Volume 125, Issue 2 e2019JA027430
Research Article
Open Access

Ground and Space Signatures of VLF Noise Suppression by Whistlers

D. R. Shklyar

Space Research Institute, Russian Academy of Sciences, Moscow, Russia

Department of Physics, National Research University Higher School of Economics, Moscow, Russia

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J. Manninen

Corresponding Author

Sodankylä Geophysical Observatory, Sodankylä, Finland

Correspondence to: J. Manninen,

Jyrki.Manninen@sgo.fi

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E. E. Titova

Space Research Institute, Russian Academy of Sciences, Moscow, Russia

Polar Geophysical Institute, Apatity, Russia

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O. Santolík

Department of Space Physics, Institute of Atmospheric Physics, Czech Academy of Sciences, Prague, Czechia

Faculty of Mathematics and Physics, Charles University, Prague, Czechia

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I. Kolmašová

Department of Space Physics, Institute of Atmospheric Physics, Czech Academy of Sciences, Prague, Czechia

Faculty of Mathematics and Physics, Charles University, Prague, Czechia

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T. Turunen

Sodankylä Geophysical Observatory, Sodankylä, Finland

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First published: 06 February 2020
Citations: 2

Abstract

VLF (VeryLow Frequency) spectrograms registered at Kannuslehto ground station, after cleaning them from strong sferics, reveal VLF noise suppression by whistlers and whistler echo trains, which consists in significant reduction in the noise spectral power after a strong whistler event. We have found similar effect in the VLF data from Van Allen Probe B taken in the equatorial region on urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0001‐shell urn:x-wiley:jgra:media:jgra55531:jgra55531-math-00023. Detailed analysis of the data shows that the whistler echo train and the VLF noise have small wave normal angles. Based on this observation, we limit our analysis to parallel (ducted) whistler wave propagation. The persistence of whistler echo train, as well as the VLF noise, suggests that in the events under discussion, plasma is unstable in the frequency range corresponding to the observed VLF noise band. In an attempt to explain the effect of VLF noise suppression, we follow up the long‐standing idea that relates this effect to the reduction of free energy in the unstable plasma distribution by whistler echo train. To develop this idea into qualitative model, we have studied the motion of energetic electrons, responsible for the noise generation, in the field of ducted whistler echo train. We show that energetic electrons that make the main contribution to the growth rate of VLF noise, during their bounce oscillations in the magnetosphere, are subject to multiple resonant impacts from the whistler echo train. These lead to energetic electron diffusion in the phase space and the corresponding reduction in free energy of the unstable distribution.

1 Introduction

In this paper we investigate the wave phenomenon in VLF frequency band which consists in transient reduction in the amplitude of VLF noise observed, first on the ground, after receiving a strong whistler. An example of this phenomenon is shown in Figure 1, which displays the spectrogram in the frequency band up to 8 kHz registered at Kannuslehto ground station in North Finland ( urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0003) on 25 December 2011. The upper panel shows the wave magnetic field power spectral density over a 4 min interval. A significant decrease in spectral amplitude of VLF noise after receiving a strong whistler is clearly visible. We will turn our attention to other panels of Figure 1 later on. Other examples of VLF noise suppression by strong whistler echo trains, registered at Kannuslehto ground station, may be found in the supporting information of the paper.

jgra55531-fig-0001
An example of spectrogram, obtained from ground station data, illustrating whistler induced suppression of VLF noise. (a) Logarithm of the total magnetic field power spectral density. (b) Parameter characterizing the wave polarization (see the text). (c) The angle between minor axis of the wave polarization ellipse and the north‐south direction.

Suppression effect under discussion was known for a long time. Based on observations on Siple Station, Helliwell et al. (1980) reported one‐to‐one correlation between hiss and optical emission ( urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0004) intensity reductions immediately following each discrete VLF event. Since optical emission was assumed to result from particle precipitation, the authors suggested that both the hiss and optical emission reductions were caused by pitch angle scattering of energetic electrons by whistlers.

A comprehensive study of whistler induced suppression of VLF noise has been performed by Gail and Carpenter (1984). The authors have established several important features of the suppression effect; in particular, they demonstrated that the effect usually occurs when the driving whistler exhibits echoes confined to the frequency band occupied by the suppressed noise. They have also shown that the recovery of the noise band to the pre‐event level takes several seconds, and this time correlates with the damping rate of the echo train. The effect of suppression of the VLF noise band produced by the whistler that triggered it has been reported by Platino et al. (2005) basing on observations by the Cluster spacecraft.

The explanation of the noise suppression suggested by Helliwell et al. (1980) and accepted by Gail and Carpenter (1984) consists in pitch angle scattering of energetic electrons by the whistler signal that leads to disruption of wave amplification in the magnetospheric interaction region. Those electrons which are scattered into the loss cone are observed as precipitation, providing the correlation between urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0005 optical emission and VLF noise suppression. We should mention that, although the pitch angle scattering may lead to reduction in the particle pitch angle anisotropy and, thus, decrease the amplification efficiency, the electrons close to the loss cone do not contribute significantly to the noise amplification due to small amplitude of interaction between these particles and parallel‐propagating whistler waves.

2 Experimental Features of Suppression Phenomenon

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; urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0006) 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0007:
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0008(1)
where urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0009 are the right‐ and left‐hand polarized horizontal magnetic field components, respectively, and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0010 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: urn:x-wiley:jgra:media:jgra55531:jgra55531-math-00115.1 hr, MLAT urn:x-wiley:jgra:media:jgra55531:jgra55531-math-001217.4°, and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0013. The electron plasma frequency urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0014 corresponding to the measured electron density was 379.5 kHz, and the electron cyclotron frequency urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0015 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-001620 s after the first reflected whistler.

jgra55531-fig-0002
Dynamic spectra of VLF emission computed from RBSP‐B waveform data for wave event observed on 22 December 2014. The sum of the power spectral densities of three orthogonal (a) magnetic and (b) electric components. (c–e). The results of multicomponent analysis of VLF wave measurements by RBSP‐B on 22 December 2014 (see text for explanation of the displayed quantities).
jgra55531-fig-0003
(a) Cold plasma density measured by RBSP‐B. (b) Power spectral densities of electric field fluctuations as a function of time observed by RBSP‐B in the frequency bands 3758–4218 Hz (red line), 4730–5309 Hz (blue line), and 5309–5957 Hz (black line).

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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0017 nT urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0018 Hz urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0019, 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0020(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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0021 and magnetic momentum urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0022 ( urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0023 are electron mass and total and transverse velocities, respectively, and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0024 is electron cyclotron frequency), this expression takes the form:
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0025(2)
Here urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0026 is the wave angular frequency, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0027 is the magnitude of the wave normal vector, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0028 is the magnitude of electron charge, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0029 is the speed of light, and
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0030(3)
where
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0031(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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0032, 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):
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0033(5)

Relation 5 determines the electron distribution function in CGS system of units, used in the paper, through the measured differential electron flux urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0034 expressed in practical units, that is, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0035 and the particle energy in keV denoted by urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0036. 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0037.

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.

jgra55531-fig-0004
Normalized growth rate versus normalized frequency for parallel‐propagating whistler mode waves calculated from RBSP‐B measurements of electron fluxes by MagEIS instrument. Left and right panels display the normalized growth rate before and after the series of whistlers shown in Figure 3.

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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-003817.4° 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-003917.4° 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0040‐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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0041, 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.

jgra55531-fig-0005
Normalized growth rate versus normalized frequency at the equator ( urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0042, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0043°) calculated from electron fluxes measured by MagEIS instrument on RBSP‐B on 22 December 2012, at 10:13:59 UT.

3 Qualitative Model of the VLF Noise Suppression by Whistlers

3.1 Main Assumptions of the Model

A general idea, which has been put forward by Helliwell et al. (1980) and Gail and Carpenter (1984) for explanation of the wave phenomenon under discussion, consists in the modification of the electron distribution function by whistler echo train in the way that reduces plasma instability causing the hiss generation. In the present study we follow up this idea and develop it into a qualitative model. A few assumptions suggested by observations that we use are as follows.
  1. VLF noise is generated in the equatorial region of the magnetosphere by unstable plasma distribution and is characterized by a quasi‐parallel direction of the wave normal vectors. The generation process is a subject of quasilinear theory, and thus, the growth rate is determined by the linear expression in which the unperturbed distribution function is replaced by varying in time phase‐averaged distribution.
  2. Whistler echo train causing the noise suppression propagates in ducted mode and can be described in the approximation of parallel propagation.
  3. At the pre‐event stage, the plasma is in a marginally unstable state, the upper frequency of the observed noise band corresponding to the boundary between unstable and stable frequency bands.
  4. In the magnetospheric region, where the strong whistler interactions with resonant electrons responsible for the noise generation take place, the electromagnetic field of the whistler echo train can be represented as a sum of three wave packets bounded in space and time, with varying frequencies and wave numbers satisfying the local dispersion relation for parallel‐propagating whistler mode waves. The frequency range of the causative whistler echo train overlaps the frequency band occupied by the suppressed noise.
  5. While the generation of VLF hiss may be described in quasilinear approximation, that is, by the linear growth rate determined by the time‐dependent distribution function, the strong whistler echo train changes the energetic particle distribution on the time scale much shorter than the quasilinear time.

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.

jgra55531-fig-0006
Upper panels. Frequency in kHz, color coded according to the color bar, in space‐time domain of three wave packets corresponding to multihop whistlers: (left column) fractional hop whistler, (middle column) second hop whistler reflected from the ground in the Northern Hemisphere, and (right column) third hop whistler reflected from the ground in the Southern Hemisphere. Lower panels. Parallel resonance energy in keV, color coded according to color bar, in space‐time domain for three wave packets corresponding to multihop whistlers. Only domains where the parallel resonance energy calculated according to nonrelativistic expression is below 500 keV are displayed.
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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0044 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0045 components, transverse to the direction of wave propagation. Thus, we will write the wave electric field in the form
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0046(6)
where urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0047 is the coordinate along the ambient magnetic field and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0048 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0049 are amplitudes and phases of the urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0050th 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0051 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:
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0052(7)
In each wave packet, the wave frequency urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0053 and the wave normal vector urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0054 depend on both coordinate urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0055 and time urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0056, but for each urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0057 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0058 they are related by the dispersion relation for parallel‐propagating whistler mode waves, namely,
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0059(8)
where urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0060 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0061 are the frequency and the wave number defined above which depend on urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0062, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0063 is electron plasma frequency that depends on urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0064, and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0065 is, as before, the electron cyclotron frequency which also depends on urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0066. 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0067 and coordinate urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0068, the wave number urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0069, up to its sign, is determined by 8. For the first and the third wave packets urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0070, while for the second wave packet urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0071. 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0072‐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: urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0073 with the constant of proportionality that provides the ratio urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0074 at the equator on urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0075.
In nonrelativistic approximation, and for parallel‐propagating whistler mode wave, the resonance between the wave and electron arises under condition urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0076. Correspondingly, electron resonance parallel energy in keV, with the account of 8, is given by
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0077(9)
where urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0078. The quantity urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0079 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0080, that is, of the direction of the wave propagation. In our case, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0081 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0082 are positive, while urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0083 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0084 and magnetic momentum urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0085 are conserved, and the particle motion can be described by the equations that follow from the unperturbed Hamiltonian
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0086(10)
where canonically conjugated variables are urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0087 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0088 and where urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0089 is the particle gyrophase. Transverse components of electron velocity are expressed in canonical variables as follows:
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0090(11)
We now write the variation of electron kinetic energy due to interaction with the whistler echo train:
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0091(12)
where urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0092 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
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0093(13)
The equations of motion which follow from the Hamiltonian 13 have the form
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0094(14)

While we have taken into account the derivative of cyclotron frequency with respect to urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0095 in the equation for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0096 that originates from the unperturbed Hamiltonian, we have neglected the derivatives with respect to urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0097 of slowly varying quantities urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0098 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0099 in the interaction Hamiltonian that contains wave amplitudes urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0100 as the factors. This is valid under conditions of applicability of GO urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0101, where urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0102 is the characteristic scale of plasma inhomogeneity. We should stress that neglecting the derivatives of the abovementioned quantities with respect to urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0103 in the equation for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0104 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0105 is an unknown function, while the quantities urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0106 should be considered as known ones. However, the equations of GO from which the quantities urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0107 should be found define first of all their derivatives, that is, the quantities urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0108 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0109. That is why, in the equations of motion 14, it is convenient to use quantities urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0110 as new unknown functions, since the function urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0111 enters the equations of motion only in combinations urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0112. Thus, instead of equation for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0113 and equations for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0114, it is more convenient to use the equations for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0115, which follow from the definitions given above and the equation for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0116:
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0117(15)

In these equations, the quantities urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0118 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0119 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0120, and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0121 in an evident form. This procedure is not appropriate in our case. First, we have three phases urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0122 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0123, parallel momentum urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0124, and the magnetic momentum urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0125 (equation 14), while the equation for particle gyrophase urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0126 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0127 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0128, 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0129 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.

jgra55531-fig-0007
Time variations of particle latitude urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0130, longitudinal velocity urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0131, and magnetic momentum urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0132 normalized to its initial value urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0133, for three particles shown in magenta, green, and blue, along their trajectories in the field of multihop whistlers and the ambient geomagnetic field. Initial parameters of the particles are, respectively, as follows: urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0134=(1.55°, 24.05°, 24.05°); urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0135( urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0136, 0.22 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0137, urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0138); urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0139; urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0140; urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0141. Particles move along urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0142 where electron cyclotron frequency at the equator urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0143.
jgra55531-fig-0008
Time variations of particle kinetic energies urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0144 normalized to initial values and derivatives of the phases of three wave packets representing multihop whistlers. Initial conditions and color correspondence are the same as in Figure 7.
For quantitative estimation of the suggested mechanism, we have calculated the average variation squared, during the duration of the whistler echo train urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0145 s, of magnetic momentum for 100 particles with a characteristic initial energy urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0146 keV and a characteristic magnetic momentum urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0147 such that urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0148 keV, uniformly distributed over gyrophases and the accessible range of initial latitudes on urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0149. These determine the particle diffusion coefficient urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0150 according to the relation
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0151(16)
where the overline means the average over the time urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0152. For the parameters of the model described above, the calculations give urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0153, so that urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0154 s urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0155.

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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0156 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0157.

As is well known, for diffusion process urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0158, thus, the diffusion coefficient urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0159 does not depend on time urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0160 for which it has been calculated, since with increasing urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0161, the quantity urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0162 increases proportionally. The characteristic diffusion time urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0163 is determined as the time for which urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0164. 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 urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0165. Substitution urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0166 for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0167 and urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0168 for urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0169 in 16 gives
urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0170(17)

Using the figures given above, we obtain the characteristic diffusion time urn:x-wiley:jgra:media:jgra55531:jgra55531-math-0171, which also provides an estimation for the relaxation time of unstable plasma distribution.

4 Concluding Remarks

We have presented experimental evidences of VLF noise suppression by strong whistlers which is observed on VLF spectrograms obtained at Kannuslehto ground station, after cleaning them from sferics, as well as on spectrograms obtained on the Van Allen Probe B satellite. The investigation of suppression effect based on satellite wave and particle measurements has been performed for the first time.

The idea that we use to explain this phenomenon can be traced back to early works by Helliwell et al. (1980) and Gail and Carpenter (1984). It consists in the assumption that, due to interaction with strong whistlers, the unstable electron distribution is modified in such a way that decreases the instability threshold. This modification should first of all be revealed at the boundary of unstable band, as is indeed observed in experiment.

Our general approach to the problem of VLF noise suppression by whistlers consists in the following. According to observations, the emission above 2 kHz that we consider has an incoherent spectrum, with low spectral intensity as compared, for instance, to that below 1 kHz. The evolution of such a spectrum is a subject of quasilinear theory. One of the main concepts of this theory consists in that the growth rate which determines the evolution of this spectrum is determined by the linear growth rate, with the unperturbed distribution function replaced by the phase‐averaged time‐dependent distribution obtained from a diffusion equation in the phase space. This diffusion is a relatively slow process since the diffusion coefficients are proportional to spectral intensity which is assumed to be small and which is fulfilled in our case as has been mentioned above. In the quasilinear theory, there are no coherent effects, such as phase trapping by strong whistler echo train, of course. We suggest that in the presence of whistler echo train that brings these effects into play, the evolution of the distribution function determining the growth rate of VLF noise takes place much faster, which permits to explain the suppression effect.

To develop these ideas, we set forth a theory of electron resonant interaction with parallel‐propagating multihop whistlers (whistler echo train). For the first time, we have derived and solved numerically the set of equations that describe electron motion in the field of three wave packets representing the multihop whistlers and the ambient inhomogeneous geomagnetic field. These equations take into account space‐time variations of frequencies and wave vectors in the packets, as well as space‐time boundedness of the wave packets. We have shown that resonant interaction with such wave packets leads to faster electron diffusion in the phase space. We suggest that this diffusion causes the modification of unstable electron distribution and is responsible for the suppression effect.

Although in a sense the VLF noise may be considered as an upper frequency part of plasmaspheric hiss, we should stress that the problem of VLF noise suppression by whistlers is related but by no means identical to the problem of plasmaspheric hiss origin (Bortnik et al., 2009; Nakamura et al., 2016; Omura et al., 2015), and in fact, the latter is out of the scope of the present paper.

Acknowledgments

This study was done in the frame of the Grant 294931 of the Academy of Sciences of Finland. D. R. S. and E. E. T. also acknowledge support from RFBR Grant 19‐02‐00179. O. S. and I. K. acknowledge support from Grants LTAUSA17070, GACR 17‐07027S, and the Praemium Academiae award from the CAS. The authors thank the PI of EMFISIS instrument C. Kletzing and the PI of MagEIS instrument H. Spence of Van Allen Probes team for the use of the data which is available online (http://cdaweb.sci.gsfc.nasa.gov/istp_public/). SGO ELF‐VLF quicklook plots are available online (http://www.sgo.fi/pub_vlf/).