2.1 Ground‐Based Observations

Let us return to Figure 1 which presents the observations of VLF noise suppression by whistlers performed at Kannuslehto ground station (KAN) in Finland (67.74°N, 26.27°E; ) on 25 December 2011. As was mentioned above, the upper panel displays the power spectral density of the wave magnetic field and illustrates the phenomenon under discussion. This spectrogram was preliminary cleaned from sferics using the method described in Manninen et al. (2016). Pay attention that a pronounced decrease in the noise intensity is observed after the second dispersed trace of the whistler echo train. Only the frequency band up to 8 kHz is displayed, although VLF emissions were recorded in the frequency range from 0.2 to 39 kHz. Magnetic field measurements were performed using two mutually orthogonal magnetic loop antennas oriented in the geographical north‐south and east‐west directions. This allows us to determine the polarization of waves which is characterized by the parameter :

(1)

where are the right‐ and left‐hand polarized horizontal magnetic field components, respectively, and are the northward and eastward projections of the wave magnetic field. This parameter, displayed in the middle panel of Figure 1, shows that both whistlers and VLF noise have left‐hand polarization, which indicates that the signals come to the Kannuslehto station over the Earth‐ionosphere waveguide (Ostapenko et al., 2010). The lower panel of Figure 1 displays the angle (with an ambiguity of 180°) between the minor axis of the wave polarization ellipse and the north‐south direction, which determines (with the same ambiguity) the direction of wave arrival at the station. Close values of the displayed quantity for both whistlers and noise suggest that they have close exit regions from the ionosphere. As for time characteristics of the noise suppression phenomenon, it becomes the most pronounced about 15–30 s after strong whistler event and lasts several tens of seconds.

2.2 Space Observations Onboard Van Allen Probes

We have not found suppression events in Cluster or RBSP data simultaneous with any one observed at Kannuslehto ground station. An independent suppression event that we have found in RBSP‐B data is discussed below.

Intense whistlers and echo trains were registered onboard RBSP‐B on 22 December 2014. VLF spectrogram up to 12 kHz obtained from magnetic (a) and electric (b) field measurements of the EMFISIS instrument (Kletzing et al., 2013) during the time period 19:23:53–19:24:17 UT when the instrument operated in the burst mode is shown in the two upper panels of Figure 2. During this time RBSP‐B was in the morning sector in the Southern Hemisphere and had the following coordinates: 5.1 hr, MLAT 17.4°, and . The electron plasma frequency corresponding to the measured electron density was 379.5 kHz, and the electron cyclotron frequency was 51.4 kHz. Both electric and magnetic receivers registered whistlers, sometimes with their echoes, and VLF noise below 6 kHz. The most intense whistlers were registered at about 19:24 UT, first a fractional hop whistler at 19:23:57 UT, which came from the Southern Hemisphere, and then three echo signals, after which a decrease in the VLF noise intensity below 6 kHz can be observed (see Figure 3b). Minimum intensity of the VLF noise was observed 20 s after the first reflected whistler.

Before proceeding with the description of the wave phenomenon under consideration, an important remark is in order. Following the original paper by Gail and Carpenter (1984), we use VLF noise and VLF hiss as equivalent terms for the emission being suppressed by whistler trains. As this emission is observed inside the plasmasphere, we should clarify the relation between this emission and the plasmaspheric hiss. The latter is usually understood as the emission in the frequency band from hundred(s) Hz to several kHz, with maximum spectral intensity below 1 kHz (see, e.g., Summers et al., 2014, and references therein). In a sense, the VLF hiss under discussion may be considered as the higher‐frequency part of plasmaspheric hiss, with its specific features, namely, noise‐like spectrum, predominantly parallel wave normal vectors, the Poynting vectors directed from the equator, and relatively low intensity as compared to that at lower frequencies. Thus, for instance, the maximum power spectral density of the wave magnetic filed for the plasmaspheric hiss at the frequency 500 Hz was about  nT  Hz , while the power spectral density of the VLF hiss at the frequencies of (4–6) kHz was about 2 orders of magnitude lower.

Suppression of the noise intensity by whistler can be clearly seen from Figure 3b which shows electric field spectral density in three frequency bands centered on 3988, 5020, and 5623 Hz in the time interval 19:20–19:28 UT. Three distinct spikes of spectral amplitude, which we associate with multihop whistlers (marked by “w” in the figure), followed by decrease in the noise amplitude are clearly seen. Minimum values of the noise intensity marked by arrows are observed (15–30) s after whistlers.

Figure 3a shows cold plasma density (Kurth et al., 2015) along the satellite trajectory, which smoothly increases between 19:21 and 19:24 UT, then remains almost constant and then slightly decreases after 19:27:30 UT. The amplitudes of whistlers and VLF noise also increase till 19:24 UT, whereupon intense whistlers are not observed, while the amplitude of noise slowly decreases. Simultaneous increase of the cold plasma density and VLF wave amplitudes (both of whistlers and VLF noise) may be related to ducting of whistler waves by density gradient of the cold plasma (Inan & Bell, 1977; Semenova & Trakhtengerts, 1980).

Ducted propagation of whistlers and VLF noise is confirmed by multicomponent analysis of VLF waves. Lower panels of Figure 2 show (c) planarity of the wave magnetic field (Santolík et al., 2002), (d) wave normal angle (Santolík et al., 2003), and (e) a spectral estimate of a polar angle of Poynting vector with respect to the ambient magnetic field (Santolík et al., 2010). One can see that fractional hop whistler propagates toward the Northern Hemisphere, while further multihop whistlers propagate by turns toward the Southern and Northern Hemisphere. At the same time, VLF noise propagates toward the Southern Hemisphere. An important result that follows from the multicomponent analysis consists in that both whistlers and VLF noise have small wave normal angles. This suggests that the observed multihop whistlers propagate in ducted mode and that VLF noise is most probably generated at the equator.

To check the assumption that resonant interaction of energetic electrons with multihop whistlers changes their distribution in such a way that the free energy of the distribution is decreased, leading to the corresponding decrease in the growth rate, we have calculated the growth rates for parallel‐propagating whistler waves using the data from MagEIS instrument on RBSP‐B (Blake et al., 2013). This growth rate is determined by the expression given in Sagdeev and Shafranov (1961). Assuming that electron distribution function determining the growth rate is the function of particle kinetic energy and magnetic momentum ( are electron mass and total and transverse velocities, respectively, and is electron cyclotron frequency), this expression takes the form:

(2)

Here is the wave angular frequency, is the magnitude of the wave normal vector, is the magnitude of electron charge, is the speed of light, and

(3)

where

(4)

is the resonance velocity at the first cyclotron resonance, the only one that exists at parallel propagation. As it is indicated above, after taking the derivatives in 3, particle parallel velocity is equated to , so that the combined derivative 3 becomes a function of the magnetic momentum.

Electron distribution function that enters the above relations is determined from the measured electron differential flux by the relation (Cornilleau‐Wehrlin et al., 1985):

(5)

Relation 5 determines the electron distribution function in CGS system of units, used in the paper, through the measured differential electron flux expressed in practical units, that is, and the particle energy in keV denoted by . We should underline that, no matter in which variables the distribution function is expressed, it is always equal to particle density in the phase space .

Normalized growth rates calculated from RBSP‐B data for two moments of time (before and after strong whistler events) are shown in Figure 4 as functions of frequency normalized to local electron cyclotron frequencies, which at that moments of time were equal to 43.85 and 49.03 kHz, respectively. We see that the decrease of growth rate after the series of three multihop whistlers is significant. Qualitative explanation of this result follows from the consideration in the next section.

While the decrease in the growth rate related to strong whistler events is clearly seen in Figure 4, the unstable frequency band does not correspond to that of the equatorial VLF noise, and the magnitude of the growth rate is quite small. There are several reasons for these. First, the observed spectrum is determined not by the growth rate but rather by the amplification factor along the wave trajectory, as well as other factors, in particular, by the reflection coefficients at the mirror points. Second, the main contribution to the wave amplification factor comes from the near‐equatorial region where the growth rate is largest and the group velocity is minimal. We presented the growth rates at the latitude 17.4° only for illustrative purposes, as we have in situ measurements of the particle distribution function in this region, to demonstrate the decrease of the growth rate after strong whistler echo train event. At the latitude 17.4° the loss cone is wider than at the equator, while the cyclotron frequency and, thus, the resonance velocity are larger. Due to these factors, outside the equator we should expect smaller values of the growth rate, but wider unstable frequency range, as the instability criterion is usually expressed in terms of the wave frequency normalized to cyclotron frequency and the effective anisotropy of the distribution function.

Unfortunately, we do not have measurements of the distribution function at the equator at the moments of time that we need, while the measurements outside the equator which we have do not permit to reconstruct the distribution function at the equator, since the particles with large perpendicular velocities that make the main contribution to the growth rate have mirror points close to the equator. That is why we have to rely upon direct calculation of the growth rate using the measured distribution function at the equator, at the same ‐shell where the suppression phenomenon has been observed but at other moment of time. In particular, during the previous turnover, at 10:13:59 UT, RBSP‐B crossed the equatorial region exactly at , and we used the measurements of hot and cold plasma components to calculate the growth rate at the equator. This growth rate is shown in Figure 5. We see that at the equator, the growth rate is 1 order of magnitude larger than outside the equator, and the frequency of the growth rate maximum corresponds much better to the observed frequency band of VLF noise.

3.1 Main Assumptions of the Model

3.2 Energetic Electron Motion in the Field of Whistler Echo Train

Since VLF noise suppression is usually observed after the second dispersed whistler trace, we will assume the wave field to consist of three wave packets: a fractional hop whistler originating from a lightning in the Southern Hemisphere (for the sake of definiteness) and two reflected whistlers, one from the Northern Hemisphere and one from the Southern Hemisphere, corresponding to the traces marked by numerals 1, 2, and 3 in the bottom panel of Figure 2. The first fractional hop whistler propagating from south to north and registered on the satellite in the Southern Hemisphere is marked by numeral 1. The trace corresponding to the wave packet reflected from the Northern Hemisphere is marked by numeral 2. The third trace corresponding to the wave packet reflected from the Southern hemisphere is marked by numeral 3. The waves that form this trace propagate to the north and are visible, although not very clearly, by blue color. The reason for this is twofold. First, the amplitude of the trace 3 is substantially lower than for the Trace 2 since the waves that compose the former have not crossed the equatorial region where the essential cyclotron amplification takes place. Second, while Traces 1 and 2 are separated by double crossing the equator, where the time delay is most significant, the time delay between Traces 2 and 3 is smaller, since during this period the signal does not cross the equatorial region. The same is seen in Figure 6. While the Wave Packets 1 and 2 are well separated in space‐time domain, the Wave Packets 2 and 3 strongly overlap. After the third trace, other whistlers reflected from the Northern Hemisphere and propagating to the south are seen in Figure 2. We do not discuss these traces and do not include them in numerical analysis of particle motion, since the effect of VLF noise suppression by whistlers is revealed after the first few traces.

Based on experimental observations, we will assume parallel propagation of all waves, which essentially simplifies the problem. In the case of parallel propagation, the wave field has only and components, transverse to the direction of wave propagation. Thus, we will write the wave electric field in the form

(6)

where is the coordinate along the ambient magnetic field and and are amplitudes and phases of the th wave packet. In the following numerical calculations, we will not consider space‐time variation of the wave amplitudes but will put them equal to 8 mV m wherever the wave packets exist. At the same time, we will use exact expressions for wave packet phases as they follow from the equations of geometrical optics (GO).

The wave frequency and the wave normal vector in each wave packet are determined in the usual way:

(7)

In each wave packet, the wave frequency and the wave normal vector depend on both coordinate and time , but for each and they are related by the dispersion relation for parallel‐propagating whistler mode waves, namely,

(8)

where and are the frequency and the wave number defined above which depend on , is electron plasma frequency that depends on , and is, as before, the electron cyclotron frequency which also depends on . The wave frequency in each wave packet calculated from the equations of GO under assumption of parallel propagation is shown in the three upper panels of Figure 6. For given frequency and coordinate , the wave number , up to its sign, is determined by 8. For the first and the third wave packets , while for the second wave packet . In order to find these quantities, we solved the equations of GO in the approximation of parallel propagation for 46 narrow band wave packets, starting at the height of 500 km in the Southern Hemisphere on ‐shell equal to 3. For all of these wave packets, the initial wave vectors were parallel to the ambient magnetic field at the starting point, while their magnitudes corresponded, according to the local dispersion relation, to the frequencies in the range 1–10 kHz. We followed the solution of GO equations until the fast propagating 10 kHz wave reached the height of 500 km in the Northern Hemisphere after two reflections from the Northern and Sourthern Hemispheres. This forms three wide‐band, space‐time bounded wave packets that represent multihop whistlers or whistler echo train. In solution of the GO equations, we used the dipolar model of the ambient geomagnetic field and the gyrotropic model of the cold plasma density: with the constant of proportionality that provides the ratio at the equator on .

In nonrelativistic approximation, and for parallel‐propagating whistler mode wave, the resonance between the wave and electron arises under condition . Correspondingly, electron resonance parallel energy in keV, with the account of 8, is given by

(9)

where . The quantity in space‐time domains where the multihop whistlers exist is shown in the three bottom panels of Figure 6.

The wave electric field in the form 6 corresponds to right‐hand polarization with respect to the ambient magnetic field independently of the sign of wave number , that is, of the direction of the wave propagation. In our case, and are positive, while is negative. The wave magnetic field can be found from the wave electric field 6 using the Faraday induction law.

In the absence of the wave field, particle kinetic energy and magnetic momentum are conserved, and the particle motion can be described by the equations that follow from the unperturbed Hamiltonian

(10)

where canonically conjugated variables are and and where is the particle gyrophase. Transverse components of electron velocity are expressed in canonical variables as follows:

(11)

We now write the variation of electron kinetic energy due to interaction with the whistler echo train:

(12)

where is electron charge. As has been shown by Shklyar and Matsumoto (2009), for resonant particles, the rate of energy variation coincides with partial derivative of the interaction Hamiltonian with respect to time. Taking this into account, and making use of 10, we come to the expression for the total Hamiltonian of the problem in the form

(13)

The equations of motion which follow from the Hamiltonian 13 have the form

(14)

While we have taken into account the derivative of cyclotron frequency with respect to in the equation for that originates from the unperturbed Hamiltonian, we have neglected the derivatives with respect to of slowly varying quantities and in the interaction Hamiltonian that contains wave amplitudes as the factors. This is valid under conditions of applicability of GO , where is the characteristic scale of plasma inhomogeneity. We should stress that neglecting the derivatives of the abovementioned quantities with respect to in the equation for by no means imply that we treat them as constant; we do take into account their variation in space and time along the particle trajectory.

In the set of equation 14, the quantity is an unknown function, while the quantities should be considered as known ones. However, the equations of GO from which the quantities should be found define first of all their derivatives, that is, the quantities and . That is why, in the equations of motion 14, it is convenient to use quantities as new unknown functions, since the function enters the equations of motion only in combinations . Thus, instead of equation for and equations for , it is more convenient to use the equations for , which follow from the definitions given above and the equation for :

(15)

In these equations, the quantities and are the functions that are found directly from the equations of GO.

We will underline the difference between the equations of motion written above and those used when considering resonant particle dynamics in one wave packet on a short time scale, that is, on the time scale of crossing one cyclotron resonance (Nunn, 1971 see also Demekhov et al., 2006; Karpman et al., 1975; Shklyar & Matsumoto, 2009). In the latter case it is often convenient to choose the sum of the wave phase and particle gyrophase as a new dependent variable (phase) and the derivative of this quantity with respect to time, which is proportional to the deviation of particle parallel velocity from the resonant value, as the second variable (momentum). Then the equation for this momentum contains the derivatives of the quantities , and in an evident form. This procedure is not appropriate in our case. First, we have three phases instead of one. Second, we consider particle dynamics on a large time scale that includes many bounce oscillations and many crossings of cyclotron resonances inside three wave packets. That is why we need to follow the variations of three phases (equation 15), as well as the variation of parallel coordinate , parallel momentum , and the magnetic momentum (equation 14), while the equation for particle gyrophase is included in the equation 15. In fact, we also follow the variation of particle kinetic energy according to equation 12. In all these equations we take into account the dependencies of quantities entering these equations on and , as it is indicated in the equations.

The solutions of the set of equation 14 for three particles are shown in Figures 7 and 8. Upper panels of Figure 7 show the variations of latitude, normalized parallel velocity, and normalized magnetic momentum along the particle trajectories. We see that, on large time scale, the particles bounce oscillates between mirror points occasionally experiencing resonant impacts from multihop whistlers. These impacts change the particle energy and magnetic momentum, the variations taking place during a very short time as compared to bounce period. One of such impacts for each particle is zoomed in the bottom panels of the figure. As one can see from Figure 8, which displays the variations of particle energy (upper panel), each impact corresponds to stationary phase point in one of the wave packets, that is, to the moment at which one of the derivatives shown in three lower panels turns to zero. We suggest that random jumps of electron energy and magnetic momentum in the course of interaction with multihop whistlers cause particle diffusion in the phase space, which leads to a decrease in free energy of the unstable distribution and the corresponding decrease of the growth rate.

For quantitative estimation of the suggested mechanism, we have calculated the average variation squared, during the duration of the whistler echo train  s, of magnetic momentum for 100 particles with a characteristic initial energy  keV and a characteristic magnetic momentum such that  keV, uniformly distributed over gyrophases and the accessible range of initial latitudes on . These determine the particle diffusion coefficient according to the relation

(16)

where the overline means the average over the time . For the parameters of the model described above, the calculations give , so that  s .

While the term “characteristic energy” just means a typical energy of energetic particles, the term “characteristic magnetic momentum” needs some explanation. We assume that plasma in the equatorial region is unstable. It means that the integral with respect to in 3 is positive. By the characteristic magnetic momentum, we understand the center of the region which makes the main contribution to the integral with respect to .

As is well known, for diffusion process , thus, the diffusion coefficient does not depend on time for which it has been calculated, since with increasing , the quantity increases proportionally. The characteristic diffusion time is determined as the time for which . During this time, diffusion covers the entire region in the phase space which makes the main contribution to the growth rate, and we should expect its essential decrease after . Substitution for and for in 16 gives

(17)

Using the figures given above, we obtain the characteristic diffusion time , which also provides an estimation for the relaxation time of unstable plasma distribution.