Quasiperiodic Emissions: Fine Structure Corresponding to a Bouncing Wave

Quasiperiodic (QP) emissions are whistler‐mode electromagnetic waves observed in the Earth's inner magnetosphere whose intensity has a nearly periodic time modulation with typical modulation periods on the order of minutes. Some events exhibit, on top of the main modulation period, an additional fine inner modulation with modulation periods on the order of seconds. We use high‐resolution multi‐component electromagnetic wave data obtained by the Van Allen Probes spacecraft to investigate one such event. Detailed wave analysis demonstrates that the fine inner modulation is due to a wave packet bouncing back and forth between the hemispheres. The presence of a density duct is important for the formation of the event, as demonstrated by the increased ratio of wave power propagating away from the equator (a tentative source region) within the duct. The main QP modulation period corresponds to the plasma number density modulation observed just outside the plasmasphere.


Introduction
The intensity of whistler-mode electromagnetic waves observed in the inner magnetosphere can be nearly periodically modulated, with modulation periods ranging from tens of seconds to minutes (Carson et al., 1965;Kitamura et al., 1968Kitamura et al., , 1969)).These waves are typically referred to as quasiperiodic (QP) emissions.They predominantly occur at frequencies between about 0.5 and 4 kHz (Hayosh et al., 2014), but can extend up to frequencies as high as 15 kHz (Farrell et al., 2022).While ground-based (Morrison et al., 1994) and low-altitude spacecraft (Hayosh et al., 2014) surveys indicate that QP emissions are primarily dayside phenomenon, satellite observations at larger radial distances reveal the emissions at essentially all local times, with a slightly larger occurrence on the duskside (Němec et al., 2018).Detailed wave analysis indicates that the wave source region is located at the geomagnetic equator (Demekhov et al., 2020;Hayosh et al., 2016;Němec et al., 2018).Non-ducted wave propagation, combined with ionospheric/magnetospheric reflections, may result in the emissions spanning a large area (Hanzelka et al., 2017;Martinez-Calderon et al., 2016;Němec et al., 2014).Multipoint observations of QP emissions indeed demonstrate that, at the time of the events, the same modulation of the wave intensity is observed over a large portion of the inner magnetosphere (Bezděková et al., 2020;Martinez-Calderon, Katoh, et al., 2020;Martinez-Calderon, Němec, et al., 2020;Němec, Bezděková, et al., 2016;Němec, Hospodarsky, et al., 2016;Němec, Santolík, Parrot, et al., 2013).The generation of QP emissions is accompanied by periodic changes of Abstract Quasiperiodic (QP) emissions are whistler-mode electromagnetic waves observed in the Earth's inner magnetosphere whose intensity has a nearly periodic time modulation with typical modulation periods on the order of minutes.Some events exhibit, on top of the main modulation period, an additional fine inner modulation with modulation periods on the order of seconds.We use high-resolution multi-component electromagnetic wave data obtained by the Van Allen Probes spacecraft to investigate one such event.Detailed wave analysis demonstrates that the fine inner modulation is due to a wave packet bouncing back and forth between the hemispheres.The presence of a density duct is important for the formation of the event, as demonstrated by the increased ratio of wave power propagating away from the equator (a tentative source region) within the duct.The main QP modulation period corresponds to the plasma number density modulation observed just outside the plasmasphere.

Plain Language Summary
The intensity of electromagnetic waves in the near-Earth space, the magnetosphere, sometimes has a nearly periodic temporal modulation on the order of minutes.The origin of such waves, so-called quasiperiodic emissions, is not yet fully understood.On top of the main modulation period, some events exhibit an additional fine inner modulation with periods on the order of seconds.We use wave propagation directions determined from the Van Allen Probes measurements to demonstrate that this shorter modulation corresponds to a wave packet bouncing in between the hemispheres.By examining the ratio of wave power propagating away from and toward the geomagnetic equator (a tentative source region), we further demonstrate that the presence of a region with enhanced density, guiding waves along a given magnetic field line, is important for the formation of the event.Additionally, the main modulation period of the event corresponds to the plasma number density modulation observed just outside the plasmasphere, possibly linked to a plasmapause surface wave.Our results, revealing the presence and origin of the fine inner structure of the waves, provide important observational constraints for models trying to explain the generation of quasiperiodic emissions.
Two principally different generation mechanisms are suggested to explain the origin of the QP wave intensity modulation.The first suggested mechanism assumes the presence of compressional ultralow frequency (ULF) magnetic field pulsations with periods corresponding to the QP modulation period, which periodically modulate the wave growth rate in the source (Chen, 1974;Kimura, 1974;Sato & Fukunishi, 1981;Sazhin, 1987;Shang et al., 2021;Watt et al., 2011).Such magnetic field pulsations are indeed observed for some QP events (Hajoš et al., 2022;Morrison, 1990;Němec, Santolík, Pickett, et al., 2013;Sato & Kokubun, 1980, 1981;Zhima et al., 2020).The second suggested mechanism is based on the idea of a flow cyclotron maser (Trakhtengerts & Rycroft, 2008).In this model, the periodic regime is formed self-consistently, without the need for an external modulating ULF wave.It arises as a result of a bouncing wave passing regularly through the source region, supplemented by a continuous replenishment of the particle anisotropy (Demekhov & Trakhtengerts, 1994;Pasmanik, Demekhov, et al., 2004).Such a model successfully reproduces some of the characteristics and dependences of QP emissions (Manninen et al., 2013;Němec et al., 2018;Pasmanik et al., 2019;Pasmanik, Titova, et al., 2004).
Events that are accompanied by coincident ULF magnetic field pulsations with matching periods may be related to the first generation mechanism.Historically, these have been termed QP type 1.Conversely, events without such coincident magnetic field pulsations, as well as those with coincident magnetic field pulsations whose periods do not match the QP modulation period, are historically termed QP type 2 (Sato et al., 1974) and potentially linked to the second generation mechanism.However, the ULF frequency spectrum can be quite complex, rendering the evaluation of the presence of coincident magnetic field pulsations problematic (Tixier & Cornilleau-Wehrlin, 1986).Moreover, QP emissions may be observed by spacecraft far from the source region, that is, in locations where the ULF magnetic field pulsations need not be present.The event classification is thus often unclear (Sazhin & Hayakawa, 1994).In some events, high-resolution measurements reveal an additional shorter modulation period, which roughly corresponds to the two-hop travel time of echoing whistlers (Bespalov et al., 2010;Engebretson et al., 2004;Manninen et al., 2014;Němec et al., 2023;Smith et al., 1998).
In the present study, we utilize high-resolution multi-component electromagnetic field data from the Van Allen Probes spacecraft to analyze a QP event that exhibits a fine inner structure.Detailed wave analysis reveals that this fine inner structure is linked to a wave element bouncing in between the hemispheres.Additionally, the cold plasma density is shown to control the event formation.The data set is described in Section 2. The obtained results are presented in Section 3 and discussed in Section 4. Finally, Section 5 contains a brief summary.

Data
The Van Allen Probes mission, previously known as the Radiation Belt Storm Probes mission, operated from 2012 to 2019.It consisted of two identical spacecraft following similar low-inclination orbits, with the perigee and apogee of approximately 1.1 and 6 Earth radii, respectively.The Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) suite, including the Waves instrument, carried out multi-component electromagnetic wave measurements up to 12 kHz (Kletzing et al., 2013(Kletzing et al., , 2023)).The EMFISIS survey mode data, organized into 65 quasi-logarithmic frequency bins, were based on a 0.5 s waveform captured every 6 s.Occasionally, active continuous burst mode measurements provided waveforms of all six electromagnetic field components sampled at 35 kHz over approximately 5.968 s, followed by a data gap of about 0.032 s.This resulted in a total duration of about 6 s for a single burst mode interval.However, the burst mode could be active for several consecutive intervals, yielding longer, nearly continuous burst mode data.Local plasma number density data were derived from the upper hybrid frequency (Kurth et al., 2015).

Results
Figure 1 provides an overview of the QP event from 28 August 2017, analyzed based on the survey mode data measured by the Van Allen Probe A. To better observe individual QP elements, the event is divided into two frequency-time spectrograms, each 20 min long, showing the power spectral density of magnetic field fluctuations.The white lines at the top indicate the time intervals when the burst mode was active.Additionally, the plasma number density is over-plotted using black curves and the scales on the right-hand side.The spacecraft was located north of the geomagnetic equator and it was moving toward lower radial distances.Neither the spacecraft latitude nor magnetic local time changes significantly during the analyzed time interval.Toward the very end of the interval, around 04:22 UT, the spacecraft crossed the plasmapause and entered the plasmasphere.This is indicated by both the sudden increase in density and the emergence of plasmaspheric hiss and higher wave intensities across the entire plotted frequency range.The QP event is observed all the time, but it is rather weak in the beginning and at the end, after crossing the plasmapause.
It is noteworthy that, just before the plasmapause crossing, large density fluctuations are observed.The wave intensity in the lower-frequency hiss part of the spectrum (up to about 1 kHz) is notably increased at the times of the individual density peaks occurring between about 04:18:30 and 04:22:00.In fact, the timing of individual QP elements within this time interval apparently corresponds to the observed density modulation.This is demonstrated in Figure 2a, which shows a zoomed view of the relevant time interval from Figure 1.The times of individual density peaks are shown by the vertical dashed lines.The corresponding time dependence of the wave intensities in individual frequency bins is shown in Figure 2b.Noticeably, for the first two intensity peaks that exhibit significant dispersion, the density peaks coincide with the wave intensity peaks at lower frequencies, while the waves at higher frequencies peak later.Additionally, it is noteworthy that the higher frequency waves associated with the second density peak appear very faint.
We use the burst mode wave data to further investigate a detailed structure and propagation of QP emissions.We perform a spectral analysis using a discrete Fourier transform with a length of 2,048 points, 50% overlapping, and averaging over three consecutive intervals.This results in frequency and time resolutions of about 17 Hz and 88 ms, respectively.Results obtained for the 1-min long time interval between 04:02 and 04:03 UT are depicted in Figure 3, but they are quite similar for the other parts of the burst mode interval as well (not shown).Figure 3a shows a power spectral density of magnetic field fluctuations.Three QP elements can be seen.Moreover, they exhibit a clear fine inner structure, being composed of short-duration subelements.
Results of a detailed wave analysis are shown in Figures 3b-3e.They are plotted only for frequency-time intervals where the power spectral density of magnetic field fluctuations exceeds 10 −8.5 nT 2 Hz −1 and the planarity of magnetic field fluctuations (Santolík et al., 2003) is greater than 0.5.This is done to ensure the results are meaningful and to select only the emissions corresponding to the event.Figure 3b shows frequency-time plot of the ellipticity of magnetic field fluctuations determined using the singular value decomposition (SVD) method  (Santolík et al., 2003).The obtained values are close to one, corresponding to a right-handed nearly circularly polarized wave.Figure 3c shows frequency-time plot of wave normal angle (θ k ) determined using the SVD method from the magnetic field spectral matrix (Santolík et al., 2003).It can be seen that the wave normal angles are rather low, around 30°. Azimuthal angles of wave vector direction (ϕ k ) are depicted in Figure 3d.These are again determined based exclusively on the magnetic field spectral matrix, in order to avoid problems stemming from the sheath impedance of the electric double probe instrument (Hartley et al., 2022).The ±180° ambiguity of the analysis is solved by reversing the azimuthal direction of waves propagating opposite the ambient magnetic field.This is determined based on the parallel component of the Poynting flux normalized by its experimental uncertainty (Santolík et al., 2001), depicted in Figure 3e.The values of the parallel component of the Poynting flux regularly alternate between consecutive subelements, indicating that when a given subelement propagates northward, the subsequent subelement propagates southward, and vice versa.This suggests that the subelements correspond to a wave element bouncing between the hemispheres.The azimuthal direction of the wave vector regularly alternates as well.Northward propagating waves have azimuthal angles close to 0° (direction "toward larger radial distances"), while southward propagating waves have azimuthal angles closer to ±180° (direction "toward lower radial distances").We note that the propagation properties of the emissions remain consistent at lower frequencies, even below the QP emission frequency range.This suggests that QP elements may, in a sense, emerge from lower-frequency hiss-like emissions.Curiously, there is essentially no hint of a dispersion, contrary to what might be expected for a multi-hop wave element.However, analogous non-dispersive periodic emissions were recognized and termed already by Helliwell (1965).
Figure 4 presents the overall results of the detailed wave analysis at frequencies between 1.5 and 2.0 kHz for the entire burst mode time interval.The results obtained for the frequency-time intervals with the parallel component of the Poynting flux oriented northward and southward are shown by the red and blue lines, respectively.Figure 4a displays histograms of the wave normal angles.Both histograms are quite similar, typically showing rather low wave normal angles (≈30°).Figure 4b displays histograms of the azimuthal angles of wave vector directions.The azimuthal angles corresponding to intervals of northward and southward propagation are quite distinct.While the waves propagating northward tend to have azimuthal angles close to 0°, the waves propagating southward tend to have azimuthal angles closer to ±180°, in agreement with the results from Figure 3.
Given that the spacecraft is located in the northern hemisphere at the time of the event, and considering the equatorial source of the emissions (Němec et al., 2018;Trakhtengerts & Rycroft, 2008), it is expected that the northward propagation at the source L-shells should be dominant.Hence, the relative power of the northward and southward propagating waves is investigated in Figure 5. Figure 5a displays the power spectral density of magnetic field fluctuations in individual frequency-time intervals as a function of L-shell.The primarily northward-propagating waves are denoted by red points, while the southward-propagating waves are marked by blue points and plotted with negative power values.Both propagation directions are observed throughout the event.However, southward propagation appears to dominate at lower L-shells, whereas primarily northward propagation is prevalent at larger L-shells.This observation is corroborated in Figure 5b, where the black curve represents a ratio of wave power propagating northward and southward in individual L-shell bins.While this power ratio is below 0.5 at lower L-shells, it rises to approximately 2.5 at higher L-shells.Significantly, this peak in the northward to southward power ratio coincides with a peak in the plasma number density, as depicted by the blue curve (a relative density increase of about 25%).This indicates that, during that time, the spacecraft was passing through a duct-like density structure.This structure is about 0.10-0.15L-shells wide, in agreement with the duct widths reported by Williams and Streltsov (2022).However, a simultaneous presence of multiple ducts is also possible.It is further noteworthy that the corresponding wave intensity is lower than at lower L-shells, closer to the plasmapause, in agreement with Figure 1.

Discussion
High-resolution multi-component wave measurements performed by the EMFISIS instrument on board the Van Allen Probes spacecraft in the continuous burst mode allow us, for the first time, not only to observe the fine inner structure of QP emissions, but also to assess the respective propagation directions.The observed wave normal angles are generally rather low (θ k < ≈45°), in agreement with former QP wave analysis results based on survey mode data (Demekhov et al., 2020;Němec et al., 2018;Titova et al., 2015).Moreover, the fine temporal resolution achieved in the present analysis reveals that the parallel component of the Poynting flux periodically alternates in correspondence with the fine inner structure of the QP event.This indicates a wave packet periodically bouncing between the hemispheres.Surprisingly, this bouncing is not accompanied by dispersion, as one might expect for whistler-mode waves.This pattern resembles the bouncing-element event reported by Demekhov et al. (2021), and some, probably nonlinear, mechanism seems necessary to compensate for the group velocity dispersion.Such a mechanism has been suggested by Bespalov (1984) and recalled by Bespalov et al. (2010), see also Trakhtengerts and Rycroft (2008).Another puzzling point concerns the mechanism of the bouncing wave reflection at low altitudes.While the northward-propagating waves have azimuthal directions of ϕ k ≈ 0°, signifying an orientation "toward larger radial distances," the southward-propagating waves exhibit nearly opposite azimuthal direc tions, implying an orientation "toward lower radial distances."This observation aligns with the wave maintaining the same L-shell during bouncing, supported by a symmetry argument.However, the specific reflection mechanism remains elusive.
At the time of the event observation, the spacecraft was located in the northern hemisphere.Assuming the wave source region at the geomagnetic equator and its subsequent attenuation during the propagation and reflections at low altitudes, the power of northward-propagating waves at the source L-shell is expected to be larger than the power of southward-propagating waves.It turns out that this is the case in a rather narrow L-shell interval, coinciding with the interval of increased cold plasma number density.This indicates that the enhanced duct-like density structure is crucial for the wave generation, in line with the flow cyclotron maser theory (Demekhov & Trakhtengerts, 1994).
The bouncing times of ducted waves periodically reflected at low altitudes depend on the exact distribution of electron density along the magnetic field line.However, these times are quite consistent with the observed time delays between wave packets propagating in the same direction, which is about 2.85 s (see Figure 3).Assuming a dipole magnetic field model, field-aligned propagation, L-shell of 3.59, and equatorial electron density of 85 cm −3 (see Figure 5b), the calculated wave bouncing times for waves with frequencies of 1.5 and 2.0 kHz are about 2.9 and 2.5 s, respectively.This is under the assumption that the electron density along the magnetic field line is proportional  to the magnetic field magnitude (Demekhov et al., 2017).However, if we assume the electron density profile along the magnetic field line as suggested by Denton et al. (2002), somewhat shorter bounce times of about 2.1 and 1.8 s are obtained, respectively.
The importance of the cold plasma number density for the wave generation is further suggested by the measurements conducted close to the plasmapause, which reveal nearly periodic variations in the plasma number density.These variations could be attributable to a plasmapause surface wave (He et al., 2020), or they may correspond to a stationary density structure; the single-point measurements performed do not allow for differentiation between the two possibilities.Curiously, the modulation period of QP emissions apparently matches the modulation period of these density oscillations, with individual QP elements corresponding to the individual density peaks.This suggests that the QP modulation of the wave intensity might be related to the density oscillations.Unfortunately, a detailed investigation of the fine inner structure and propagation of QP emissions near the plasmapause is not possible due to the absence of the continuous burst mode.However, in some sense, the aforementioned duct-like structure observed at somewhat larger distances may be rather similar to these near-plasmapause density variations.The density within the structure might in fact exhibit the same modulation as the density near the plasmapause.We note, in particular, that each of the three minor local density maxima observed within the duct-like structure roughly coincides with the start of a QP element at the lowest frequencies (≈1.5 kHz).The association of QP emissions and plasmapause surface waves might explain why QP emissions occur predominantly in the dusk sector (Němec et al., 2018) after periods of enhanced geomagnetic activity (Hayosh et al., 2014), as these are the locations/periods where plasmapause surface waves occur most frequently (Feng et al., 2023;Hao et al., 2023).

Conclusions
We presented a detailed analysis of a whistler-mode QP event observed by the Van Allen Probes spacecraft.The nearly continuous high-resolution multi-component wave measurements allowed us to reveal that, in addition to the main modulation, the QP event features a fine inner modulation related to the wave packet bouncing between hemispheres.We further highlight the importance of plasma density structures for the wave generation.The wave generation is related to the presence of an enhanced-density duct, and the main event period corresponds to the plasma density modulation just outside the plasmasphere.

Figure 1 .
Figure 1.Frequency-time spectrograms of power spectral density of magnetic field fluctuations corresponding to the analyzed quasiperiodic event measured on 28 August 2017 by Van Allen Probe A. The white lines at the top mark the time intervals when the burst mode data are available.The black curves and the right-hand ordinates show the in-situ plasma number density.

Figure 2 .
Figure 2. (a) A zoomed view of the part of Figure 1 around the plasmapause crossing.The vertical dashed lines mark the times of the density peaks.(b) Corresponding time dependence of wave intensities in individual frequency bins.The respective frequency ranges are shown on the right-hand side.

Figure 3 .
Figure 3. Frequency-time plots corresponding to a 1-min long part of the burst mode interval from Figure 1 between 04:02 and 04:03 UT.(a) Power spectral density of magnetic field fluctuations.(b) Ellipticity of magnetic field fluctuations.(c) Wave normal angle.(d) Azimuthal angle of wave vector direction.(e) Parallel component of the Poynting flux normalized by its experimental uncertainty.

Figure 4 .
Figure 4. Histograms of wave parameters obtained over the entire burst mode time interval from Figure 1 in the frequency range between 1.5 and 2.0 kHz.The results obtained for northward/southward propagating waves are shown by the red/blue lines, respectively.(a) Wave normal angle.(b) Azimuthal angle of wave vector direction.

Figure 5 .
Figure 5. (a) Power spectral density of magnetic field fluctuations in individual frequency-time intervals in the frequency range between 1.5 and 2.0 kHz as a function of L-shell.The results obtained for northward/southward propagating waves are shown by the red/blue points and with positive/negative values, respectively.(b) Ratio of power of magnetic field fluctuations in the frequency-time intervals with predominantly northward and predominantly southward propagation is shown by the black curve and the ordinate on the left-hand side.In-situ measured plasma number density is shown by the blue curve and the ordinate on the right-hand side.