Introduction

Electromagnetic ion cyclotron (EMIC) waves are the primary wave mode responsible for scattering and heating of cold ions (e.g., Kitamura et al., 2018;Ma et al., 2019;Yuan et al., 2016) and precipitating relativistic electrons in the inner magnetosphere (see, e.g., Angelopoulos et al., 2023;Bashir et al., 2022a;Drozdov et al., 2022;Millan & Thorne, 2007;Usanova et al., 2014;Yahnin et al., 2017, and references therein). These waves can be generated by hot, transversely anisotropic ions injected from the plasma sheet and drifting duskward (e.g., Bashir et al., 2022a;L. Chen et al., 2010;Jun et al., 2019, and references therein). This localized region of EMIC generation leads to the prevalence of EMIC wave driven relativistic electron precipitation at the nightside/dusk flank (see Thorne et al., 2006). These relativistic electron losses are well studied theoretically (see reviews by Lee et al., 2021;Thorne et al., 2006, and references therein) and observationally (e.g., Blum et al., 2015;Capannolo et al., 2019Capannolo et al., , 2023;;Yahnin et al., 2016). At dayside, EMIC wave generation is associated with ion anisotropic heating by solar wind compression of the magnetosphere (N. Liu et al., 2022;Xue et al., 2023). The extreme examples of such solar wind driven EMIC wave generation are magnetosphere impact by interplanetary shock waves (Blum et al., 2021;Li et al., 2022;Yan et al., 2023) and solar wind pressure pulses (Xue et al., 2022). Alternatively, EMIC waves can also be generated by another mechanism on the dayside. This mechanism involves intense, compressional ultra-low frequency (ULF) waves that may magnetically trap anisotropic, hot-ion populations and transport them into the inner magnetosphere. Such ULF waves are often generated in response to the magnetopause impact by solar wind/magnetosheath transients (e.g., M. D. Hartinger et al., 2013Hartinger et al., , 2014;;Wang et al., 2018;Wang, Nishimura, et al., 2019). Another ULF mode capable of hot ion trapping and transport is the drift mirror mode (e.g., Hasegawa, 1969): nonlinear mirror mode waves are local magnetic field depletions filled with transversely anisotropic hot ions (Balikhin et al., 2009;Cooper et al., 2021;Rae et al., 2007;Soto-Chavez et al., 2019). Nonlinear mirror waves are generated around the magnetopause and can transport the trapped ion population earthward (due to the convection electric field), where a higher cold background density changes the stability conditions of this population and may drive EMIC wave generation within mirror modes (Kitamura et al., 2021;Loto'Aniu et al., 2009;S. Liu et al., 2019;Z. Y. Liu et al., 2022;Yin et al., 2022). In contrast to EMIC waves generated by plasma sheet ion injections (e.g., Jun et al., 2019Jun et al., , 2021) ) and thus parameterized by geomagnetic indexes and solar wind conditions (e.g., H. Chen, Gao, Lu, & Wang, 2019; Gamayunov et al., 2020;Ma et al., 2015;Ross et al., 2021;Usanova et al., 2012), the ULF-associated EMIC waves are not expected to strongly correlate with geomagnetic conditions; instead, they should be mostly determined by dayside magnetosphere interaction with mesoscale (or even ion kinetic scale) transients. Thus, this EMIC wave population may be smoothed out in an index-based empirical model and requires alternative parameterization for further incorporation into simulations of the radiation belt dynamics in the inner magnetosphere. Such ULF-associated EMIC waves may play an important role in relativistic electron precipitation at higher L-shells, well outside the plasmapause (see statistics of relativistic electron precipitation events at high-L in Capannolo et al., 2022). Moreover, the small scale of mirror mode structures (the primary source of ULF-modulated EMIC waves (Bashir et al., 2022b;Shahid et al., 2024)), relative to the typically large-scale EMIC wave generation region (Blum et al., 2016(Blum et al., , 2017) ) may explain the temporally and spatially localized EMIC-driven electron precipitation in observations (Shumko et al., 2022). Additionally, EMIC waves are also responsible for the scattering and isotropization/precipitation of ring current ions (e.g., Jordanova et al., 2001Jordanova et al., , 2007;;Khazanov et al., 2007). Therefore, EMIC wave modulation/generation within mirror-mode structures indicates the solar wind impacts on ring current dynamics.

This study aims to investigate the characteristics of ULF-modulated EMIC waves statistically, their spatial distribution (in L-shell, MLT), and the associated plasma conditions. We analyze 5 years (2017)(2018)(2019)(2020)(2021) of THEMIS E, (Angelopoulos, 2008) measurements in the inner magnetosphere and near-Earth plasma sheet. The collected data set contains 330 intervals (each interval represents a period with observations of multiple EMIC bursts associated with ULF waves; the beginning/end of intervals are determined by spacecraft entrance/exit into/ from the region of EMIC wave correlation with ULF waves) with multiple EMIC wave bursts modulated by compressional ULF waves, with a total of 1,555 individual bursts (each of them is within the ∼ 1/2 period of the simultaneously observed ULF wave). Section 2 describes typical events from our data set, whereas Section 3 provides statistical properties of EMIC waves and surrounding plasma characteristics. The development of the empirical model is discussed in Section 4 and the estimation of pitch angle scattering based on the empirical model is provided in Section 5. The results are discussed and concluded in Section 6.

Spacecraft Instruments and Data Set

We examine data from three inner probes of the THEMIS mission at an apogee of ∼12 Earth radii (R E ) . These probes are equipped with identical instruments (Angelopoulos, 2008). For investigation of EMIC waves, we use: Fast Survey (1/4s sampling) measurements of the fluxgate magnetometer (FGM, see Auster et al., 2008), spin averaged (3s) measurements of ion and electron <30 keV spectra and moments by the electrostatic analyzer (ESA, see McFadden et al., 2008), cold plasma density derived from the spacecraft potential (Bonnell et al., 2008). Details of the method of plasma density evaluation from the spacecraft potential can be found in (Pedersen et al., 1998), whereas estimates of uncertainties of this method in application to THEMIS data set are provided in (Nishimura et al., 2013). Following this uncertainty estimation procedure, we compare the density from the spacecraft potential and the density moment from the ESA during intervals without cold plasma and radiation belt contamination (see panels (b) in Figure 1 and Figure S4). However, this estimate works for the plasma sheet region, where the ESA data set is reliable. In contrast, we may not provide the same estimate for the regions with a significant population of cold (<3 eV) electrons or high spacecraft potential. Note that Pedersen et al. (2008) estimated the uncertainties for a wide range of plasma parameters, showing that the uncertainties of this method may reach 100% under some conditions. It is important to carefully check these uncertainties before applying this method to analyze density-based parameters in EMIC wave interactions with charged particles, as they are crucial for accurately interpreting the results.

The ion temperature was derived from ESA measurements. We should note that the hot ion population beyond the ESA energy range (>30 keV) may contribute to the temperature for events with sufficiently hot ions. To verify this contribution, we evaluated the temperature from the combined measurements of ESA and the solid state telescope (SST; >50 keV) (Angelopoulos et al., 2008): THEMIS data set includes temperatures evaluated with merged ESA and SST spectra, including the interpolation of the 30 -50 keV region (see details in Runov et al., 2015;Turner et al., 2012). Comparison of the temperatures derived from ESA and from combined ESA and SST measurements (see Figure 2 in Artemyev et al., 2021, for details of such comparison) shows that the temperature based on ESA underestimates the actual temperature less than 30%. For most events, this underestimate is less than 20%. Therefore, in this study, we use the ion temperature derived from ESA measurements.

For our statistical analysis, we have used 5 years of data from THEMIS-E and our event selection criteria include: (a) search for quasi-periodic EMIC waves with at least three wave intensity peaks within 1 hr, (b) determine the ULF wave frequency range (within [0.1, 20] mHz) exhibiting good correlation between ULF magnetic field minima and EMIC wave intensity peaks. Only events exhibiting such correlations are included in the data set. Note that EMIC waves in Earth's magnetosphere are observed in different bands depending on the existence of hot and cold heavy ions and are most often characterized into the H + , He + and O + bands (Chen, Zhu, & Zhang, 2019;Engebretson et al., 2018;Jun et al., 2021;Sigsbee et al., 2023). There has also been increased interest in studying EMIC waves in additional bands created by the presence of minor ions, including N + (Bashir & Ilie, 2018, 2021), He ++ (Engebretson et al., 2018;Lee et al., 2019;Miyoshi et al., 2019), and O ++ (Yu et al., 2021). However, in this study, we consider only H-band EMIC waves with frequency range f ∈ [f cp /4, f cp ] , where f cp is the proton gyrofrequency. Note that some of the H-band EMIC waves that we observe may also contain waves in the He ++ band in the frequency range f ∈ [f cp /2, f cp /4] . This is possible because THEMIS can also sample EMIC waves in magnetospheric regions where solar wind entry can supply He ++ ions and the instability threshold conditions for generating the He ++ band waves are satisfied (Lee et al., 2019). This choice of considering H-band may be justified by the fact that (a) most of the events of ULF-modulated EMIC waves belong to a large L-shell dayside magnetosphere, where populations of cold plasmaspheric heavy ions are small and (b) studying the heavy ions band is beyond the scope of this paper, mainly because the THEMIS ESA instrument cannot resolve the heavy ions, especially hot ions (a previous work attempted to use the THEMIS ESA to resolve the presence of cold heavy ions (J. H. Lee & Angelopoulos, 2014)), and limit our ability to justify the existence of heavy ions and necessary threshold conditions. However, we plan to perform a comprehensive statistical study in the future for the occurrence of ULF-wave modulated EMIC waves including other EMIC wave bands (i.e., He + , O + , & He ++ ) utilizing wave and plasma measurements from missions equipped by mass spectrometers (e.g., Magnetospheric Multiscale Mission (Burch et al., 2016;Young et al., 2016) and Cluster (Escoubet et al., 2001;Rème et al., 2001)) capable of measuring those ions in wide energy range in the magnetosphere.

Figure 1 shows three typical events from the data set collected. All are from the near-equatorial (B z is the dominant component; see Panels (a)) transition region between the plasma sheet and the inner magnetosphere. The electron plasma frequency to electron gyrofrequency ratio, f pe / f ce ∈ [5, 10], is typical for the near-Earth plasma sheet (see Panels (b)). Note that f pe / f ce derived from the spacecraft-potential density is much larger Journal of Geophysical Research: Space Physics ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ <B 2 w (t)> √ , (d) EMIC wave mean frequency normalized to f cp , (e) width of EMIC wave mean frequency normalized to f cp , (f) EMIC wave mean wave normal angle, and (g) width of the wave normal angle distribution. than f pe / f ce based on ESA density measurements, because a significant fraction of cold electrons is not measured by the ESA (McFadden et al., 2008). Using magnetic field measurements, B, we evaluate compressional fluctuations in the ULF frequency range: δB ∥ = (B -〈B〉) ⋅ 〈B〉/|〈 B 〉|. Panels (c) show quasi-periodic δB ∥ variations with amplitudes of a fraction of nT, that is, δB ∥ /|〈 B 〉| ∼ 10 -3 . To evaluate δB ∥ , we use the frequency range [0.1, 20] mHz (i.e., δB ∥ = (B -〈B〉) ⋅ 〈B〉/|〈 B 〉| denotes the averaging of 7 min). For each event, we additionally check if changing the lower and upper-frequency limits (within the [0.1, 20] mHz range) may improve the δB ∥ correlation with the EMIC wave intensity, and adjust these limits to maximize this correlation. Such compressional ULF waves propagate in a hot, transversely anisotropic ion background (see Panels (d, e)), and even with their low amplitude can drive EMIC wave generation (see details in Shahid et al., 2024). Fast Fourier Transform (FFT) analysis is used to identify EMIC waves and minimum variance analysis for quantitative determination of the wave ellipticity and wave normal angle. Panels (f) show quasi-periodic EMIC wave bursts correlated with local minima of δB ∥ , which are usually associated with an enhanced hot transversely anisotropic ion population contributing to the pressure balance (e.g., Balikhin et al., 2009; Soto-Chavez et al., 2019). Although EMIC waves may also be associated with δB ∥ maxima (see Loto'Aniu et al., 2009), our statistics include only EMIC wave bursts embedded in δB ∥ minima.

Wave ellipticity and polarization (panels (g, h)) confirm that these are classical, field-aligned EMIC waves. These wave properties are estimated using the MVA technique (Sonnerup & Scheible, 2000). Note that although we determine polarization and ellipticity for each wave burst, we do not use these properties for event selection.

Figure 2 zooms in on several EMIC wave bursts from Figure 1. All bursts are associated with local minima of ULF δB ∥ , and such a type of correlation has been found for all events in our data set. The mean wave intensity

4f cp , quite typical for EMIC waves in the inner magnetosphere (see, e.g., Kersten et al., 2014;Zhang et al., 2016). For each wave burst, we also calculate the wave

f )df averaged over the burst interval giving the wave frequency width df (t) = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ <df 2 (t) > √ , the mean wave normal angle WNA, and the dispersion of wave normal angles, dWNA (both are obtained by the MVA technique (Sonnerup & Scheible, 2000) and averaged over the wave burst interval same as frequency). The wave bursts shown in Figure 2 are quite narrow-banded with df (t)/〈 f (t) 〉 < 1/5 and weakly oblique with WNA ≈ 20 • ± 10 • . Some of the EMIC waves have linear polarization and oblique wave normal angles. Not all of the bursts shown are consistent with "classic" field-aligned, lefthand circularly polarized EMIC waves. We determine these characteristics for all wave bursts in our data set.

Although the correlation between ULF waves and EMIC wave bursts is well seen in examples shown in Figures 1 and 2 (each EMIC burst can be associated with a local minimum of the ULF field), this correlation cannot be quantified by the correlation coefficient, because both ULF and EMIC wave amplitude signals contain multiple frequency bands. The magnitude of EMIC wave bursts and ULF minima vary within the event, and this radically reduces the directly calculated correlation coefficient. Therefore, we collect a data set of our events using the alternative approach. For each event, we plot spectra of ULF wave and EMIC wave amplitudes, and select only events showing the main spectra peak in the same frequency (despite such spectra can contain multiple additional peaks corresponding to ULF and EMIC amplitude variations from one minimum (burst) to another one (see examples in Figure 2)). We then visually check that for all selected events each burst of EMIC waves is associated with a local minimum of δB ∥ (more than 20% reduction of δB ∥ ).

Figure 3 shows the histograms for the L-shell and MLT distributions for all data sets: ULF-modulated EMIC waves are mostly observed at high L-shells (L > 8) at the dayside sector (from 10 to 16), where compressional ULF waves are predominantly observed in response to solar wind transients and magnetopause oscillations (M. Hartinger et al., 2011;M. D. Hartinger et al., 2013;Wang et al., 2018;Wang, Nishimura, et al., 2019). At dawn and dusk flanks, ULF-modulated EMIC waves are only observed at high L, whereas at the dayside such waves can be found even at L ∼ 5 -6. Note that in the text we use dawn/day/dusk MLT sectors for MLT ∈ [0, 8], [8,14], and [14,24]. In Figure 3, at L ∼ 5 the number of events is quite small on this grid and is statistically insignificant. That's why for our statistical results in Figures 567, especially for comparison of observed and fitted parameters, we use only bins with at least 5 events. We have also provided the results without limiting them to five events in Figures S1-S3. . Such an obliqueness may affect relativistic electron scattering by EMIC waves (Gamayunov & Khazanov, 2007;Khazanov & Gamayunov, 2007) and should be quite Journal of Geophysical Research: Space Physics e, f) and note that df (t) = Δf (t)), are typical for EMIC waves in the inner magnetosphere (Kersten et al., 2014;Zhang et al., 2016).

Statistical Properties of ULF-Modulated EMIC Waves

Figure 5 shows the spatial distribution of all the wave and background plasma/magnetic field characteristics.

There is a clear dawn-dusk asymmetry of f pe / f ce and ion temperature T i calculated from the ESA instrument for ULF-modulated EMIC events: f pe / f ce is approximately 10 at the dawn flank and increases to 20 at the dusk flank; T i is about 3 keV at the dawn flank and increases to 5 keV at the dusk flank. These results are consistent with the general statistics from the Cluster observation of f pe / f ce associated with EMIC waves (Allanson et al., 2016), that is, ULF-modulated EMIC waves and EMIC waves not associated with ULF waves are observed under the same plasma conditions.

The root mean square amplitude is slightly larger on the dawn flank, 〈B w 〉 ∼ 200 pT, than on the dusk flank, 〈B w 〉 ∼ 50 pT. Ion β i = 8πnT i / B 2 0 maximizes at flanks, β i ∼ 10, and has a minimum on the dayside and at low L, and β i < 1. Ion temperature anisotropy, the key parameter for wave generation, has a maximum at dawn flank, A i = T i,⊥ / T i,∥ ∼ 1.6; this is consistent with generally more anisotropic ion populations associated with EMIC wave observations in noon/dawn flank (see Anderson et al., 1992;J. H. Lee & Angelopoulos, 2014;Min et al., 2012). This peak of A i most likely explains the dawn flank peak of 〈B w 〉. Note that although ULF wave amplitudes may have some MLT dependence, for our data set of ULF-modulated EMIC waves, the observed ULF amplitudes do not show any maximum at the dawn flank. Therefore, variations of the EMIC wave properties should be associated with the local plasma parameters. The maximum anisotropy on the dawn flank cannot be associated with plasma sheet injections, because the injected ions drift at the dusk flank, where most of the anisotropy is released for the generation of EMIC waves (Jun et al., 2019(Jun et al., , 2021)). Therefore, Figure 5d likely shows the selection effect: dawn flank is characterized by lower f pe / f ce ratio, and thus resonant ion energies are higher and fluxes are lower for EMIC wave generation; this implies that a higher level of anisotropy is required for the generation of EMIC waves at dawn flank.

Panels (h-l) in Figure 5 show the distribution of the main wave characteristics. The mean wave frequency <f / f cp > and the frequency width df / f cp do not vary much with MLT: for example, there is only a slightly lower <f / f cp > ≈ 0.4 on the dusk flank, in comparison with <f / f cp > ≈ 0.5 on the dawn side. However, there is a clear gradient of wave normal angle with MLT: waves at the dusk flank are more field-aligned with WNA ≈ 35 • than in the dawn flank, where WNA ≈ 55 • . This is consistent with the general THEMIS statistics of EMIC waves (Min et al., 2012), that is, WNA is likely dictated by local properties of hot ion populations responsible for EMIC wave 1) and coefficients (Table 1) obtained from fits.

generation and does not depend on ULF-modulation (see also discussion in Anderson et al., 1992). Interestingly, the wave normal angle variation, dWNA, does not depend on MLT, but increases at lower L-shells, from dWNA ∼ 5 • at L ∼ 12 to dWNA ∼ 10 • at L ∼ 5.

Empirical Model

Distributions of the main wave characteristics shown in Figure 5 can be used to estimate the ion diffusion rates that would describe plasma sheet and ring current ion precipitation via resonant scattering by ULF-modulated EMIC waves. To enable the evaluation of diffusion rates, we develop emprical model for the ULF-Modulated EMIC wave properties (wave amplitude <B w > , mean wave frequency <f / f cp >, and frequency distribution width <Δf / f cp > , wave normal angle WNA, and its width dWNA, and the plasma parameters (plasma-to-electron cyclotron frequency ratio, from ESA f pe (ESA)/ f ce and spacecraft potential f pe / f ce , Ion plasma beta β i , ion temperature anisotropy A i ) as a function of MLT and L as.

that capture main L and MLT gradients of wave characteristics. Note that these empirical model only works within the region of ULF-modulated EMIC wave observations: MLT ∈ [5, 20], L ∈ [5, 12], with two boundaries MLT -= 15 -L and MLT + = 10 + L. The fitting parameters a i , i = 0 to 5 are obtained using the non-linear least-squares fitting method. Table 1 shows the coefficients of these empirical model for each quantity. We also present the statistical properties in Figure 6 using our empirical model and coefficients given in Table 1. The results obtained from the empirical model agree well with the parameters observed in Figure 5.

Estimation of Pitch Angle Diffusion Rate

We use the empirical model for the observed EMIC wave characteristics and plasma parameters (i.e., f pe / f ce from spacecraft potential) to quantify wave contribution to plasma sheet and ring current ion losses. This was done by Journal of Geophysical Research: Space Physics 10.1029/2023JA032290

parametrizing the wave characteristics so that they can be applied to evaluate how efficiently these waves contribute to the pitch angle diffusion of the energetic ions.

We use a simplified formula by assuming the waves as field-aligned, although some of them are weakly oblique (see Summers et al., 2007, for equations of diffusion coefficients in the quasi-linear theory):

, and erf is the error function with σ = 4/3 for ions, c is the speed of light in vacuum, Ω cp = 2πf cp , Ω ce = 2πf ce that is, f cp = e|B 0 |/ m p c and f ce = e|B 0 |/ m e c are the proton cyclotron and electron cyclotron frequencies, m e and m p are the rest masses of the electrons and proton, and e is the unit charge. Also, x = f / f cp , y = kc/2πf cp , E is the dimensionless particle kinetic energy given by

w / B 2 0 is the ratio of the energy density of the wave magnetic field to that of the background field, that is, the relative wave power; x m = <f m / f cp >( = 0.35 is fixed), δx = Δf / f cp and f pe / f ce are used from observed statistics. x, y, F(x, y) = dx/dy are determined from the cold plasma dispersion of EMIC waves (Stix, 1962). The diffusion rate D αα evaluated at the loss-cone pitch-angle, α LC (L), determines the rate of wave-driven ion precipitation into the atmosphere (Kennel & Petschek, 1966). This diffusion rate can be compared to the so-called strong diffusion rate (Summers & Thorne, 2003)

that determines the maximum possible precipitation rate in the diffusion regime (Kennel, 1969;Kennel & Petschek, 1966). Although a significant fraction of ULF-modulated EMIC bursts are observed within the plasma sheet, where ion β approaches one or can be large (see Figure 5c), for statistical evaluation of diffusion rates we use cold plasma dispersion relation. The hot plasma effects are known to change the resonant energies for EMIC wave interaction with electrons (e.g., Bashir et al., 2022;H. Chen, Gao, Lu, & Wang, 2019;Ni et al., 2018), and in the next studies these effects should be investigated for EMIC wave-ion resonances as well. The same comment is valid for our assumption of field-aligned wave propagation: the observed non-zero WNA indicates that the obliquity should be included in the further investigation of efficiency of ULF-modulated EMIC waves (see Gamayunov & Khazanov, 2007;D.-Y. Lee et al., 2018, for discussion how wave obliquity modifies the efficiency of relativistic electron scattering by EMIC waves).

Figure 7 shows the diffusion rates D αα and D αα / D SD at four ion energies and α LC (L). At MLT ∈ [5, 15] and L > 10, ULF-modulated EMIC waves provide very strong D αα ∼ 10 -2 s -1 , above the strong diffusion limit (D αα / D SD > 1). Thus, when these waves are present (i.e., when sufficiently strong ULF waves transport hot anisotropic ions and EMIC waves), they will drive the modulated ion (proton) losses for a wide energy range. Such losses can be observed as day-side proton aurora forms (e.g., Berchem et al., 2003;Lorentzen & Moen, 2000;Sigernes et al., 1996), and some of these forms are indeed observed poleward from the proton aurora oval, that is, within the plasma sheet (see review by Frey, 2007). One such form, that may be associated with EMIC wave-driven precipitations of ions near the magnetopause, where ULF waves are driven by solar wind transients and can modulate EMIC waves, is so called High-Latitude Dayside Aurora (e.g., Murphree et al., 1990). Sub-auroral day-side proton precipitations (equatorward from the auroral oval) also may have transient character and be driven by EMIC waves (Fuselier et al., 2004). Although such transient proton precipitations can be potentially driven by ULF-modulated EMIC waves, their L-shells are lower than the main statistics of our data set (see Figure 3a), and further investigation of low L-shell ULF-modulation of EMIC waves is needed to connect this wave activity with sub-auroral proton precipitations.

Discussion and Conclusions

In this study, we examine a specific class of EMIC waves modulated by compressional ULF waves. The likely scenario for such EMIC wave generation includes the day-side magnetopause impact by solar wind transients that drive compressional ULF waves (presumably drift mirror mode waves Balikhin et al., 2009;Cooper et al., 2021;Rae et al., 2007;Soto-Chavez et al., 2019), and these ULF waves transport the hot, transversely anisotropic ion population to lower L-shells, where the ion anisotropy is released in the form of EMIC wave generation (see examples in Kitamura et al., 2021;Loto'Aniu et al., 2009;Z. Y. Liu et al., 2022;Yin et al., 2022). This scenario is confirmed by the prevalence of ULF-modulated EMIC waves at large L-shells (L ∈ [8, 12]) on the day-side MLT ∈ [10,16]. Therefore, we may suggest that such ULF-modulated EMIC waves constitute an additional class of EMIC waves, supplementing the two most investigated classes: EMIC waves generated on the dusk flank by plasma sheet-injected ions and EMIC waves generated at low L-shells on the day-side by compressionally heated ions (e.g., Jun et al., 2019Jun et al., , 2021;;N. Liu et al., 2022;Xue et al., 2022). The main feature of this new EMIC wave class is the periodicity of EMIC emission.

These ULF-modulated EMIC waves are moderately oblique (WNA ∈ [30

• , 60 • ]), quite narrow-banded (Δf /f ∼ 0.3 with f / f cp ∈ [0.4, 0.55]), and moderately intense (B w ∈ [50, 300] pT). Due to their insufficiently high wave amplitudes, these waves resonate with protons in the quasi-linear regime. The main ion population responsible for the generation of ULF-modulated EMIC waves is typical plasmasheet ions with energies E R ∈ [5,15] keV. Due to the relatively small plasma density at high L, the frequency ratio f pe / f ce during ULFmodulated EMIC waves is about ∈ [5,10]. This ratio controls the resonant energies for electrons interacting with EMIC waves, and when f pe / f ce ∈ [5, 10] EMIC waves cannot scatter relativistic or energetic electrons, but only ultra-relativistic electrons that are usually scarce in such high L-shells. Therefore, ULF-modulated EMIC waves are not effective in driving electron losses. However, these waves are very effective in scattering ions (pitch-angle diffusion rate reaches the strong diffusion limit for [10, 100] keV ions). Therefore, ULF-modulated EMIC waves can produce quasiperiodic ion precipitation that contributes to the day-side proton aurora (Berchem et al., 2003;Frey, 2007;Lorentzen & Moen, 2000;Sigernes et al., 1996). Further transported by ULF waves to lower L-shells, hot anisotropic ion population may generate EMIC waves within the outer radiation belt, where these waves are expected to serve as the seed population for intense coherent EMIC waves (see discussions in Gamayunov & Engebretson, 2021, 2022).

Statistics of ULF-modulated EMIC waves demonstrate one more mechanism of direct solar wind impact on the Earth's magnetosphere (see discussions of other mechanisms in M. D. Hartinger et al., 2013;M. D. Hartinger et al., 2014;Wang et al., 2018;Wang, Nishimura, et al., 2019), which underlines the importance of including solar wind transients into models of inner magnetosphere dynamics and magnetosphere-ionosphere coupling.