As described in earlier work [10], the interaction of the beam-cyclotron mode ω = Ω_e + k·V_b (V_b < 0) with the whistler wave is of decisive importance for the formation of the frequency gap. The basic effect is the mode splitting, combined with the existence of a “forbidden” zone, which occurs in the frequency and wave number range where both modes cross, see also [34]. Similar splits are known from other wave modes, e.g. the ion-cyclotron waves in proton plasmas with small admixtures due to heavy ions ( [15] [16]). If due to kinetic effects the modes merge, enhanced damping generally remains as a characteristic signature of the ongoing interaction. Examples of mode splitting using the fluid approach of cold plasmas are shown in Figure 1(a) and Figure 1(b) for two beam velocities and propagation angles. As seen there, the interaction of the beam-cyclotron mode ω = Ω_e + k·V_b (V_b < 0) with the whistler wave leads to mode splitting around the point of intersection. No wave propagation exists in the resulting frequency gap whose width increases with increasing beam density. The intersection point of both modes can easily be calculated. Using the approximated whistler dispersion relation x² = y/(cosθ− y) with x = kc/ω_e, y = ω/Ω_e and U_b = V_b/V_Ae (V_Ae is the electron Alfven velocity, V_Ae = c.Ω_e/ω_e, ω_e: electron plasma frequency), the wave number of the intersection point is obtained as solution of the third-order equation

U b cos θ x 3 + ( cos θ − 1 ) x 2 + U b cos θ − 1 = 0 (1)

In Figure 1(c) and Figure 1(d), the coordinates of the intersection point are shown for two beam velocities.

The excitation of whistler waves by electron beams has been studied in laboratory [35], and space ( [36] [37]), followed by numerous analytical and numerical investigations (e.g. [38] [39] [40]). In the following, the arising beam instability itself is not of interest, but only the frequency and wave number at which maximum instability may occur. It is assumed that this happened at the ω − k points where the beam mode ω = k·V_b and the whistler mode cross each other. Using, as before, the approximated whistler dispersion relation x² = y/(cosθ− y) with x = kc/ω_e, y = ω/Ω_e and U_b = V_b/V_Ae, the two cutting points between the beam mode ω = k·V_b (k > 0, V_b > 0) and the whistler mode are obtained by simple analytics as

x ± = 1 2 U b [ 1 ± { 1 − ( 2 U b ) 2 } 1 / 2 ] (2)

Accordingly, the frequency of the lower (−) and upper (+) beam instability versus the propagation angle θ is given by

ω ± = 0.5 Ω e A ± cos θ (3)

A ± = 1 ± { 1 − ( 2 U b ) 2 } 1 / 2 (4)

As seen in Figure 2(a) (θ = 50˚) and Figure 2(b) (θ = 20˚) the beam mode intersects the whistler mode at two points. The dependence of the corresponding wave number and frequency versus θ for two beam velocities V_b (0.3V_Ae, 0.45V_Ae) is shown in Figure 2(c) and Figure 2(d), respectively. Considering the upper beam instability for warm electron plasma with V_b~0.5V_Ae (U_b~0.5) one gets A₊~1 and thus ω₊~0.5Ω_ecosθ. That means the frequency follows almost the Gendrin angle. In the case of cold electrons with V b / V A e ≪ 0. 5 , the frequency would follow the resonance cone angle, i.e. A₊~2.

The main aim of this subsection is to illustrate how the whistler wave activity is decisively influenced by the simultaneous interaction of both beam modes. One has to have in mind how the individual beam mode interacts with the whistler wave: The beam mode ω = k·V_b (V_b > 0) is responsible for the beam instability, the beam-cyclotron mode ω = Ω_e + k·V_b (V_b < 0) causes mode splitting and increased damping, respectively, in the interaction area. Our suggestion is: The interaction of both effects is the cause of the gap formation at ω = Ω_e/2. As first step, the intersection of both beam modes is considered. From y = xU_bcosθ and y = 1 − xU_bcosθ one gets immediately: x = 0.5/(Ucosθ) and y = 0.5; that means,

both modes always intersect at ω = Ω_e/2. Inserting the coordinates of this point into the whistler dispersion relation x² = y/(cosθ− y), the simple relation between the (normalized) beam velocity U_b and the propagation angle θ results: U_b = 0.5 (2cosθ − 1)^1/2/cosθ. These relationships are illustrated in Figure 3.

From Figure 3(d) one may conclude that for beam velocities V_b~±0.5V_Ae the frequency gap at ω = Ω_e/2 will be formed by quasi-parallel waves. In case of V_b ≲ 0.3V_Ae, on the other side, mainly waves with θ~60˚ are involved. For the illustration of both cases, fluid dispersion analysis as described in [40] has been applied. The beams are described by water-bag distributions in order to include thermal effects. Results are shown in Figure 4 for the parameters marked in Figure 3(d) by stars and full rectangles: θ = 20˚, V_b/V_Ae = ±0.49 in (a, b), θ = 52˚, V_b/V_Ae = ±0.38 in (c, d). It is noteworthy that the essential effects of the interaction, which are discussed in more detail below in the context of the kinetic theory, are already contained in the fluid approach: These are: 1) Two areas of beam instability with maxima at the intersections of the beam mode ω = k·V_b (V_b > 0) with the whistler wave and 2) the formation of the frequency gap at the frequency ω = Ω_e/2, where the beam-cyclotron mode ω = Ω_e + k·V_b (V_b < 0) intersects the whistler mode.

We start our dispersion analysis with a plasma configuration that is similar to that in former PIC simulations, e.g. in [13] [24] [25]. A two-electron plasma is used with a cold and warm population which are denoted by the subscripts “c” and “w”. Their densities and temperatures (given by the plasma beta β c , w = 2 μ 0 n e k T | c , w | / B 0 2 ) are n_c/n_e = 0.9, β_c = 2·10⁻⁴ and n_w/n_e = 0.1, β_w = 0.1, respectively. n_e is the total electron density. For the warm electron component an anisotropy of A_w = T_w/T_||w−1 = 2.0 was taken by which the whistler waves are driven; the cold electrons are isotropic, i.e. A_c = 0.

In Figure 5, results of dispersion analysis for two plasma configurations represented by the electron distribution functions in the upper panels (a) and (e) are shown. The difference between the two distributions is that in (e) the Maxwell distribution in (a) has been expanded by a downstream beam/plateau (b/p) population. The parameters of this minor electron component marked by the subscript “b” are: n_b/n_e = 0.003, V_b/V_Ae~0.47, T_b/T_w = 0.04. (The effect caused by an upstream population is considered subsequently.) In both cases, whistler waves are excited by the temperature anisotropy of the warm population. The color plots of panels (b) and (f) show the growth rate of the whistler instability (γ/Ω_e) versus the wave number kc/ω_e and the propagation angle θ.

Comparing in Figure 5 both color plots, it is evident that the addition of the downstream b/p population generates an area in the k-θ space in which whistler damping instead of wave growth occurs. This ‘damping region’ is located along the dotted white line in panel f) which represents the θ-variation of the wave number according to Equation (1) at which the b/p cyclotron mode ω = Ω_e + k·V_b (V_b < 0) intersects the whistler mode, see also Figure 1(c).

The most remarkable effect of the cyclotron resonance damping is the arising frequency gap at ω~Ω_e/2 which is marked in panel (g). This can also be seen in the maximum growth rate in panel (h) which is split into two lines compared to the single maximum (panel (d)) of the pure Maxwellian distribution.

Summarizing the view in Figure 5, one can say that the gap formation at ω~Ω_e/2 is based on the resonance between the cyclotron mode of the downstream b/p population and the whistler mode. To demonstrate this effect from a slightly different perspective, the dispersion characteristics for the electron configurations of Figure 5(a) and Figure 5(e) are shown in Figure 6 for two propagation angles. The electron distribution functions are plotted again in the top panels (a) and (e). In the panels below the growth rate γ/Ω_e is drawn versus the frequency for the propagation angles θ = 10˚ (middle panels) and θ = 45˚ (bottom panels). The gap formation in panel f) is seen. As mentioned before, it appears at the intersection of the b/p cyclotron mode (dashed-dotted line) with the whistler mode (solid line) of the total electron plasma. In the same way, the cyclotron damping due to the downstream b/p population causes that the weak growth rate of the pure Maxwellian plasma for θ = 45˚ (panel (c)) completely disappears in panel (g).

Up to now, modifications of the whistler dispersion properties by a downstream b/p population have been considered. As we shall see, the existence of an upstream b/p population results in a source of wave excitation due to beam instability. The different roles of the two electron populations are shown schematically in Figure 3. The whistler mode and the two beam modes that are associated with the up- and downstream electrons are plotted there.

Returning to the kinetic dispersion analysis, the completely new situation at the presence of an upstream b/p population with regard to the whistler wave excitation is shown in Figure 7. As before, a temperature anisotropy of A_h = 2 is assumed. Looking first to the left panels of Figure 7, serious modifications compared with the situation in Figure 5 are evident. The most pronounced signature is the fact that the region of unstable waves is not more restricted to quasi-parallel propagation, but it extends to large propagation angles with a maximum at about 60˚. At the transition from quasi-parallel to oblique propagation, the wave number of maximum growth rate shifts from kc/ω_e~1 to kc/ω_e~1.6 and remains almost constant for θ ≳ 40˚. The instability in the range of nearly constant

wave number (kc/ω_e~1.6) has the character of a beam instability which is caused by the interaction of the beam/plateau mode ω~k·V_b (V_b > 0) with the whistler mode. A beam speed of V_b/V_Ae~0.45 is required to get from Equation (2) the above determined wave number. Accordingly, the frequency plotted as a solid line in Figure 7(c) varies with the propagation angle θ (for θ > 30˚) as ω~0.75·cosθ.

Another remarkable feature is the appearance of the lower-frequency beam mode, seen in color plot of Figure 7(b)) as a blue strip at kc/ω_e~0.5 and plotted in panels (c) and (d) as dotted lines. In agreement with the prediction of Equations (3) and (4), its frequency varies with θ as ω~0.25·cosθ. This mode reaches its maximum at θ~60˚ with a growth rate which is about one order less than that of the “upper beam instability” with kc/ω_e~1.6. In the right panels of Figure 7 the damping effect of the cyclotron mode—as described before—becomes evident for the case that the downstream beam/plateau population is added. As seen in Figure 7(f), the “absorption region” in the k − θ space is again located

along the dotted white line which represents the k versus θ variation of the intersection point between the cyclotron mode ω = Ω_e + k·V_b (V_b~−0.5V_Ae) and the whistler mode. As in Figure 6(e), in the range of quasi-parallel waves the frequency gap at ω~Ω_e/2 (kc/Ω_e~2) is formed. Additionally, a second frequency gap may arise for obliquely propagating waves at θ ≳ 50˚. It is caused by the simultaneous interaction of the two beam/plateau modes with the whistler mode, as sketched in Figure 3(b). Otherwise, the signatures are similar as in the left panels: 1) constant wave number in the range of beam-generated waves, 2) frequency variation with cosθ, 3) occurrence of the lower-frequency beam instability.

To better illustrate how the growth rate is dependent on the frequency, Figure 8 shows the growth rate for three propagation angles, θ = 10˚, 45˚, and 60˚. The

gap formation seen in Figure 8(f) is a common feature for quasi-parallel whistler waves driven by the temperature anisotropy at the presence of a down-stream b/p population of resonant electrons.

The second frequency gap at θ~45˚ (Figure 8(g)), on the other hand, is based on the simultaneous existence of the two beam/plateau populations, the upstream population causing wave excitation and the downstream population, which ensures the frequency-selective damping. This mechanism can unambiguously identified by comparing the rate of growth γ/Ω_e versus frequency ω/Ω_e in the right and left panels. The dotted curves in the panels (b)-(d) and (f)-(h) around ω ≲ 0.2Ω_e belong to the (lower-) beam instability which arises at the low-frequency cutting point of the b/p mode ω~k·V_b with the whistler mode.

Although its maximum amplitude is about one order of magnitude below that of the upper beam instability, it seems to be relevant for space observations, as we discuss later.

To underline the great relevance of the b/p populations, in Figure 9 results of

the dispersion analysis are shown for the case that—in comparison to the situation in Figure 7—the resonant energetic electrons are the only source of whistler waves. Such a situation can be imagined after the original instability due to the temperature anisotropy has relaxed and the associated waves are damped away. The lack of temperature anisotropy instability in Figure 9(b) and Figure 9(f) is indicated by the absence of whistler wave activity with kc/ω_e~1 in the range of quasi-parallel propagation (θ ≲ 30˚). The growth rate for obliquely propagating waves generated by the b/p instability, on the other hand, is only weakly influenced in comparison with the situation in Figure 7. The frequency follows the expression (2) and is located between the Gendrin and the resonance cone angle. The maximum growth rate occurs at about 60˚. By the reduced growth rate in the range around θ ≲ 50˚ compared with the values in Figure 7, there are no upper-band waves and the frequency gap of obliquely propagating waves, seen in Figure 7(f), almost disappears. Indeed, the lower-frequency b/p instability (ω ≲ 0.2Ω_e) is nearly unaffected by the absence of the temperature anisotropy.

Finally, slightly different distribution functions of resonant electrons have been used in order to investigate their influence on the b/p instability. In Figure 10(a), two b/p distributions with V_b~0.5V_Ae are shown which have the same density (n_b/n_e = 0.005), but differ slightly in shape by shifting the left border of the beam/plateau (generated by superposition of Maxwellians) a little to smaller values. In this way, a transition from a beam (df/dv ≳ 0, solid curve) to a real plateau (dashed curve) with df/dv ≲ 0 has been realized. As expected, that has no effect on the ω (real)-θ variation shown in Figure 10(b). However, a significant change in the growth rate (Figure 10(c)) is obvious. At transition to a plateau a shift of the wave activity to larger propagation angles, associated with an amplitude reduction, takes place. As a remarkable feature one has to note that for a strict plateau with df/dV≲ 0 a weak instability (dashed curve in Figure 10(c)) is still retained. Which distribution function of resonance electrons exists can only be obtained from kinetic simulations, ultimately 2D, which take into account all relevant plasma parameters such as electron plasma beta and temperature anisotropy.

The first point of our discussion concerns the gap formation. Depending on the source of the whistler waves involved, two types of frequency gaps are to be distinguished; one for quasi-parallel waves driven by temperature anisotropy and the other for oblique waves which are generated by the upstream beam/plateau population. In both cases, the gaps are caused by the cyclotron damping of the b/p mode ω~Ω_e + k·V_b (V_b < 0) which belongs to the resonant downstream electrons (V_b < 0) and crosses the whistler mode at ω~Ω_e/2.

Gaps of quasi-parallel waves which are based on self-consistent deformations of the electron distribution function have been studied in recent papers, beginning with the 2D PIC simulations of [13]. They have shown that selective reduction of the whistler wave growth rate is caused by a deformation of the electron distribution function at that point in the velocity space where the resonant interaction between the unstable wave and the electrons takes place and a plateau is formed. In the paper [14], gap formation, seen in 1D PIC simulations, has been interpreted by splitting the anisotropic electrons into two energy ranges, which are separated by a more isotropic one as a consequence of parallel resonant Landau acceleration. The physical relevance of the beam/plateau related cyclotron mode is first considered in the paper of [10] in which, furthermore, the excitation of oblique whistler waves and related gap formation (at the absence of temperature anisotropy) by the combined action of resonant up- and downstream electrons have been analysed. It should be noted, however, that the cyclotron damping of the whistler waves near Ω_e/2 by the added b/p population is equivalent to a local suppression of the initial temperature anisotropy at cyclotron resonance, in agreement with the results of the numerical simulations in [13] [14].

In paper [32] the characteristics of whistler waves excited by temperature anisotropic electrons whose distribution function is a combination of a bi-Maxwellian and minor beam/plateau-like modifications have been investigated by both linear theory and PIC simulations. Special emphasis was directed on the influence of the beam/plateau parameters (density, velocity) on the gap position. In the subsequent papers [24] [25] parts of these predictions have been checked by 1D/2D PIC simulations. For fixed propagation angle of θ = 20˚, the authors confirmed our above conclusions after which a frequency gap of quasi-parallel propagating waves can only be formed in warm plasmas in the range 0.05 ≲ β_w ≲ 0.2.

The outcome of our analysis of two types of gaps seems to be very important with regard to interpretation of the magnetospheric gap observations. In [4] have been classified gap events and found out one class which have in common that the gap occurs when θ rises above the Gendrin angle θ_G or even starts to spread out over a broad range of angles (up to 55˚) the closer it gets to Ω_e/2. Similarly, published spectra in which the frequency gap at Ω_e/2 occurs around θ = 60˚, close to the resonance cone angle [44]. Furthermore, the statistical analysis in [7] revealed that banded emission exists in a broad range of propagation angles reaching from the Gendrin angle [45] up to the resonance cone. These results fully confirm our conclusions about the essential role of the resonant b/p populations for both wave generation and gap formation.

Finally, it should be mentioned that whistler wave emission over a broad frequency range without a gap has been observed in e.g. [4] [7] and [46]. Looking at the ω-θ distribution of the no-gap and banded whistler waves in Figure 4 in [7], one gets the impression that both kinds of waves have the same origin. The simplest reason for the partially observed lack of gaps seems to be a missing downstream b/p population that rules out frequency-selective cyclotron damping as the causative mechanism. Hereby, asymmetries by plasma drift and spatial inhomogeneities may play a role. Further, it is noteworthy that most chorus wave observations mentioned above that exceeded 0.5Ω_e exhibits a weak low-frequency line with a nearly constant frequency of ω~0.2Ω_e. Since the corresponding wave normal angle is not given there, unfortunately, it remains unclear whether a beam/plateau population is the possible origin of this weak wave activity.

Another challenging point with respect to our theoretical studies is the debate about the wave normal angle distribution of the waves in the lower band. In addition to the peak within about 20˚, a second peak was observed at oblique angles near the resonance cone, e.g. in [28] [47] [48] and in [7]. While the quasi-parallel waves can be related to temperature anisotropy, the source of the oblique waves of the lower band remained largely open. In this context, we refer to the simple relationships (3) and (4), according to which the frequency of the b/p generated waves versus the propagation angle is given by ω = 0.5Ω_eAcosθ, where A varies between A = 1 (Gendrin angle) and A = 2 (resonance cone). Besides, our dispersion analysis whose main results are reviewed in Figure 11 shows that the growth rate of beam-generated whistler waves (ignoring temperature anisotropy) is located at θ~60˚. Thus, assuming A~1.5 as a middle between Gendrin and resonance cone angle, the associated frequency would be ω~0.38Ω_e. Such a type of waves based on simultaneous observations of low-energy electron streams (beams and/or plateaus) have been discussed in the literature, e.g., in [14] [28] [49] [50] [51] and in [29]. It has been found in [28] that measured electron distribution functions having a plateau/beam-like shape are directly correlated with a positive growth rate of whistler waves at very oblique propagation (θ ≳ 60˚). Additionally, weaker whistler activity at ω < 0.2Ω_e is mentioned there which in our view may arise owing to the lower-beam instability, see e.g. Figure 9(j) and Figure 9(o).

An obvious relation between lower-band chorus waves and beam-like distributions has been described in [29]. As a clear signature, the measured plateau at a velocity of V_b~V_Ae/2 is correlated with whistler waves at ω/Ω_e~0.23 propagating close to θ = 70˚ (denoted there as Event II waves). Also there, a wave at a lower frequency (ω/Ω_e~0.13)—not further discussed there—is simultaneously observed which again seems to us is another indicator of the presence of a beam that is responsible for both lower-band emissions. For an understanding of the process by which the electron distribution with a beam/plateau is generated, one has to mention that quasi-parallel whistler waves in the lower band also exist (Event I waves). To drive the Event I waves unstable by reasonable temperature anisotropy, according to the dispersion analysis (see Figure 11(m)), an electron plasma beta of β_w ≳ 0.4 is required.

The described in [29] observation is directly related to the earlier conclusions in [31], where it was explained the statistically found rarity of simultaneous observations of intense parallel and oblique waves in the lower band by means of beam/plateau formation. The following mechanism is suggested: Initial temperature anisotropy generates quasi-parallel whistler waves associated with a modified distribution function owing to resonant electrons. Subsequently, very oblique waves (θ ≳ 60˚) are amplified, whereas the parallel waves are supressed by increasing damping.

Further the existence of the beam/plateau modes ω = k·V_b seems to be an important aspect in the formation of the so-called QE-MBC (quasi-electrostatic multiband) events, as described in [5]. In the case of the primary excited lower-band chorus with propagation angles of about 70˚, the observation of the second and third harmonic (see their Figure 2) suggests beam excitation, whereby the energy (ω) and impulse (k) conditions for three-wave interaction are immediately fulfilled, similar as for the harmonics of the electron plasma frequency at beam-plasma interaction [20]. Similarly, beam modes seem to be involved in the analogous EM (electromagnetic)-MBC events which occur at quasi-parallel propagation.

Finally, it would be worth adding that in the present work we have exclusively assumed that the beam/plateau is formed by resonant electrons of the initially unstable whistler waves. However, plateau formation can also occur through other types of waves, such as kinetic Alfvén waves, time domain structures or ion outflow, even before a whistler-mode wave was excited [30].

With the theoretical evidence that whistler waves can be generated by resonant beam/plateau populations, the question arises to what extent this mechanism is involved in the formation of chorus waves which occur in form of rising and falling tones, e.g. in [2] [52] [53] [54]. A comprehensive classification of different chorus events has been done in [4] who distinguished five groups among the most common rising tones: group 1: crossing 0.5Ω_e with θ < θ_G (Gendrin angle) in lower band, group 2: frequency gap in emission at 0.5Ω_e, group 3: crossing 0.5Ω_e, with θ_G < θ < θ_r (resonance cone), group 4: the lower band only (<0.5Ω_e), and group 5: the upper band only (>0.5Ω_e).

The events of group 1 and group 2 in [4] belong to the quasi-parallel waves (θ < 30˚), and they are related to our studies in Section 3.1 after which unstable waves covering 0.5Ω_e are generated by temperature anisotropy, provided the plasma beta of the hot electron component (β_h) is in the range 0.05 ≲ β_h ≲ 0.3. As further shown in Figure 5 and Figure 6, to form a gap the presence of a downstream beam/plasma population with velocities V_b~0.5V_Ae is required. Our view that the waves of group 1 and 2 have the same origin is supported by the following observation in [4]: For both groups almost identical traces in the θ-ω plane of the scanned bursts outside the gap area have been obtained; see Figure 4(a) and Figure 4(b) therein.

All chorus bursts of group 2 have in common that the frequency gap occurs when θ rises above the Gendrin angle θ_G. For some events, θ starts to spread up to about 55˚ if it gets close towards the gap. In our view, this behavior indicates the presence of hot electron plasma (β_h ≳ 0.3) whose maximum growth rate owing to temperature anisotropy is shifted below Ω_e/2 and thus a frequency gap can only be formed by resonant b/p populations at oblique wave propagation. Such a situation has been shown in Figures 11(k)-(m). In Figure 12, as modification of Figure 8(m) the propagation angle θ versus the frequency ω/Ω_e is plotted, including the Gendrin and resonance cone angle θ_G and θ_rc, respectively. As seen, it has much similarity with Figure 3(c) in [4]. The strongest emission is outside the Gendrin angle θ_G in the range of quasi-parallel propagation (θ ≲ 30˚) and with a frequency of ω~0.4Ω_e. The wave activity betweenθ_G and θ_rcwhich is caused by the (upstream) b/p population stretches from about ω/Ω_e = 0.25 to about 0.7 whereby the angle θ drops from about 70˚ down to about 30˚.

The group 3 events of [4] can be classified in a similar way. These exhibit no pronounced gap and are characterized by constantly raising frequency as the angle

θdecreases, with θ_G < θ < θ_rc always applying. That means, the picture is similar as in Figure9, however, without wave activity below the Gendrin angle θ_G. This brings us back to Section 3.2 in which whistler wave generation by upstream beam/plateau populations has been analysed. The main characteristic of these waves is their ω-θ variation which according to Equation (2) follows as ω = 0.5Ω_eAcosθ where A varies between A = 1 (Gendrin angle) and A = 2 (resonance cone) depending on the beam/plateau velocity, see Equation (3). Further, in the representations of riser elements investigated in [55], especially in their FigureA1, there is a striking observation that seems to be closely related to our investigations. In addition to the measurements of lower-band risers of quasi-parallel waves, subsequent chorus events in distances of about one second that spread over angles between 60˚ and 90˚ are seen as a signature of beam-driven waves.

One point that has not been considered so far concerns the fine structure of the chorus events. It has already been analyzed in the past (e.g. [53] [56] [57] [58] and has recently received new impetus from research in [55] and [59] [60] on lower-band chorus risers, which present themselves as a series of monochromatic wave packets.

Due to the fact that for sufficiently hot electron plasmas (β_e > 0.025) the phase velocity of the unstable whistler waves is very close to 0.5VA_e, the inflection point where phase and group velocity coincide and have a maximum, the interpretation comes to mind that these “coherent” waves are connected to whistler oscillitons, a type of solitons ( [61] [62]). This class of solitons arises due to the periodic momentum exchange between the electrons and protons, mediated by Maxwell stresses, giving rise to wave packet structures. Wave packets for classical whistler waves appear due to nonlinear beating between wave motions of the electrons and protons due to a small difference of their phases. In plasma with electron beams a beating can occur between wave motions of the bulk and beam electrons. The feature of the inflection point with maximum of the group velocity on the dispersion curve is that wave packets will be gathered there. The phase mixing of waves in the vicinity of the inflection point with maximum of the phase speed is very small. This concept has been applied to the interpretation of STEREO radiation belt and auroral hiss observations ( [19] [40]).

Besides the observations in [55], who showed that some short wave packets seem to have nearly constant frequency, it means they are nearly monochromatic with df/dt~0, the statistic of short chorus wave packets in [59] [60] more often show a significant variation of the wave frequency over their duration. Such short wave packets have been interpreted, in agreement with nonlinear chorus wave simulations [63], as result of a superposition of two waves whose frequencies are separated by Δω/Ω_e~0.06. In fact, more recent studies using realistic simulations of non-linear chorus wave generation have shown that two superposed intense waves are often simultaneously generated with a minimum frequency difference imposed by the criterion for resonance non-overlap [64] in good agreement with satellite observations, explaining the formation of short chorus packets [65] [66].