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Of all publications in the section: 19
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Article
Kronberg E., E.E. Grigorenko, Malykhin A. Y. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2019. Vol. 124. P. 5056-5069.

The dissipation processes which transform electromagnetic energy into kinetic particle energy in space plasmas are still not fully understood. Of particular interest is the distribution of the dissipated energy among different species of charged particles. The Jovian magnetosphere is a unique laboratory to study this question because outflowing ions from the moon Io create a high diversity in ion species. In this work, we use multispecies ion observations and magnetic field measurements by the Galileo spacecraft. We limit our study to observations of plasmoids in the Jovian magnetotail, because there is strong ion acceleration in these structures. Our model predicts that electromagnetic turbulence in plasmoids plays an essential role in the acceleration of oxygen, sulfur, and hydrogen ions. The observations show a decrease of the oxygen and sulfur energy spectral index γ at ̃30 to ̃400 keV/nuc with the wave power indicating an energy transfer from electromagnetic waves to particles, in agreement with the model. The wave power threshold for effective acceleration is of the order of 10 nT2Hz-1, as in terrestrial plasmoids. However, this is not observed for hydrogen ions, implying that processes other than wave-particle interaction are more important for the acceleration of these ions or that the time and energy resolution of the observations is too coarse. The results are expected to be confirmed by improved plasma measurements by the Juno spacecraft.

Added: Mar 5, 2020
Article
Petrukovich A. A., Lukin A. S. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2018. Vol. 123. No. 4. P. 2872-2883.

In a statistical model plasma sheet By primarily depends on interplanetary magnetic field (IMF) By and geodipole tilt τ. With 11 years of Geotail measurements we investigate a role of several other parameters with a linear regression model. Optimal averaging window of IMF input, maximizing correlation and regression coefficients, is found to be 2.25 hr. Influence of IMF  and local Bz on IMF penetration (regression with regard to) and the deviations from the predefined warp deformation are at the level 5–10% relative to the primary model coefficients. The IMF penetration beyond 25 RE is somewhat larger for northward IMF, while closer to the Earth it becomes somewhat larger for southward IMF. These smaller effects turned out to be rather uneven across the tail, making reliable quantification and physical interpretation not always possible. The major reasons of difficulties are uneven coverage and internal correlations in the input space ( ByτBz ) due to combination of spacecraft orbit and neutral sheet dynamics, effects of coordinate transformations, etc. In particular, origins of extremely large IMF  penetration (order of 30–50% above the average one) for some years and tail locations remain not fully clear. A larger multispacecraft data set covering all seasons in all spatial zones is necessary to further advance in this study.

Added: Jan 31, 2020
Article
V. E. Shaposhnikov, Zaitsev V. V., Litvinenko G. V. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2018. Vol. 123. No. 11. P. 9395-9405.

The ion cyclotron waves instability near the frequencies of ionic cyclotron harmonics is studied under conditions typical for the Jovian kilometer radio emission source. This instability is caused by the presence of the ions with a nonequilibrium loss cone-type distribution function in the source. Expressions for the growth rate of the ion cyclotron wave’s instability are obtained, and it is shown that the amplification of these waves significantly increases when the frequency of the lower-hybrid resonance coincides with one of the ion cyclotron harmonics. Possible mechanisms for conversion of the ion cyclotron waves excited in the Jovian kilometer radio emission source into electromagnetic radiation are discussed. It is shown that the conversion of ion cyclotron waves, due to their coalescence with high-frequency plasma waves, is possible only into ordinary electromagnetic waves, while the process capable of providing conversion of ion cyclotron waves into fast extraordinary electromagnetic waves is scattering by suprathermal electron fluxes. In the latter, despite the thermal spread of electron velocities, the radiation that leaves the source region is concentrated in a narrow frequency range Δ𝜔 ≪ 𝜔near the local gyrofrequency 𝜔 𝜔Be, and the polarization of this radiation corresponds to the extraordinary wave. The estimates of the electron energy, which is necessary for such conversion, have shown the possibility of the process realizing under conditions characteristic for the Jovian kilometer radio emission source.

Added: Nov 16, 2018
Article
Shapoval S., Le Mouël J., Shnirman M. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2017. Vol. 122. No. 12. P. 11,874-11,887.

This paper explores some features of the dynamics of daily sunspot numbers on scales from days to years. We define higher and lower frequency energy components of the series that are related to periods ranging over 1-6 days and 6 days-2 years respectively. The lower frequency component is found to follow the solar activity, but the maxima of the higher frequency component are unexpectedly lower during the last epoch of high solar activity than during the preceding epoch of low solar activity. We also consider the birthrate of sunspot groups as another indicator of quickly varying components of the solar activity and show that it is the general growth of solar activity in the 1930-1940s which drives up this birthrate. We propose an auto-regressive model that captures the opposite trends exhibited by the two representatives of the high-frequency content, accurately reproduces the evolutions of the lower and higher frequency energy components, and replicates the shape of the curve representing the daily sunspot numbers. The three following hypotheses underlie the model construction: (1) proxy series of solar activity can be modeled by a random process with a modulated noise; (2) sunspot's birth and disappearance rates, both following the solar cycle, determine properties of this process; (3) the births of sunspots are positively correlated in time during epochs of high solar activity. We find that the mean birthrate varies as a power function of the mean lifetime. Derived constraints could contribute to narrowing the choice of a proper solar dynamo model.

Added: Jan 30, 2018
Article
D.R. Shklyar. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2017. Vol. 122. P. 640-655.

We study the problem of energy exchange between waves and particles, which leads to energization of the latter, in an unstable plasma typical of the radiation belts. The ongoing Van Allen Probes space mission brought this problem among the most discussed in space physics. A free energy which is present in an unstable plasma provides the indispensable condition for energy transfer from lower energy particles to higher-energy particles via resonant wave-particle interaction. This process is studied in detail by the example of electron interactions with whistler mode wave packets originated from lightning-induced emission. We emphasize that in an unstable plasma, the energy source for electron energization is the energy of other particles, rather than the wave energy as is often assumed. The way by which the energy is transferred from lower energy to higher-energy particles includes two processes that operate concurrently, in the same space-time domain, or sequentially, in different space-time domains, in which a given wave packet is located. In the first process, one group of resonant particles gives the energy to the wave. The second process consists in wave absorption by another group of resonant particles, whose energy therefore increases. We argue that this mechanism represents an efficient means of electron energization in the radiation belts.

Added: Nov 26, 2017
Article
Malova H. V., Mingalev O. V., Grigorenko E. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2015. Vol. 120. No. 6. P. 4802-4824.
Self-consistent kinetic (particle-in-cell) model of magnetotail thin current sheet (TCS) is used to understand the formation of self-consistent sheared magnetic structures. It is shown that shear configurations appear in TCS as a result of self-consistent evolution of some initial magnetic perturbation at the current sheet center. Two general shapes of shear TCS components are found as a function of the transverse coordinate: symmetric and antisymmetric. We show that TCS formation goes together with the emergence of field-aligned currents in the center of the current sheet, as a result of north-south asymmetry of quasi-adiabatic ion motions. Ion drift currents can also contribute to the magnetic shear evolution, but their role is much less significant, their contribution depending upon the normal component Bz and the amplitude of the initial perturbation in TCS. Parametric maps illustrating different types of TCS equilibria are presented that show a higher probability of formation of symmetric shear TCS configuration at lower values of the normal magnetic component.
Added: Feb 16, 2017
Article
Shklyar D. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2020. Vol. 125. P. 1-14.

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 L-shell ∼3.

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.

Added: Mar 12, 2020
Article
Grigorenko E. E., Shuvalov S., Malova H. V. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2017. Vol. 122. No. 10. P. 176-193 .

Numerous studies of the current sheets (CS) in the Earth’s magnetotail showed that quasi-adiabatic ion dynamics plays an important role in the formation of complicated multilayered current structures. In order to check whether the similar mechanisms operate in the Martian magnetotail, we analyzed 80 CS crossings using MAVEN measurements on the nightside of Mars at radial distances ~1.0–2.8RM. We found that CS structures experience similar dependence on the value of the normal component of the magnetic field at the neutral plane (BN) and on the ratio of the ion drift velocity outside the CS to the thermal velocity (VT/VD) as it was observed for the CSs in the Earth’s magnetotail. For the small values of BN, a thin and intense CS embedded in a thicker one is observed. The half-thickness L of this layer is ~30–100 km ≤ ρH+ (ρH+ is a gyroradius of thermal protons outside the CS). With the increase of BN, the L also increases up to several hundred kilometers (~ρO+, ρO2+), the current density decreases, and the embedding feature disappears. Our statistical analysis showed a good agreement between L values observed by MAVEN and the CS scaling obtained from the quasi-adiabatic model, if the plasma characteristics in Martian CSs are used as input parameters. Thus, we may conclude that in spite of the differences in magnetic topology, ion composition, and plasma thermal characteristics observed in the Earth’s and Martian magnetotails, similar quasi-adiabatic mechanisms contribute to the formation of the CSs in the magnetotails of both planets

Added: Oct 25, 2017
Article
Zlotnik E. Y., V. E. Shaposhnikov, Zaitsev V. V. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2016. Vol. 121. No. 6. P. 5307-5318.

The origin of a fine structure as quasi-harmonic parallel drifting stripes of enhanced brightness (zebra pattern) in the dynamic spectrum of Jovian kilometric radiation is discussed. A possible interpretation of the observed structure based on the effect of double plasma resonance (DPR) in the Jupiter magnetosphere is analyzed. It is shown that the observed features of the zebra pattern cannot be attributed to the DPR effect at electron cyclotron harmonics. The proposed scheme consists of excitation of ion cyclotron waves at the low hybrid frequency in the ion DPR regions and succeeding coalescence of these waves with a longitudinal wave at the upper hybrid frequency. The source parameters necessary for matching the expected and observed properties of the Jupiter zebra pattern are discussed.

Added: Nov 18, 2016
Article
E.E. Grigorenko, Runov A., Angelopoulos V. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2019. Vol. 124. No. 3. P. 1883-1903.

We study characteristics of ion and electron beams observed during 101 crossings of the near-separatrix region by Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon's Interaction with the Sun (ARTEMIS) spacecraft in the magnetotail. We found that accelerated ion beams are observed under any level of geomagnetic activity. A duration of earthward moving ion beams is statistically longer (≤ 10 min) than a duration of tailward ion beams (≤ 4 min), which can be due to the transient character of ion acceleration in the vicinity of the near-Earth neutral line (NENL). Energetic characteristics of earthward and tailward ion beams are similar indicating similar acceleration conditions at ion kinetic scales at both sides of an X line independently of its location. Conversely, electron velocity distributions observed near magnetic separatrix earthward of the distant neutral line (DNL) differ from those observed tailward of the NENL. Earthward of the DNL a scattered and thermalized electron population without energetic field-aligned beams is observed near the separatrix. On the contrary, tailward of the NENL field-aligned electron beams accelerated to a few kiloelectron volts are detected. These observations show that near DNL the electron scattering and thermalization dominate over the direct acceleration, whereas stronger electric fields in the NENL produce substantial population of field-aligned kiloelectron volt electrons.

Added: Mar 5, 2020
Article
Katushkina O., Izmodenov V.V., Alexashov D. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2016. Vol. 121. No. 1. P. 93-102.

We perform a new analysis of the Lyman alpha data obtained by Voyager 1 during the spatial scans in 1993–2003 while Voyager 1 was at 53–88 AU from the Sun. These data are the important source of information on the hydrogen distribution in the outer heliosphere. A sophisticated global kinetic-MHD model of the heliospheric interface and a radiative transfer model are used for the analysis. It is shown for the first time that the ratio of the Lyman alpha intensities detected in the downwind and upwind lines of sight in the outer heliosphere is sensitive to the configuration (peak value and location) of the hydrogen wall. The hydrogen wall is a source of Doppler-shifted backscattered Lyman alpha photons, so it can be seen from inside the heliosphere. Therefore, Voyager 1/ultraviolet spectrometer (UVS) Lyman alpha data can be used for remote sensing of the hydrogen wall. We show that our current global model of the outer heliosphere, which is consistent with many other measurements including Lyman alpha data from both Voyager 1 and 2 in 1980–1993, provides a systematically larger downwind to upwind intensity ratio compared with the UVS data in 1993–2003. In order to decrease the ratio, a higher and/or closer hydrogen wall is needed.

Added: Jan 26, 2018
Article
Shklyar D. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2019. Vol. 124. No. 5. P. 3522-3531.

For the first time an evaluation of the whistler rate around the Earth is performed using results from the neural network aboard the microsatellite DEMETER. It is shown that the rate of whistlers with low dispersion calculated all around the Earth as a function of longitude vary between 1 and 6 s−1 during nighttime (22.30 LT) and between 0.5 and 0.7 s−1 during daytime (10.30 LT). The whistler rate is anticorrelated with the F10.7‐cm solar flux. A decrease by 25% of the solar flux corresponds to an increase of 62% (26%) of the averaged whistler rate calculated for the entire Earth during nighttime (daytime). Using this averaged whistler rate, the global lightning rate is estimated to be of the order of 123 s−1 (27 s−1) during nighttime (daytime). The main conclusion concerns the precipitation of the electrons in the radiation belt by interaction with the whistlers. It is shown that the decrease of the lightning activity at solar minimum (shown with the help of the Schumann resonances) is largely counterbalanced by the increase of the whistler rates in the upper part of the ionosphere due to the decrease of the ionospheric absorption.

Added: Jun 17, 2019
Article
Lukin A.S., Artemyev A. V., Petrukovich A. A. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2019. Vol. 124. No. 7. P. 5027-5041.

One of the most important sources of magnetospheric plasma is particle entry through the distant magnetotail boundary, the nightside magnetopause. This entry mechanism depends on the magnetopause configuration. Off the equator, the strong lobe magnetic field renders the magnetopause a tangential or a rotational discontinuity, and thus the magnetosheath field orientation predominantly controls particle entry through magnetic reconnection. At the equatorial, distant tail magnetopause, however, the magnetic field's control of particle entry is diminished because the plasma beta there is large on both sides of the boundary. Thus, transport there can be significantly different from that at the dayside and off-equatorial magnetopauses. Using observations from two Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon's Interaction with the Sun probes, we investigate plasma transport mechanisms around the distant equatorial magnetopause.We find that transport occurs as a series of abrupt transitions in density, ion and electron temperatures, and ion kinetic energy of spatial scales as small as a typical plasma sheet ion gyroradius. Analysis of the particle phase space density reveals that an energy-selection mechanism controls electron transport across the magnetopause, whereas ion transport is likely controlled by spatial diffusion driven by low-frequency magnetic field fluctuations.We discuss the importance of these fluctuations for the magnetopause structure (e.g., the thickness of the transitions in plasma density, ion and electron temperatures, and ion kinetic energy).

Added: Jan 31, 2020
Article
Shustov P., Zhang X., Pritchett P. L. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2019. Vol. 124. No. 1. P. 342-359.

Magnetic energy release during magnetic reconnection in the magnetotail leads to fast plasma flows transporting thermal energy toward the inner magnetosphere or deep tail. The interaction of such flows with the ambient plasmas is controlled by forces at the flow's leading edge, manifested as a sharp enhancement of the south‐north component of magnetic field there, which has been called the dipolarization front. In this study, we examine the kinetic plasma structure of equatorial magnetic field perturbations observed behind dipolarization fronts. Using statistical observations of dipolarization fronts in the near‐Earth magnetotail by Time History of Events and Macroscale Interactions during Substorms mission, we find sub‐ion scale (scale is below ion gyroradius) magnetic field depressions (magnetic holes), mostly around the equatorial plane, drifting dawnward. They are populated by hot, transversely anisotropic electrons, likely heated around the front. Combining spacecraft observations, analytical estimates, and particle‐in‐cell simulations, we suggest that these holes result from the ballooning/interchange instability at the dipolarization front. They may represent the nonlinear stage of magnetic field perturbations associated with front instability, which trap hot electrons behind the front. We also discuss the possible role of these holes in scattering and heating electrons and ions in the dipolarized magnetotail.

Added: Jan 31, 2020
Article
Балюкин И. И., Bertaux J., Quemerais E. и др. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2019. Т. 124. № 2. С. 861-885.

The Earth’s hydrogen exosphere Lyman-α radiation was mapped with the SWAN/SOHO instrument in January 1996, 1997 and 1998 (low solar activity). The use of a hydrogen absorption cell allowed to disentangle the interplanetary emission from the geocoronal one and to assign the absorbed signal almost entirely to the geocorona. The geocorona was found to extend at least up to 100 Earth Radii (R E ) with an intensity of 5 Rayleigh, an unprecedented distance well exceeding the recent results of LAICA imager (∼50 R E ), and encompassing the orbit of the Moon (∼60 R E ). We developed a numerical kinetic model of the hydrogen atoms distribution in the exosphere which includes the solar Lyman-α radiation pressure and the ionization. The radiation pressure compresses the H exosphere on the day side, producing a bulge of H density between 3 and 20 R E which fits observed intensities very well. The SWAN Lyman-α distribution of intensity was compared both to LAICA (2015) and to OGO-5 (1968) measurements. Integrated H densities of SWAN at a tangent distance of 7 R E are larger than LAICA/OGO-5 by factors 1.1-2.5, while we should expect a stronger effect of the radiation pressure at solar max. We discuss the possible role of H atoms in satellite orbits to explain this apparent contradiction. An onion-peeling technique is used to retrieve hydrogen number density in the exosphere for the three SWAN observations. They show an excess of density versus models at large distances, which is likely due to non-thermal atoms (not in the model).

Added: Feb 7, 2019
Article
Vaisberg O., Berthellier J., Moore T. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2016. Vol. 121. No. 12. P. 11750-11765.

Increasing the temporal resolution and instant coverage of velocity space of space plasma measurements is one of the key issues for experimentalists. Today, the top‐hat plasma analyzer appears to be the favorite solution due to its relative simplicity and the possibility to extend its application by adding a mass‐analysis section and an electrostatic angular scanner. Similarly, great success has been achieved in MMS mission using such multiple top‐hat analyzers to achieve unprecedented temporal resolution. An instantaneous angular coverage of charged particles measurements is an alternative approach to pursuing the goal of high time resolution. This was done with 4‐D Fast Omnidirectional Nonscanning Energy Mass Analyzer and, to a lesser extent, by DYMIO instruments for Mars‐96 and with the Fast Imaging Plasma Spectrometer instrument for MErcury Surface, Space ENvironment, GEochemistry, and Ranging mission. In this paper we describe, along with precursors, a plasma analyzer with a 2π electrostatic mirror that was developed originally for the Phobos‐Soil mission with a follow‐up in the frame of the BepiColombo mission and is under development for future Russian missions. Different versions of instrument are discussed along with their advantages and drawbacks

Added: Dec 19, 2018
Article
Shklyar D. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2018. Vol. 123. P. 7077-7088.

U-shaped spectrograms observed by the DEMETER satellite in the equatorial region of the upper ionosphere are presented and explained for the first time. These spectra are characterized by the latitude-dependent upper cutoff frequency that increases with the latitude during nighttime. The observations suggest that this very low frequency wave phenomenon is closely connected with the anomalous increase in cold plasma density at and below the satellite. We show that restricting the analysis to propagation effects is insufficient to explain U-shaped spectra, and other effects, in particular, the collisional wave damping should be taken into consideration. To calculate the wave damping, we use the method of successive approximations. In the first approximation, we calculate the ray trajectory and the wave normal vector along it, using the equations of geometrical optics neglecting collisions. In the second approximation we calculate the collisional wave damping along the precalculated trajectory.We show that the effects of very low frequency wave propagation and attenuation in the equatorial region of the upper ionosphere explain the main features of the phenomenon under discussion. 

Added: Oct 31, 2018
Article
Katushkina O., Quemerais E., Izmodenov V.V. et al. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2017. Vol. 122. No. 11. P. 10921-10937.

In this work, we present for the first time the Lyman α intensities measured by Voyager 1/UVS in 2003–2014 (at 90–130 AU from the Sun). During this period Voyager 1 measured the Lyman α emission in the outer heliosphere at an almost fixed direction close to the upwind (i.e.“ toward the interstellar flow). The data show an unexpected behavior in 2003–2009: the ratio of observed intensity to the solar Lyman α flux is almost constant. Numerical modeling of these data is performed in the frame of a state-of-the-art self-consistent kinetic-MHD model of the heliospheric interface. The model results, for various interstellar parameters, predict a monotonic decrease of intensity not seen in the data. We propose two possible scenarios that explain the data qualitatively. The first is the formation of a dense layer of hydrogen atoms near the heliopause. Such a layer would provide an additional backscattered Doppler-shifted Lyman α emission, which is not absorbed inside the heliosphere and may be observed by Voyager. About 35 R of intensity from the layer is needed. The second scenario is an external nonheliospheric Lyman α component, which could be galactic or extragalactic. Our parametric study shows that ∼25 R of additional emission leads to a good qualitative agreement between the Voyager 1 data and the model results.

Added: Jan 26, 2018
Article
D.R. Shklyar, Balikhin M. JOURNAL OF GEOPHYSICAL RESEARCH-SPACE PHYSICS. 2017. Vol. 122. P. 10072-10083.

Equatorial noise in the frequency range below the lower hybrid resonance frequency, whose structure is shaped by high proton cyclotron harmonics, has been observed by the Cluster spacecraft. We develop a model of this wave phenomenon which assumes (as, in general, has been suggested long ago) that the observed spectrum is excited due to loss cone instability of energetic ions in the equatorial region of the magnetosphere. The wavefield is represented as a sum of constant frequency wave packets which cross a number of cyclotron resonances while propagating in a highly oblique mode along quite specific trajectories. The growth (damping) rate of these wave packets varies both in sign and magnitude along the raypath, making the wave net amplification, but not the growth rate, the main characteristic of the wave generation process. The growth rates and the wave amplitudes along the ray paths, determined by the equations of geometrical optics, have been calculated for a 3-D set of wave packets with various frequencies, initial L shells, and initial wave normal angles at the equator. It is shown that the dynamical spectrum resulting from the proposed model qualitatively matches observations.

Added: Nov 24, 2017