Reﬂection of low-frequency fast magnetosonic waves at the local two-ion cutoff frequency: Observation in the plasmasphere

. We investigate the reﬂection of low-harmonic fast magnetosonic (MS) waves at the local two-ion cutoff frequency ( f cutHe + ). By comparing the wave signals of the two Van Allen Probes, a distinct boundary where wave energies cannot penetrate inward are found in time-frequency domain. The boundary is identiﬁed as the time series of local f cutHe + . For a certain frequency, there exists a spatial interface formed by f cutHe + , where the incident waves should be reﬂected. The waves with small incident angles are more likely to penetrate the thin layer where the group velocity reduces signiﬁcantly, and being 5 trapped (cid:58)(cid:58)(cid:58) then (cid:58)(cid:58)(cid:58)(cid:58) slow (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) down (cid:58) in a period of several to tens of seconds before the reﬂection process complete. The cutoff reﬂection scenario can explain the intense outward waves observed by Probe-A. These results of MS reﬂection at f cutHe + may help to predict the global distribution of MS waves and promote the understanding of wave-particle dynamics in the radiation belt.

and Thorne, 2010) or the bounce resonance (Shprits, 2016;Tao and Li, 2016). Similar mechanisms can also result in a parallel acceleration and can lead to the butterfly pitch angle distribution for MeV electrons (Xiao et al., 2015;Li et al., 2016b, a;Lei 20 et al., 2017;Yang et al., 2017).
Recent studies found that the frequency of the wave peak occurrence increases from approximately 2 f H + at 2 R E to 21 f H + at 5 R E (Boardsen et al., 2016). Previous observations have demonstrated the propagation of MS waves in the radiation belt (Santolík et al., 2002;Su et al., 2017). The high harmonics with a few hundred Hz that are generated in the plasmasphere, can penetrate into the low altitude of approximately 700 km (Gulelmi et al., 1975;Santolík et al., 2016). Additionally, Chen and 25 Thorne (2012) considered the perpendicular propagation in the central symmetrical medium and predicted the trapping of MS waves in the vicinity of the plasmapause, which has been supported by subsequent observations (Ma et al., 2014;Yuan et al., 2019). A systematic study by Posch et al. (2015) revealed a wide MLT distribution, both inside and outside the plasmapause.
Furthermore, Liu et al. (2018b) found that the multiple fine-scale of density irregularities where the WKB approximation is violated also effectively blocked MS propagation. Teng et al. (2019) further revealed a wide distribution of MS waves below 30 f H + , with a higher intensity in the high-density afternoon section.
Theoretically, if the background plasma consists of at least two ion components, then the R-X mode waves below f H + should approach the cutoff frequency, which is dependent on the ion abundance ratios and magnetic field, and is independent on either wave normal angle or plasma density (Smith and Brice, 1964;Stix, 1992). The refractive index and wavenumber are close to zero near the cutoff frequency, and thus the wave should be reflected as a consequence of Snell' law, as seen by 35 the low-frequency hiss observed . :::::::: Previous :::::: studies :::: have :::::::::::: demonstrated ::: the :::::::: reflection :: of :::: hiss ::::: waves : in the high latitude region (Gurnett and Burns, 1968;Santolík and Parrot, 1999;Chen et al., 2017). To date, there is a lack of evidence for the reflection of MS waves near the local two-ion cutoff frequency (or the helium cutoff frequency, referred to as f cutHe + ) in the outer radiation belt (or in the vicinity). We present direct evidence for such a reflection process in the current study.

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The data of two Van Allen Probes were used to analyze the wave behavior (Mauk et al., 2013). 64 Hz magnetic field data from the triaxial fluxgate magnetometer (MAG) of the Electric and Magnetic Field Instrument and Integrated Science (EMFISIS) suite (Kletzing et al., 2013) were processed through the fast Fourier transform (FFT, without detrending) and using the method of singular value decomposition (SVD) (Santolík et al., 2003), to obtain the wave normal angle and wave ellipticity. The 32 Hz electric field data from the Electric Field and Waves (EFW) instrument (Wygant et al., 2013) were used. As there are only two 45 available components, the third component was estimated based on E · B = 0. The FFT was performed on the electric field and magnetic field (resampled to 32 Hz) to obtain the cross-power spectra, and the Poynting vector was then obtained (Santolík et al., 2010).
The electron density n e which was derived from the upper hybrid frequency (Kurth et al., 2015) measured by EMFISIS HFR, the ambient magnetic field B 0 which was measured by EMFISIS MAG, and the proton (H + ) flux data measured by the 50 Helium, Oxygen, Proton, and Electron (HOPE) instrument (Funsten et al., 2013) and by the Radiation Belt Storm Probes Ion Composition Experiment (RBSPICE) instrument (Mitchell et al., 2013) were used for the calculation of growth rates.. The RBSPICE fluxes were divided by a factor of 3 to eliminate the mismatch with HOPE fluxes (Min et al., 2017;Kistler et al., 2016). Following the works of Kennel (1966) and Chen et al. (2010b), we can evaluate the wave temporal growth rate: 55 and the corresponding convective growth rate: Here, D (0) and D (1) are the real and imaginary parts of the dispersion relation (Chen et al., 2010b, a). D (1) s depends on the phase space density F and its deviation with respect to energy and pitch angle. The technique details follow Su et al. (2018); Liu et al. (2018a); Wang et al. (2019).

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Simulations were carried out based on a raytracing code, where the transformation of the coordinate systems followed Horne (1989), and the adaptive step size was derived by restricting the deviation between the locally calculated wavenumber and the integrated wavenumber. Two kinds of plasma conditions were used for comparison. The first condition combined the Plasma density in the Inner magnetosphere Neural network-based Empirical (PINE) density model (Zhelavskaya et al., 2016(Zhelavskaya et al., , 2017, and the TS05 magnetic field model (Tsyganenko and Sitnov, 2005). In the second condition, the observed n e and B 0 were simply replicated in the MLT to obtain the central symmetric two-dimensional distributions.
3 Event on 01 March 2017 :::::::::: Observation ::: of ::: the ::::: Wave ::::::::: Reflection 3.1 Observation of the Wave Reflection indicating that the two groups of waves belong to the first harmonics and second harmonics, respectively. The frequency of each patch increases slightly within half an hour, the time scale of which is significantly larger than the rising tone structure (Fu et al., 2014), indicating its association with the injection of the wave sources, rather than nonlinear generation. A sharp low 75 boundary can be found just above the helium gyrofrequency (f He + ), indicating the existence of the f cutHe + . From Figure 1b and 1f, the plasma density n e of approximately 2000 cm −3 indicates the immersion of probes inside the deep plasmasphere.
4q : e) is larger than that without reflection (Figure 4r : f). This can explain why some outward waves measured by Probe-A are dominant in Region-III. For the further revelation of the cutoff reflection process, a ray with a small incident angle (thick blue 150 line) is analyzed in Figure 4s ::: 5m−4x :: 5r. At the beginning (at group time < 18 seconds), neither the variation of n e nor that of B 0 deflect φ obviously. V g increases with B 0 as a consequence of the increasing of the slope of the dispersion curve ( Figure   A1). At group time ∼ 18 seconds, the ray starts to be trapped :::: slows ::::: down : in the region where V g decreases (shaded region in (e,k) Total magnetic field power P . (f,l) Spectrum-weighted averaged azimuthal angle of Poynting flux φ S .

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The second event was observed on 20 July 2015. From Figure 5, the waves display coherent harmonic structures, indicating a relatively narrow source region. The first harmonic recorded by Probe-B is below local f H + at L ∼ 5.0 − 5.6. Meanwhile, Probe-A was more inward than Probe-B, and thus observed the first three harmonics below f H + . Similarly with the previous event, the vanishing of the first harmonic just above the local f cutHe + is noticeable, comparing the spectra of Probe-A with Probe-B at 14:30 UT, suggesting the potential reflection of waves. Most of the waves of Probe-A demonstrate a westward 165 orientation, while the waves of Probe-B demonstrate a eastward orientation. The outward waves gradually overwhelm inward waves at larger MLTs (eastward) at locations of Probe-B. These phenomena are further demonstrated in the quantitative analysis in the last two panels. Most waves observed by Probe-A have φ S below zero, while most waves of Probe-B have φ S above zero. Considering that Probe-A was at an MTL ∼ 15.0 and Probe-B was at an MTL ∼ 18.0, the source region may be located at an MTL between 15.0 and 18.0 in an outer L-shell. As Probe-B moved eastward and the source drifted westward, the observed waves are gradually dominated by outward Poynting fluxes, which are most likely to be reflected in the vicinity of the local f cutHe + at L ∼ 3.9.

Conclusions and Discussions
Low-harmonic MS waves are frequently observed near the radiation belts (Balikhin et al., 2015;Boardsen et al., 2016;Teng et al., 2019) and have a potentially important dynamic influence on relativistic electrons (Maldonado et al., 2016;Yu et al., 175 2019). The propagation of low-harmonic MS waves is thus important because it controls the wave distribution. In the present study, the reflection of low-harmonic MS waves in the vicinity of the local two-ion cutoff frequency f cutHe + within the plasmasphere is studied. The results can be summarized as follows: 1. In the event on 01 March 2017, several wave patches were identified as the first harmonic MS waves. In a time-frequency region (Region-I) which was below the local f cutHe + for Probe-A and above local f cutHe + for Probe-B, the waves 180 were intense for Probe-B but vanished for Probe-A. According to calculation and simulation, for waves with a certain frequency, there exists an interface in space at f cutHe + , where the refractive index (RI) decreases to zero, and incident waves should be reflected as a consequence of Snell's law. There also exists a spatial layer within which the group velocity decreases significantly, and the waves with small incident angles should be trapped ::: slow ::::: down : for several to tens of seconds before the reflection process complete. 4. In the event on 20 July 2015, some intense first-harmonic waves were observed by Probe-B above the f cutHe + but were not measured by Probe-A below the f cutHe + within the same time-frequency region. Outward waves were observed by In Region-II, the azimuthal angels observed by Probe-B are almost perpendicular to the radial axis. The source of the rays with these angles are to the west of Probe-A from simulations. However, the observed upstream waves were in east of Probe-A.
This may be due to the separation of waves by the density dip along MLT in reality, similar to the mechanism in Figure 4b. 200 Liu et al. (2018b) found that the multiple fine-scale density irregularities can block MS propagation. The event in the present study, however, penetrate deep into the plasmasphere, which may be because the waves are initially generated inside the plasmasphere and are not influenced by the multiple fine-scale structures. In fact, most of the waves near f cutHe + are actually reflected before their wavelengths (minimum λ ∼ 300 km for the case of small incident angle according to Figure 4t :: 5n) become comparable with the scales of density irregularities (generally > 0.05 R E according to Figure 1b and 1f).

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For the inward propagated high-harmonic waves, with the increase in the ambient magnetic field or decrease in the plasma density, the refractive index decreases accordingly, and waves should be reflected, as studied by Chen and Thorne (2012) of MS waves and therefore promote the understanding of wave-particle dynamics in the outer radiation belt.

Appendix A
Here we demontrate the reasonability of neglecting the minor content of oxygen ion in deriving the MS mode dispersion relation. In the cold plasma, the following Stix parameters are helpful to investigate the dispersion relation (Stix, 1992):

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here ω ps = (n s q 2 s /m s ε 0 ) 1 2 and Ω s = q s B/m s are respectively the plasma frequency and the gyrofrequency of a particle species s.
The cutoff frequency Ω cut is the frequency where the phase velocity equals to zero (Smith and Brice, 1964), and can be obtained by setting the Stix parameter L equal to zero: Considering the charge neutrality condition, the determination of full ion abundance ratios requires the values of at least two characteristic frequencies (except gyrofrequencies) to be known in H + , He + and O + plasma. However, if the O + abundance η O+ is much lower than the H + abundance η H+ and if the focused mode belongs to the H + band which has a much larger frequency than the oxygen gyrofrequency Ω O+ , then we have the following relations: Therefore, if the three-ion (H + , He + and O + ) plasma are approximated as two-ion (H + and He + ) plasma, i.e., the terms with oxygen plasma frequency ω pO+ are dropped in Equation (A1), the Stix parameters will have only a negligible change. Under such an approximation, a group of ion abundance ratios (η H+ and η He+ ) can be obtained by substituting the observed value of ω cutHe+ /Ω H+ into Equation (A2). Consequently, the approximated dispersion relations for the modes in H + band can be 235 found.

Appendix B
Here we provide the proof that the directions of the wave vector and Poynting flux are the same for a perpendicular MS wave.
In the magnetized cold plasma, the frequency for a plane wave is the function of magnetic field B 0 , electron density n e , wave normal angle ψ and wavenumber k: 240 ω = ω(B 0 , n e , ψ, k).
Following Stix (1992), the group velocity can be expressed as: Here µ is the refractive index. Expressing the dispersion relation in the form: where R, L and P are the Stix parameters, and S = (R + L)/2. Considering a small angle β, and Expanding the left side of Equation (B4) around π/2, and considering that µ 2 ∼ RL/S P for the fast magnetosonic mode branch, we can obtain the following relation: here β = π/2−ψ. As tan α ∼ 0 when β ∼ 0, the direction of group velocity and wave vector are the same for the perpendicular MS waves.