Trapping and Amplification of Unguided Mode EMIC Waves in the Radiation Belt

Electromagnetic ion cyclotron (EMIC) waves can cause the scattering loss of the relativistic electrons in the radiation belt. They can be classified into the guided mode and the unguided mode, according to wave's propagation behavior. The guided mode waves have been widely investigated in the radiation belt, but the observation of the unguided mode waves have not been expected. Based on the observations of Van Allen Probes, we demonstrate for the first time the existence of the intense unguided L‐mode EMIC waves in the radiation belt according to the polarization characteristics. Growth rate analyses indicate that the hot protons with energies of a few hundred keV may provide the free energy for wave growth. The reflection interface formed by the spatial locations of local helium cutoff frequencies can be nearly parallel to the equatorial plane when the proton abundance ratio decreases sharply with L ‐shell. This structure combined with hot protons may lead to the trapping and significant amplification of the unguided mode waves. These results may help to understand the nature of EMIC waves and their dynamics in the radiation belt.

vectors should quickly turn oblique where the refractive indexes are larger, as a consequence of Snell's law, as demonstrated by simulations  and observations . In the vicinity of plasmapause, the tendency of turning oblique can be counteracted by negative density gradient , which facilitates the wave growth (Chen et al., 2009;de Soria-Santacruz et al., 2013), although the effect is limited (Fraser & Nguyen, 2001;Halford et al., 2015;Tetrick et al., 2017). At approximately 10 20    , poleward waves should approach the local bi-ion frequencies (Rauch & Roux, 1982), where the wave energy could be reflected , absorbed by heavy ions (Horne & Throne, 1997; or transmitted (Hu et al., 2010), depending on the abundance ratios of heavy ions (Hu et al., 2010).
Refractive index distribution is approximately isotropic for unguided mode waves (Rauch & Roux, 1982), therefore the group velocity should follow the wave vector and not be guided by the magnetic field. As the wave vector and Poynting flux could turn oblique very quickly, the unguided mode waves are likely to propagate to larger L-shell rapidly . Kim and Johnson (2016) performed 2-D full-wave simulation and found that the unguided waves propagate to the stronger magnetic field can be reflected at cutoff frequency and propagate to the outer magnetosphere. Moreover, as the wavenumber of the unguided mode is relatively larger than that of the guided mode ( Figure S1), the resonant energy may exceed the peak energy of the substorm injected hot H  , which does not favor the wave generation. Therefore, it is supposed that the inner magnetosphere does not support the amplification of intense unguided mode EMIC waves in general . Yuan et al. (2019) reported a He  band wave event associated with density variation, wave mode analyze suggesting that the waves belong to the guided mode. EMIC waves with quasi-constant frequency crossing a range of L-shell have been reported previously (e.g., Paulson et al., 2014Paulson et al., , 2017Tetrick et al., 2017;G. Wang et al., 2019, and others). One explanation for the events in G. Wang et al. (2019) is that the waves emitted from the sources may pass through the crossover frequency and bi-ion frequency, turning from the guided L-mode waves to the unguided R-mode waves (Hu et al., 2010;G. Wang et al., 2019). Such a process can also explain the origin of the R-mode waves observed on the ground (Feygin et al., 2007).
Up to now, there is a lack of observation that confirm the existence of the unguided L-mode EMIC waves (or the Class-II waves, as shown in Figure S1) according to the polarization characteristics. However, the hot H  with hundreds of keV can inject into the ring current during storms (Zhao et al., 2015), theoretically providing the energy source for the generation of unguided mode waves. Moreover, the smaller group velocities of unguided mode waves may lead to the larger convective growth rates, which can favor the wave generation (Horne & Thorne, 1994). The motivation of the present study is based on the following questions: Can we find the distinguishable unguided mode EMIC waves that are locally generated below cH f  ? If so, what's the difference in the generation mechanism between the guided and unguided mode waves? Here we identify the existence of the intense unguided mode EMIC waves in the earth's radiation belt by presenting Van Allen Probe (Mauk et al., 2013) observations. We will propose that the generation of these waves is associated with a new type of wave trapping mechanism, which is controlled by the spatial variation of ion abundance ratios.

Extraction of Wave Properties
The magnetic field data from MAG, and the electric field data from EFW are used to analyze the wave properties. The 64 Hz magnetic field data are processed through a 1024-point fast Fourier transform (FFT, without detrending) to obtain the magnetic spectral matrix. Singular value decomposition (SVD) analysis (Santolík et al., 2003) is performed on the magnetic spectral matrix to estimate the wave normal angle, azimuthal angle, and wave polarization (Santolík et al., 2003). The 32 Hz two-dimensional electric field data from EFW are used to estimate the vector electric field, and the spin axis-aligned component is estimated with the assumption that 0   E B . The error may be large when z B is small, which is not the case for the parallel waves measured by Van Allen Probes as their spin axes are approximately in the equatorial plane (Mauk et al., 2013). A 512-point FFT is carried out for both the electric field and the magnetic field WANG ET AL. 10.1029/2021JA029322 2 of 18 (resampled to 32 Hz) to obtain the cross-power spectra, and thus, we obtain the Poynting vector (Santolík et al., 2010). The technical details are the same as G. Wang et al. (2019).

Plasma Conditions for Calculating Wave Growth Rate
The linear growth rate described in the appendix can be estimated based on observations. The derivatives of the phase space density (F j p  / 2 ) with respect to the energy E and pitch angle  for each ion species can be obtained based on observation of the Helium, Oxygen, Proton, and Electron (HOPE) mass spectrometer (Funsten et al., 2013) and the Radiation Belt Storm Probes Ion Composition Experiment (RBSPICE) instrument (Mitchell et al., 2013). As the major ions H  dominate the wave growth, here the phase space density F is estimated approximately from the H  flux collected by HOPE and RBSPICE. HOPE measures the H  flux ( H j  ) with the energy from several eV to 50 keV, while the RBSPICE measures H j  from 50 keV to  600 keV. These two energy ranges cover the main structures of the ring current H  and are sufficient to evaluate the linear instability. The HOPE flux is multiplied by a factor of three so that the mismatch between the low-energy part of RBSPICE and the high-energy part of HOPE can be eliminated Min et al., 2017). The distribution of ( , ) F E  is fitted along pitch angle  for each fixed energy channel, and the required derivatives are then approximated with the help of B-spline interpolation. The details regarding this technique follow (Liu et al., 2018).The zero-order (0) D and first-order (1) D of the dispersion matrix are also depend on the background magnetic field, the plasma number density e n and the ion abundance ratios  s s e n n  / (where s denotes the particle species) besides phase space density F (Kennel, 1966). The magnetic field is measured by the triaxial fluxgate magnetometer (MAG), the local plasma density is measured by the High Frequency Receiver (HFR) (Kurth et al., 2015) of the Electric and Magnetic Field Instrument and Integrated Science (EMFISIS) suite (Kletzing et al., 2013). The ion abundance ratios are determined via the plasma diagnostics as described in the appendix.

Overview of the Event
As shown in Figure 1a, the event was observed by Probe A on December 15, 2015, during the recovery phase of a magnetic storm, after at least three successive prominent substorms where AE  500 nT. In Figure 1b-c, the hot H  fluxes enhance in four distinct energy ranges: below 10 keV, 10  30 keV, 30  100 keV and 100  600 keV. The peak energy of the hot H  for each energy range decreases with L-shell, indicating the energy-dispersive substorm injected ions (Zhao et al., 2015). From Figure 1d, the satellite passed through the plasmapause at 2.87 L  , and the background electron number density e n measured by HFR dropped from 3 100 cm   at just outside the plasmapause to 3 20 cm   at 3.7 L  . The conspicuous H  band waves appear from 3.1 L  to 3.7 L  , as exhibited in Figure 1e. These waves with quasi-constant frequencies appear to be intense just outside the plasmapause and decay gradually with the increasing L-shell.

Wave Analyses and Ion Abundance Ratios
The wave properties are presented in Figure 2. In general, the time series of the wave spectra can be divided into three distinct regions based on the observational characteristics, as marked by the blue boxes. The Region A waves appear at MLAT ~−1.5, have the strongest magnetic spectral density PSD B , associated with the quasi-parallel wave normal angle  and small compressional ratio PSD B /PSD B , indicating that they were propagating roughly along the ambient magnetic field. Furthermore, the general direction (shown by the diamonds, which are the spectrum-weighted averages) of the Poynting fluxes appears to be northward for the inner waves, and be southward for the outer waves (Figure 2f), and almost all these intense waves possess a slight inward orientation (Figure 2e). The results of the ellipticity B  show predominately left-hand polarization, indicating that the waves belong to the L-mode. In Region B, the waves show a decrease in PSD B of approximately one order. Noticeably, the B  spectra show a sharp boundary which separates the left-hand and right-hand polarization, indicating the pass-through of the He  crossover frequency crHe f  . Independent of the polarization, the waves with different values of the wave normal angle or compressional ratio are highly mixed. The outermost waves are presented in Region C, with a wave power 2 orders WANG ET AL.

10.1029/2021JA029322
3 of 18 of magnitude smaller than that of the waves in Region A. The wave vectors are almost perpendicular to the ambient field, the compressional ratios are relatively large, and the polarizations are predominantly right-handed, indicating these are the oblique R-mode waves.
With increasing L-shell value, the relative frequency f / cH f  increases for the observed H  band waves with a quasi-constant frequency, and the polarization mode thus changes from L-mode to R-mode. Such a phenomenon indicates that the waves belong to the unguided mode (Rauch & Roux, 1982), as illustrated in Figure 2j. Here, we perform the plasma diagnostics to verify the wave mode and estimate the ion abundance ratios s  . As shown by Figures 2c and 2a     for the given ion abundance ratios. The colored solid and dashed lines trace the local crossover frequencies crHe f  and cutoff frequencies cutHe f  . (j) Dispersion relations under the observed plasma conditions. The red curve represents the observed wave mode. The letter "L" and "R" denote the left-hand and right-hand polarizations, respectively. in these inner L-shell under the assumption of H  ~ 83% and He    17%. Considering that the background plasma density decreases prominently with L in the range where the wave exists (from 100 3 cm  at L  3.1 to 50 3 cm  at L  3.3 according to Figure 1d), one can assume that s  may also vary with L, leading to the rapid change of cutoff frequencies. Assuming that the cutoff frequencies lie just below the lowest frequencies of the observed intense waves (the yellow dashed curve in Figures 2c and 2h), a new group of s  can be obtained by numerically solving Equation A9: 92% H  and 8% He  . As shown in Figure 2h, the * B  spectra contain the lower-frequency waves observed before 03:20 UT under this group of ion abundance ratios. In addition, a group of s  is provided in Figure 2i for comparison, where both the cutoff frequency and the polarization distribution deviate from the observations significantly. These results confirm that the polarization reversal boundary is the time series of crHe f  in the unguided mode, and suggest that the H  abundance decreases from at least 92% at where the polarization reversal takes place.
The field-aligned and anti-field-aligned Poynting fluxes and the strong wave power, strongly indicate that there are overlaps between Region A and the wave source region (Allen et al., 2013;Loto'Aniu et al., 2005;Vines et al., 2019). The waves in Region B and Region C can be the leaked waves from the source, or the locally excited waves with oblique wave normal angles. We will explore the origin of these waves in the following section.

Linear Growth Analyses
To reveal the generation mechanism of the observed unguided mode waves, we first present the results of the linear convective growth rate in Figure  (7 e-foldings), to satisfy the growth of the observed intense waves from the background noise level. This corresponds to an integrated traveling distance in the source adding up to 11 E R .

Potential Wave Trapping
The waves within the inner L-shell are intense and predominated by field-aligned wave vectors and Poynting fluxes but become weaker and more oblique outside. Moreover, the waves observed in all these regions have almost a constant frequency range. This evidence strongly implies that the waves are generated inside and propagated outward. The growth rate results further suggest this assumption by eliminating the possibility of oblique excitation for the outer right-hand waves. However, the most challenging problem is that the convective growth rate is too small for the observed strong waves generated in any potential source region with a limited spatial range. According to the previous analysis, an integrated traveling distance of ~11 E R in the source region is needed for the full growth of the parallel waves. It is reasonable to associate the convective growth process with some kind of wave trapping mechanism. As the hot ions are circling around the earth in the narrow range of the L-shell, the azimuthal propagation of the waves may provide the required integrated distance in one-half of the drift circle.
with the help of the observations. As demonstrated in Figure 4a-b, in a plasma environment with constant s  , the reflection interface where the refractive indexes RI decrease to zero is approximately perpendicular to the equatorial plane, because the value of cutHe f  / cH f  remains the same while the value of cutHe f  decreases with radial distance but increases with latitude due to the variation of the magnetic field. As shown by Kim and Johnson (2016), the unguided waves will be reflected at G  along the resonant surface as shown in (e). Here  is the pitch angle, v  is the perpendicular velocity of H  , and * 1H G  is the integrating factor in the growth rate expression.
reflection interfaces in both sides of the equatorial plane may produce a trapping zone, within which some of the unguided mode waves will be trapped and experience the significant amplification to the observed intensity. In addition, as shown in Figure 2b, the plasma number density experiences a sharp change at approximately 3.48 L  , which may result in a dramatic variation in s  with L, and thus form a quasi-parallel or even negatively inclined reflection interface at the radial distance E 3.2 3.3 R   . This is consistent with the strongest waves observed at 3.2 3.3 L   near the equator. Moreover, both the L-mode waves and the R-mode waves in Region B possess almost the same magnitude of wave power (Figure 2), suggesting that they are the leaked waves from the inner trapping region (polarization reversal process does not influence the wave power). Following such a scenario, one can predict that the waves observed away from the trapping region should propagate in the direction pointing from the trapping region to the observed position, which can explain the quasi-perpendicular waves observed in Region C.

Reliability of Linear Theory for Explaining Observations
The estimated values of linear growth rates are too small to explain the observed wave intensity without the wave trapping. Blum et al. (2009) developed the two-parameter proxy of EMIC growth based on linear theory (Gary, 1993), and found an association between EMIC growth and relativistic electron loss during storms. The linear growth proxy can well predict the statistical wave enhancement regions (Allen et al., 2016) as well as the specific wave events (J.-C. Zhang et al., 2014). However, Saikin et al. (2018) found that the calculated wave amplitudes are too low compared to the observation. As far as our information goes, the following reasons may be responsible for the mismatches between calculations and observations: nonlinear effects (Omura et al., 2010); inaccurate plasma conditions, for example, the bi-Maxwellian distribution assumption, the assumed ion abundance ratios, the model predicted electron density, and the neglecting of the hot ions of a few hundred keV (  The coordinate system is established in the Earth's dipole field: the horizontal axis eq r represents the radial distance to the dipole axis in the equatorial plane, while the z axis is along the dipole axis. Solid black curves trace the background magnetic field estimated with the help of the T89 model. convective growth (Thorne & Horne, 2007;Khazanov et al., 2007); the marginally stable assumption (Gary & Lee, 1994;Yue et al., 2019;. Nonlinear effects tend to suppress wave amplitude for the intense waves (Min et al., 2015;Omura et al., 2008), thus are not likely to lead to the underestimation of wave growth in the linear approach. In the present calculations, the ion distributions are based on observations, the assumed ion abundance ratios are derived from observed crossover frequency and cutoff frequency, and the hot ions of a few hundred keV are considered, therefore, all the plasma condition parameters in the linear growth rate are more realistic.
For the guided mode, the wave behaviors near bi-ion frequency are complex, and the proportions of energies that experience absorption, transmission, or reflection are highly dependent on the ion abundance ratios (Horne & Throne, 1997;Hu et al., 2010;. Therefore, the ambiguity of ion abundance ratios may lead to the improper estimation of the integrated wave gain. In addition, the present study indicates that the spatial variation of ion abundance ratios may also influence the propagation of guided waves near the bi-ion frequency, which may be a candidate, besides the oblique excitation by heavy ions (Denton et al., 1992;Gamayunov et al., 2018;Hu & Denton, 2009;Hu et al., 2010), to explain the origin of previously reported oblique linear or right-hand EMIC waves in the inner plasmasphere Grison et al., 2016;Min et al., 2012;Saikin et al., 2015;G. Wang et al., 2019) or away from general peak occurrence regions (Wang, Huang, et al., 2017). Unlike the guided mode, the convective growth for the unguided mode is relatively simple, because the refractive index is approximately independent of the wave normal angle, which allows us to predict the wave behavior solely based on the global plasma conditions. Furthermore, as the wave vectors are approximately parallel to the magnetic field for the rays with parallel group velocity, Landau damping should not be prominent (Thorne & Horne, 1992) (Landau damping rate should be zero for parallel k [Kennel, 1966]); and as the wavenumbers are small near cutoff frequency, cyclotron damping by heavy ions should also be insignificant . Yue et al. (2019) demonstrated an upper limit of growth rate for EMIC waves based on the observed ion distributions, indicating the importance of local growth, that is, the disturbance intensity of the wave source may be dependent on the time integration. This scenario of the local growth of wave source makes the trapping and convective growth "unnecessary," for the explanation of the observed wave intensity. The problem that either the local growth or the convective growth determines the observed wave amplitude in the source region cannot be exactly solved based on the present observations, however, we can still do some exploration. According to X. Yu and Yuan , it typically needs a period of tens of minutes for a wave source to reach the saturation state without injection of free energy. Considering the western drift of hot ions with a speed of several km/s (for 10 keV), for the typical EMIC event with a scale size of 0.5L  (Blum et al., 2017), the time scale of the refresh of anisotropy should be comparable with that of the saturation. In the present event recorded by Probe A on December 15, 2015, considering that the drift speed of 200 keV protons at 3.2 L  is approximately 30 km/s, and the calculated growth rates are relatively small, the observed hot proton distributions are more likely to be in the "fresh" state. From Figure 3, the values of the calculated convective growth rates are in the same order of magnitude in the selected time-frequency region, does not match exactly the variation of the observed wave intensity, further indicating that the anisotropy is not significantly relaxed within the source region (otherwise the calculated growth rate should be relatively smaller in the region of intense waves) (Gary & Lee, 1994;. Furthermore, the growth rates of the guided mode waves (please see Figure S3) are in the same order of magnitude as that of the unguided mode waves (Figure 3). As the guided mode waves should experience imperfect reflection and wave vector obliquity, which may substantially reduce the repeated convective amplification Hu et al., 2010;Loto'Aniu et al., 2005), if the actual generation process of the observed unguided mode waves is also not significantly influenced by the reflection and convective amplification, the guided mode waves are supposed to be observed with the same intensity of the unguided mode waves. Considering that only the unguided mode waves are observed, and that the intense waves are just located within the predicted trapping region, the process of trapping and convective amplification may be important to facilitate the wave generation. WANG ET AL.

Spatial Inhomogeneity of Ion Abundance Ratios
The spatial distributions of the ion abundance ratios s  in the inner magnetosphere have considerable uncertainty. The statistics using Dynamics Explorer-1 (DE-1) showed that the He H /     ratio decreases with L below 2 L  but increases with L over 2 5 L   (Goldstein et al., 2019). The magnetoseismology study by Takahashi et al. (Takahashi et al., 2006) revealed that the average ion mass during the quiet period is  2.0  3.0 outside the plasmasphere (identified by e n larger than 100 3 cm  ) but is lower than 2.0 inside the plasmasphere. Therefore, there is a general trend that the proton abundance increases from outside the plasmasphere to just inside the plasmasphere. In the event recorded by Probe A on December 15, 2015, the value of H   decreases from 92% at just outside the plasmapause ( according to Huba and Krall (2013) are approximately 5 %  10% in the outer plasmasphere, corresponding to the present estimated values near the plasmapause. Therefore, the present results are consistent with the previous studies that the substantial spatial variation of ion abundance ratios may exist in the vicinity of the plasmapause. In addition, the O  abundance ratio O   has been neglected in the main analyses (based on Appendix A). In a more reality case of 5 O %  (Bashir & Ilie, 2018;Miyoshi et al., 2019), as demonstrated in Figure S4, the trend of decreases of H   (from 89% to 82%), the peak values of linear growth rate, and the predicted trapping region have no substantial differences with the corresponding results neglecting O  .

How Prevalent are the Unguided Mode EMIC Waves?
During the period from September 2013 to May 2018, another obvious event of unguided mode waves with a distinguishable crossover frequency was recorded by Probe A on November 30, 2015, when the satellite was outside the plasmasphere (please see Figure S5). A similar trend of the spatial variation of ion abundance ratios can be obtained based on plasma diagnostics ( Figure S6), indicating the potential existence of wave trapping. The intense L-mode waves are observed along with the enhancement of hot H  with the energies of hundreds of keV, while the positive convective growth rates are obtained ( Figure S7). The magnitude of largest growth rates is approximately several times of 7 1 10 m   , which is inadequate for the waves to be amplified to the observed magnitude (more than 7 e-foldings) in an integrated distance less than 1 E R . As the waves are observed near the equatorial plane, one can thus predict that the wave trapping mechanism may have played an important role in the amplification of the observed waves.
The event in the previous section was recorded by Probe A. In fact, Probe B also recorded a wave event in the same area ~ 2 h earlier. As shown in Figure 5, the intense H  band waves were observed at 3.16, 8.5 L MLT   and 1.2 MLAT   , which were within exactly the same area of Probe-A-event. Comparing Figures 1 and 5, the background plasma conditions are also similar for these two probes during their respective passes, that is, waves were observed just outside the plasmasphere, with the enhancement of hot protons of hundreds of keV. In the event of Probe B, a weak "tail" toward the higher L-shell and larger value of f f / cH can also be observed. This "tail" can be explained by the outward propagation of the unguided mode waves. In this event, however, no obvious boundary of R-mode and L-mode can be distinguished, as shown in Figure 6. Therefore, we cannot determine the mode of these waves based on the observations. It is possible that the configuration of the potential trapping region restricts the intense waves below crossover frequencies. However, as the plasma conditions are the same as that of Probe-A-event, and the weak 'tail' may indicate the propagation toward higher L shell, it is reasonable to suppose that these waves are the unguided mode waves, being trapped and amplified with the same mechanism as the waves in Probe-A-event ( Figures S8  and S9 show the growth rates). If so, as these waves do not have an obvious right-hand polarization, they cannot be distinguished from the guided mode waves, maybe they are common in the radiation belt and are intermixed with the guided mode waves in the previous studies. The exact measurements of ion abundance ratios are needed for identification of these different modes, in the cases with the crossover frequencies unidentifiable.
WANG ET AL.

Summary
From previous studies, EMIC waves can scatter relativistic electrons (e.g., Blum et al., 2015;Cao et al., 2017;Miyoshi et al., 2008;Ni et al., 2015;Omura & Zhao, 2013;Rodger et al., 2015;Su et al., 2017;Usanova et al., 2014;Z. Wang et al., 2014;Wang, Su, et al., 2017;X. J. Zhang et al., 2016;Zhu et al., 2020, and others), lead to the decay of ring current ions (Daglis et al., 1999;Fuselier et al., 2004;Su et al., 2014), or heat the cold electrons Yuan et al., 2014). The present study clearly demonstrates the existence of unguided mode EMIC waves in the radiation belt, and propose a trapping and amplification mechanism for explaining their origin. The results are summarized here: plasmapause. With increases in the L-shell and the relative wave frequency f / cH f  , the wave vectors became oblique, and an obvious boundary where the wave polarization reversed from left-handed to right-handed was observed. Plasma diagnostics verified that the waves belonged to the unguided mode, and the proton abundance ratio may decrease with the L-shell. 2. Weak linear growth rates contributed by hot protons with a few hundred keV were obtained in the regions of intense parallel L-mode waves, where a source-integrated distance of approximately 11 E R is needed for the convective growth of the L-mode waves to the observed wave power. 3. The reflection boundary formed by the spatial distribution of the local helium cutoff frequencies was investigated. Under the condition of a sharp decrease in the proton abundance ratio with increasing L -shell, the reflection boundary can be nearly parallel to the equatorial plane. As the unguided L-mode WANG ET AL.
10.1029/2021JA029322 12 of 18 waves should be reflected at the reflection boundary, this structure combined with hot protons of a few hundred keV may serve as an amplifier for the generation of the observed class-II waves.
These results may have potential importance for understanding the dynamics of the Earth's radiation belt.

Appendix A: Plasma Diagnostics
In the cold plasma, the following Stix parameters are helpful to investigate the dispersion relation (Stix, 1992): Here,   / are respectively the plasma frequency and the gyrofrequency of a particle species s.
The crossover frequency cr f and the cutoff frequency cut f can connect the observations to the dispersion relation.
cr f is the frequency where the L-mode and R-mode phase velocities are equal (Smith & Brice, 1964) and can be obtained by setting the Stix parameter L equal to R: Therefore, if the three-ion (H  , He  and O  ) plasma are approximated as two-ion (H  and He  ) plasma, that is, the terms with oxygen plasma frequency pO   are dropped in Equation A1, the Stix parameters will have only a negligible change (example in Figure S2). Under such an approximation, a group of ion abundance ratios ( H   and He   ) can be obtained by substituting the observed value of  crHe . Consequently, the approximated dispersion relations for the modes in H  band can be found.
As the wave magnetic field vector is always perpendicular to the wave vector (i.e., 0   B k ) for a plane electromagnetic wave, it is reasonable to define the polarization state in terms of the wave magnetic field, which is perpendicular to k. The polarization state should include two aspects: rotation sense and polarization ellipse. As the sense of wavefield rotation about the ambient magnetic field 0 B remains the same for a certain polarization mode, it is convenient to define the rotation sense in terms of 0 B . As the wave magnetic field vector lies in the plane perpendicular to k, it will convey more complete information by calculating the polarization ellipse in k-centric frame. Based on these considerations, the polarization state can be represented WANG ET AL.
x y B R L P n B P R L n where  is the wave normal angle. The field amplitude can be transformed into the coordinate system with the z-axis along k: the theoretically predicted ellipticity * B  is then obtained considering that x B  and y B  are axes of the polarization ellipse.
In summary, the ion abundance ratios are estimated as follows: first, estimate the potential crossover frequency from the observed polarization of H  band waves; then, estimate the H  and He  abundance ratios by Equation A1; finally, compare the observed ellipticity B  with the calculated theoretical ellipticity * B  for verification.
The cutoff frequency cut f is the frequency where the phase velocity equals to zero (Smith & Brice, 1964), and can be obtained by setting the Stix parameter L equal to zero: Theoretically, the vanishing of waves below the lowest frequency of the observed wave power spectrum (denoted as min f ) may be simply a result of a decrease of wave growth, which is due to the hot ion distribution and resonant conditions rather than the effect of the cutoff frequency. Therefore, the lower limit of the H  abundance ratio can be estimated by inserting the relation min cut f f  into Equation A9.

Appendix B: The Convective Growth Rate for Unguided Mode Around H  Gyrofrequency
For a given ion phase space density distribution ( , ) F v v   , the growth rate depends on the integration of the terms containing   Kennel et al. (Kennel, 1966), here, we derive a simplified form of the convective growth rate for the unguided mode near H  gyrofrequency to better demonstrate how the ( , ) F v v   distribution leads to wave growth. In the environment of the radiation belt, the Stix parameter P dominates L, R and refractive index I R . One can thus simplify the imaginary part of the first order of the dispersion function: Following (Chen et al., 2010), express the magnitude of convective growth rate i k as It can be proven that the product of the factors outside the integral is greater than zero, therefore, the values of * 1s G represent the contribution to wave growth by hot ions with a given energy range.