frequency ratio suggests that the electron temperature anisotropy is of the order of e e c a k À 1 % 0X1 for all the events. On the other hand, we can take advantage of the whistler resonance condition v kres f e k k f f e À 1 2 which holds at both the upper and lower frequency limits. Writing it explicitly for these two limits we have In addition, the whistler dispersion relation imposes a condition on the parallel wave length Taken at the upper frequency cut-o, the resonance condition sets a lower limit on the parallel velocity of the resonant electrons v kmin v ee 1 À f kmax f e f e f kmax 1a2 5 which we estimate as v kmin % 2X7v ee , roughly three times the local electron AlfveÂ n speed (note that the waves and particles are antiparallel for resonance). In terms of the anisotropy the former expression reads Similarly, the lower frequency cut-o sets an upper limit on the resonant speed. De®ning a f min af max as the ratio of the measured frequency cut-os, one obtains v kmax av kmin a À1a2 e e 1 À a 1 3a2 7 Inserting the observed values yields v kmax % 1X53v kmin % 4X14v ee 8 The estimated anisotropy suggests that the perpendicular energy E c of the resonant particles is comparable to their parallel energy, E c % 1X1E k . The magnetic energy per electron is given by E f m e v 2 ee a2 yielding for the minimum parallel resonant energy E kmin % 7X3E f , and for the perpendicular energy of the resonant particles at instability threshold, E cmin % 8X0E f . Hence, the total energy of resonant electrons at threshold is E min % 15X3E f . Proceeding along the same lines for the resonant electrons at maximum resonant energy, we  with E max aE min % 2X35 only. Figure 7 shows the region of resonance in the v c -v k plane. The resonant region is a narrow rectangular domain instead of the vertically extended resonant strip expected in usual whistler resonance interaction. The vertical dashed line at v k v ee is the low energy threshold boundary above that whistler instability may set on for positive anisotropy. Clearly the resonant region is suciently far above this boundary. Resonance seems to be possible only in a very narrow domain in velocity space and only a limited group of nearly monochromatic energy electrons contributes to wave growth. One possible reason is that these are just the electrons which can be trapped in the mirror wave ®eld. The solid lines in the ®gure show the slight deviation of the electron iso-density contours from isotropy for a weak anisotropy of e e % 0X1.
In order to check if the assumption of trapping is satis®ed we can use the constancy of the magnetic moment l E c af both at the ®eld maximum and minimum. This leads to which is roughly satis®ed in the observations of the mirror modes for electrons mirroring close to the ®eld maximum but still inside the mirror wave bottle. The ratio of particle energy to magnetic energy per particle gives an idea about the value of the electron b e % EaE f , which is b e % 10 in our case. Together with the estimated anisotropy e e % 0X1 and referring to the numerical solutions of the whistler dispersion relation (cf., Gary, 1993) one ®nds from Fig. 8 that the maximum growth rate c of the whistler instability in our case is From the de®nition of the whistler growth rate (see Hasegawa, 1975;Treumann and Baumjohann, 1997) we obtain Inserting the measured values for the frequency ratio, f af e % 0X1, taking v k % 3X5v ee , and using c % 10m e am i 5X4 Â 10 À3 for protons, we obtain for the fraction of resonant electrons that contributes to wave growth n res an 0 % 0X03 13 implying that about 3% of the total electron population trapped in the mirror mode bottle develops the slight electron anisotropy which excites low frequency lion roars within the mirror modes. This observation raises the question why it is particularly the small group of electrons in the resonant region that drives the lion roar whistlers unstable. This question is not easily answered without measurement of the actual electron distribution function. Since such measurements are not available, we refer to the overall anisotropy of the electron distribution in this paper, as shown in Fig. 7. We show below that the whistlers have relatively large wave intensities. Under such conditions electrons in the resonant region will undergo diusion from high v c towards higher v k . This may lead to the formation of a weak residual bulge on the electron distribution located in the resonant strip, similar to the ion-bulge distribution proposed for the nonlinear state of the mirror mode by Kivelson and Southwood (1996). The main electron distribution is then essentially isotropic. The packet form of the emissions suggests that the whistler mode lion roar emissions reach the nonlinear state. In fact, the measured amplitude of df % 1 nT corresponds to a magnetic wave energy density of E wf jdfj 2 a2l 0 % 4 Â 10 À13 Jm À3 14 and a whistler wave electric ®eld of di wi % v ee df w % 1 mVm À1 15 or a corresponding electric wave energy density of E w % 0 jdij 2 a2 % 5 Â 10 À14 Jm À3 16 Compared to an estimated plasma thermal energy density of nk f % 10 À10 Jm À3 , these values correspond  Gary, 1993). For the measured weak anisotropy e e 0X1 and 1`b e`1 0 the growth rate is of the order of the ion cyclotron frequency c % x i . The curves are drawn for f pe af e 100, close to that expected in the magnetosheath to fractions of 4 Â 10 À3 and 5 Â 10 À4 , respectively. Waves of such an intensity will not only lead to the above mentioned quasilinear diusion of electrons with excess perpendicular velocity towards parallel velocities. They will necessarily lead to nonlinear eects like modulational instability. Whistler wave amplitudes of 1 mV/m are rather high, of the same order as the convection electric ®eld in the magnetosheath. It is therefore not unreasonable to assume that the original whistlers excited will nonlinearly evolve into packets being trapped in the modulations of the plasma. Since it is known that whistlers are trapped in overdense regions, the lion roar packets may be the wave signature of plasma compressions inside the mirror mode. Unfortunately, the lack of plasma measurements does not allow to check this suggestion. The observation of glitches (Fig. 2) and gradual variations (Fig. 3) in the emission bands poses an interesting but dicult to resolve question. The easiest interpretation is the assumption of a time variation in the electron anisotropy, e e . Such a variation may be caused by a decrease of the length of the mirror wave bottle during the time it takes the satellite to cross the bottle, which causes an increase in the parallel energy of the mirroring electron component, E k , and thus a decrease in e e and a shift to lower frequencies. The eective decrease in e e is very small, from f max af e 0X1 to f max af e 0X05 a decrease of only De e 0X01 is required.
A similar decrease may also cause the apparent glitch in frequency in Fig. 2. If one assumes that essentially the same azimuthally drifting electron component is responsible for the emission, a shrinkage of the bottle in length during the passage of the satellite from left to right, causing a decrease in anisotropy of the above order, would explain the frequency glitch. This interpretation is, however, not unique. One cannot exclude the possibility that the two wave bands result from excitation in dierent remote parts of the bottle or even in dierent bottles. Since the waves are ideally parallel propagating waves, they could as well have migrated from a neighboring bottle into the one where they are observed.