Fine Structure of Magnetospheric Magnetosonic Waves: 1. Elementary Rising‐Tone Emissions Within Individual Harmonic

The present study uncovers the fine structures of magnetosonic waves by investigating the EFW waveforms measured by Van Allen Probes. We show that each harmonic of the magnetosonic wave may consist of a series of elementary rising‐tone emissions, implying a nonlinear mechanism for the wave generation. By investigating an elementary rising‐tone magnetosonic wave that spans a wide frequency range, we show that the frequency sweep rate is likely proportional to the wave frequency. We studied compound rising‐tone magnetosonic waves, and found that they typically consist of multiple harmonics in the source region, and may gradually become continuous in frequency as they propagate away from source. Both elementary and compound rising‐tone magnetosonic waves last for ∼1 min which is close to the bounce period of the ring proton distribution, but their relation is not fully understood.


Introduction
Magnetosonic waves, also known as equatorial noises (Russell et al., 1970;Santolík et al., 2002) or ion Bernstein mode waves (Min & Liu, 2016), are ubiquitously observed in Earth's magnetosphere (Hrbáčková et al., 2015;Ma et al., 2013Ma et al., , 2016Ma et al., , 2019)).Magnetosonic waves originate from proton ring distributions (∂f/∂v ⊥ > 0, where f is the phase space density and v ⊥ is the proton perpendicular velocity) with the ring velocity comparable to the Alfven velocity (Chen et al., 2010(Chen et al., , 2011;;Curtis & Wu, 1979;Ma et al., 2014).They play an important role in radiation belt dynamics by accelerating energetic electrons via Landau resonance (Bortnik & Thorne, 2010;Horne et al., 2007), producing electron butterfly pitch angle distributions from ∼100 keV to several MeV both in the inner and outer radiation belts (Li, Ni, et al., 2016, Li, Bortnik, et al., 2016).Magnetosonic waves also interact with electrons via bounce resonance (Chen et al., 2015;Li et al., 2015;Shprits;2016), which contributes to the formation of the electron butterfly distribution (Maldonado et al., 2016).Recent studies show magnetosonic wave modulation of energetic electrons in the dipolarization front, providing a new mechanism for electron energization therein (Fu et al., 2020(Fu et al., ). et al., 2002)), and the harmonic waves are coherent over a spatial scale of at least 3-5 wavelengths (Balikhin et al., 2015).Such a harmonic structure has been successfully explained by proton ring instabilities, which excite magnetosonic waves close to multiples of proton gyrofrequency at wave normal angles close to 90° (Balikhin et al., 2015;Chen et al., 2015;Min et al., 2018).Transfer function analysis and bicoherence analysis suggest that nonlinear interaction is negligible in the formation of the harmonic structure (Walker et al., 2020).
Magnetosonic waves sometimes exhibit periodic rising-tone features with a duration of ∼1 min, a repetition of a few minutes (Boardsen et al., 2014;Fu, Cao, Cully, et al., 2014;Fu, Cao, Zhima, et al., 2014) and a frequency sweep rate typically of ∼1 Hz/s.Magnetosonic waves occasionally consist of two series of periodic rising-tone waves known as "zipper-like" magnetosonic waves (Li et al., 2017).Burst mode waveform measured by Cluster satellites shows rising-tone magnetosonic waves that consist of multiple harmonics (Němec et al., 2015), and a case study suggests that the sweep rate maximizes at the equator (Walker et al., 2016).Van Allen Probe also observed the harmonic structure of rising-tone magnetosonic waves (Němec et al., 2020).These works investigated the magnetic field, plasma density and proton distribution variation in association with rising-tone magnetosonic waves, and no evidence of a modulation source has been clearly identified.Several hypotheses have been proposed to explain these rising-tone magnetosonic waves.By performing particle-in-cell simulations, Sun et al. (2020) indicated that the proton ring distribution excites magnetosonic waves at low frequencies in the beginning.As the waves scatter the ring distribution and lead to a new ring distribution at lower energies, magnetosonic waves are excited at higher frequencies, and this process occurs on a timescale comparable to observations.
Instruments with advanced capabilities are able to uncover phenomena that were previously hidden or invisible, leading to scientific discoveries or innovation.The recently released burst-mode waveforms measured by the EFW instrument onboard Van Allen Probes provides unprecedented opportunity for revealing fine structures of a variety of waves.In this paper, we report a new structure of magnetosonic waves, namely, the elementary risingtone emissions within each harmonic.We also compare these elementary rising-tone magnetosonic waves with the traditionally known rising-tone ones, which we name as compound rising-tone magnetosonic waves, and discuss their possible origin.

Data Description
The Van Allen Probes (Mauk et al., 2013) were equipped with two instruments that measure electromagnetic waves with high cadence.The Waves instrument's Wave Form Receiver (WFR), a part of the Electric and Magnetic Field Instruments Suite and Integrated Science (EMFISIS) (Kletzing et al., 2013), provides surveymode wave spectra over a frequency range of 10 Hz-12 kHz.It provides 3-D electric and magnetic waveform measurements at a cadence of 35,000 samples/s, and each episode of waveform data lasts for ∼6 s.Another instrument measuring waves is the Electric Field and Waves (EFW) instrument (Wygant et al., 2013), which consists of four deployable antennas that extend up to 100 m in space and two fixed antennas mounted on the spacecraft body.The EFW measures waveforms at cadences varying from 512 to 16,384 samples/s, and it provides continuous measurements of waveforms up to several hours, much longer than the 6-s WFR waveform.With high cadences and long durations, the EFW waveforms can produce unprecedented resolution in frequency, and thus are ideal for revealing the detailed structures of magnetosonic waves.This study also investigates proton distributions provided by the Helium Oxygen Proton Electron (HOPE) (Funsten et al., 2013;Spence et al., 2013) instruments to assist understanding the generation of magnetosonic waves.

Elementary Rising-Tone Emissions Embedded in Each Wave Harmonic
Figure 1 exhibits magnetosonic waves measured by Van Allen Probe A from 02:30 UT to 02:52 UT on 2 December 2015, when the satellite was in the pre-noon sector. Figure 1a shows the survey-mode magnetic spectrum measured by the WFR instrument.Figures 1b and 1c show the magnetic and electric spectra produced by Fast Fourier Transformation (FFT) of the EFW burst-mode waveforms.A window of 131,072 samples (8 s) and a shift window of 16,384 samples (1 s) were chosen for the FFT to optimize the frequency resolution (1/8 Hz) and temporal resolution.The EFW measurements show a harmonic structure above 19 f cp (equatorial proton gyrofrequency).The wave normal angle (Figure 1d) and the ellipticity (Figure 1e) calculated using the Means (1972) method show that those harmonic waves were approximately perpendicularly propagating and linearly polarized, consistent with the essential characteristics of magnetosonic waves.
The 20th-26th harmonics exhibit an internal fine structure, that is, each harmonic consists of a series of risingtone emissions that has not been reported previously, to the best of our knowledge.These emissions grow from frequencies close to multiples of f cp , and vanish before their frequencies rise to a higher harmonic.Each emission typically lasts for 30-70 s.The repetition period is about 18 s and does not vary over different harmonics.The temporal resolution of proton distribution measurements is 22 s, and is unable to resolve whether the proton distribution exhibits a periodic pattern.The frequency sweep rate is about 4 Hz/min and varies slightly over time.
The mechanism proposed by Sun et al. (2020) for compound rising-tone waves cannot explain the elementary magnetosonic waves, because the rising-tone structure is observed over all harmonics simultaneously.The discrete rising-tone emissions observed in each harmonic are somewhat analogous to the rising-tone emissions that make up chorus waves (Burtis & Helliwell, 1969).The excitation of chorus waves was believed to be associated with nonlinear interactions while different mechanisms have been proposed (e.g., Fu, Cao, Cully, et al., 2014;Omura et al., 2008;Tao et al., 2021).The rising-tone emissions in each magnetosonic wave harmonic imply a nonlinear excitation mechanism of those magnetosonic waves.
While the nonlinear excitation of these elementary rising-tone magnetosonic waves will be a future study, we apply the linear theory of wave growth for a basic understanding of their generation and spectral range.Figure 1f shows the perpendicular proton phase space density (PSD) derived from proton differential fluxes.Figure 2a shows the detailed perpendicular and parallel PSDs averaged over 02:45-02:50 UT, showing a ring distribution around 10-30 keV in both parallel and perpendicular directions.Assuming a nominal pitch angle range of [18°, 90°], the bounce periods of these ring protons (Lyons & Williams, 1984) are in the range of 33-87 s, which is very close to the durations of the elementary rising-tone emissions, but is significantly longer than their repetition period.
We use a combination of Gaussian and Kappa distributions to fit the proton PSD averaged from 02:45 UT to 02:50 UT.The averaged density provided by the EFW is 15 cm 3 , which is obtained by fitting the spacecraft potential to the density derived from the upper hybrid resonant frequency (Jahn et al., 2020;Kurth et al., 2015).Figure 2b shows the calculated linear wave growth rates (Kennel, 1966), demonstrating that magnetosonic waves can be readily generated at frequencies above 18 f cp .The observation shows rising-tone emissions generally between n f cp and ∼(n + 1/2) f cp , especially for harmonics from 20 f cp to 30 f cp .In contrast, the growth rates in Figure 2b show wave generation at frequencies centered around multiples of f cp .The linear growth theory provides an explanation on the initial excitation of the wave harmonics, while the exact frequency range of each harmonic at saturation amplitudes is not well captured, and a nonlinear mechanism for the rising-tone structure remains a topic for future studies.
Figure 3a shows another elementary rising-tone magnetosonic wave on 20 May 2012, in which the rising-tone feature is observed over a broad frequency range, from 4 f cp to over f LHR .These waves are perpendicularly propagating (Figure 3b), but they are right-hand polarized (Figure 3c), unlike most of magnetosonic waves.The proton ring distribution around 12 keV (Figure 3d) possibly supports the generation of these waves.These waves form a few series, and each series consists of elementary ring-tone emissions around multiples of f cp which last for around 1 min.The frequency sweep rate is larger at higher frequencies, and is approximately proportional to their frequency.For instance, the waves observed between 15:14 UT and 15:15 UT show a frequency sweep rate of ∼0.34 Hz/min at 8 f cp , while it exhibits a sweep rate of ∼0.80 Hz/min at 16 f cp , and ∼1.63 Hz/min at 32 f cp .The linearly increasing sweep rate is probably not associated with nonlinear harmonic effect (e.g., Huang et al., 2020), because the wave of 31 f cp and other prime-number harmonics is obviously not the harmonic of waves at lower frequencies.Besides, the waves at 14 f cp are more intense than the waves at 7 f cp , and are not harmonics of waves at 7 f cp .The mechanism that leads to a sweep rate that increases with wave frequency is yet to be understood.We note that these waves are not compound rising-tone waves because the harmonics of each series are observed almost simultaneously.
In addition to elementary rising-tone magnetosonic waves, the EFW instrument onboard the Van Allen Probe also recorded elementary falling-tone magnetosonic waves.Figure 4 shows Van Allen Probe A observations on 20 August 2017.A series of falling tone magnetosonic waves is observed from the 14th harmonic to the 29th harmonic.Between the 16th and the 25th harmonic, magnetosonic waves exhibit a periodic compound rising-tone structure.We note without delving into details that the falling tone sweep rate in this case is likely proportional to the harmonic number, like that in the rising-tone waves shown in Figure 3.The observation of falling-tone magnetosonic waves adds another interesting similarity between the morphologies of chorus waves and magnetosonic waves, providing additional insights into the waveparticle interaction.

Compound Rising-Tone Magnetosonic Waves Consisting of
With the high resolution EFW data, we are able to investigate whether compound rising-tone magnetosonic waves can intrinsically be decomposed into multiple harmonics.Figure 5a shows the WFR survey-mode magnetic spectra measured on 31 August 2013.Periodic rising-tone magnetosonic waves were observed between 11:12 UT and 11:25 UT.The EFW burst-mode wave magnetic and electric spectra, shown in Figures 5b  and 5c, clearly indicate that each of those rising-tone emissions consists of multiple harmonics.The compound rising-tone magnetosonic waves are perpendicularly propagating and linearly polarized (Figures 5d and 5e).They are observed at multiples of f cp , suggesting that those magnetosonic waves are likely locally generated.A proton ring structure in PSD is seen just above the Alfven energy (Figure 5f), consistent with a local generation mechanism.
We note that the waveform sampling rate for this case is 1,024 samples/s, much lower than the case shown in Figure 1.Hence, the data is unable to reveal fine structures inside each harmonic if any.The periodicity is about 1 min, and each group lasts for about 1.6 min, close to the bounce period of ring protons that excite those waves.While these compound rising-tone magnetosonic waves have similar temporal scales to elementary rising-tone waves, their sweep rate is about 70 Hz/min, much higher than that of elementary ones.Our detailed study shows that these periodic occurring of compounding rising-tone waves are not modulated by magnetic field, plasma density, or energetic proton distribution, consistent with previous studies (e.g., Boardsen et al., 2014;Fu, Cao, Zhima, et al., 2014;Němec et al., 2020).
Figure 6 shows a different event recorded by Van Allen Probe A on 12 December 2012.Again, we show WFR survey-mode spectra (Figure 6a) and EFW burst-mode spectra (Figures 6b and 6c) for a comparison.The wave normal angle (Figure 6d) and the ellipticity (Figure 6e) indicate that all the waves shown in this figure are magnetosonic waves.The EFW sampling rate is 512 samples/s for this case and is not sufficient to resolve the possible elementary rising-tone structure.
Rising-tone magnetosonic waves are seen above 120 Hz.They show a clear harmonic structure from 13:30 UT to 13:50 with a frequency gap between adjacent harmonics of about 8 Hz, which slowly increases over time.Assuming a dipole background magnetic field, the estimated source region of those waves, based on generation at harmonics of the local f cp , is at L = 3.9.The harmonic structure was clear when the satellite was at L = 3.7-4.2,possibly close to the source.At earlier times, as the satellite approached large L shells, the harmonics gradually vanished, the wave intensities became weaker, and the spectra became continuous over frequency.One possibility could be that the waves were generated with the same periodicity in the source region that spanned a certain area.At locations farther from the source, waves from a broad source region mix together, resulting in a continuous rising-tone spectrum.Another possible mechanism could be the spectral broadening effect due to self-phase modulation.This effect occurs when a short pulse travels through a medium with varying reflective index, which produce a phase shift, leading to the change of the pulse's spectrum (Stolen & Lin, 1978).The periodicity of these rising-tone magnetosonic waves is ∼1.4 min and is unchanged during the propagation, while the frequency sweep rate slightly increased from ∼160 Hz at L = 3.9-∼180 Hz/min at L = 4.6.

Conclusions
This paper uncovers the elementary rising-tone structure of magnetosonic waves in Earth's magnetosphere, which is only plausible using the recently released burst-mode waveform measurements by the instrument onboard Van Allen Probes.We summarize the new findings as follows.
1. Magnetosonic waves have fine structures within the harmonics, that is, each harmonic may consist of a series of elementary rising-tone emissions.This suggests a nonlinear excitation of magnetosonic waves.2. A case study shows that the frequency sweep rate of the elementary rising-tone magnetosonic wave is likely proportional to the wave frequency.3. Elementary falling-tone magnetosonic waves are also recorded.The periodicity is close to that of elementary rising-tone waves.4. Compound rising-tone magnetosonic waves typically exhibit harmonic structures in the source region, and may gradually develop a continuous spectrum over frequency as they propagate further away from the source.The elementary and compound rising-tone magnetosonic waves have similar durations of about 1 min, which is close to the bounce period of the ring protons.The relation between the two rising-tone structures is yet to be understood.
The fine structures we uncover in this paper provide new insights into the complex nature of magnetosonic waves and their generation mechanism as well as their impact on ring current ions and electrons.In the future, we plan to carry out a statistical study with all available EFW data to explore under what favorable conditions and in what regions elementary rising-tone MS waves are typically observed.

Figure 1 .
Figure 1.Van Allen Probe measurements during 02:30-02:52 UT on 2 December 2015.(a) The survey-mode magnetic spectrum measured by the WFR instrument (b)-(c) The magnetic and electric spectra derived from the EFW burst-mode waveforms by FFT.(d) The wave normal angle and (e) the ellipticity of waves calculated by Means (1972) method.(f) The perpendicular PSD of protons and the Alfven energy E A .

Figure 2 .
Figure 2. Wave linear growth calculation.(a) Proton PSD distribution measured by the HOPE instrument.(b) The calculated linear growth rate resulting from the interpolated proton PSD.

Figure 3 .
Figure 3. Elementary rising-tone magnetosonic waves observed on 20 May 2013.(a) The wave magnetic intensity.(b) The wave normal angle and (c) the ellipticity.(d) The perpendicular PSD of protons.

Figure 5 .
Figure 5.The same as in Figure 1 but showing compound rising-tone magnetosonic waves observed on 31 August 2013.

Figure 6 .
Figure 6.The same as in Figure 5, showing an observation of non-local magnetosonic waves measured on 17 December 2012.