Statistical Properties of Pc4‐5 ULF Waves in Plasmaspheric Plumes

Ultra‐low‐frequency (ULF) waves emerge as pivotal factors in elucidating the mechanisms that drive the intricate dynamics of radiation belt electrons within the plasmasphere and plasmaspheric plumes. Utilizing THEMIS data from September 2012 to September 2017, we conducted a comprehensive statistical analysis of Pc4‐5 ULF waves within and outside the plasmaspheric plume. Our findings reveal a distinctive dawn‐dusk asymmetry in occurrence rate and wave power of poloidal mode waves in the absence of the plume, resembling the toroidal mode asymmetry observed. Poloidal mode waves exhibit a higher likelihood of formation within the plume, while the toroidal mode waves show the opposite trend, contributing to the elevated dusk‐side occurrence rate of poloidal mode waves. Moreover, both wave modes within the plume demonstrate lower peak frequencies compared to waves outside the plume. The global distribution of wave power within the plume suggests higher power at noon than on the dusk side.


Introduction
The Earth's magnetosphere is a highly intricate and dynamic system, shaped by the interaction of solar wind particles and energy, as well as internal dynamical processes.In the realm of magnetosphere and space weather research, a paramount goal lies in achieving robust predictive capabilities, with a particular focus on the highenergy particle population enveloping Earth's magnetic field, known as the radiation belts (Baker, et al., 2007).Current empirical observations (Claudepierre et al., 2013;Lotoaniu et al., 2006;Mann et al., 2004;Tan, Shao, et al., 2011;Zong et al., 2009) and cutting-edge theoretical modeling and simulations (Fei Y. et al., 2006) concur that a key source of radiation belt electrons originates from their inward radial transport driven by resonant interactions with Pc5 Ultra-Low-Frequency (ULF) waves spanning the 1-10 mHz range.In essence, these ULF oscillations facilitate the transfer of energy from the magnetosphere to the ionosphere, resulting in the acceleration of high-energy particles, modulation of auroral luminosity, mediation of reconnection processes, and instigation of substorms (e.g., Baumjohann & Glaßmeier, 1984;Keiling, 2009;Lessard et al., 1999;Rae et al., 2014;Ukhorskiy et al., 2005;Zong et al., 2009Zong et al., , 2017)).Consequently, attaining a comprehensive understanding of the global distribution of ULF wave characteristics bears profound significance in unveiling the intricate coupling mechanisms between the solar wind, magnetosphere, and ionosphere.
The sources of ULF waves can be broadly classified into two main groups: external sources, which are linked to the solar wind (e.g., Claudepierre et al., 2008;Shi et al., 2013;Tian et al., 2016), and internal sources, which are associated with plasma instabilities (e.g., Southwood et al., 1969;Zong et al., 2017).Depending on the oscillation direction of magnetic field lines, the ULF waves can be further subdivided into poloidal mode (with radial oscillations in magnetic field) and toroidal mode (with toroidal oscillations in magnetic field).Plasmaspheric plumes, which extend beyond the plasmasphere toward the dayside magnetopause, are noteworthy for their occurrence due to the influence of enhanced convection electric fields and erosion from the external plasmasphere (e.g., Darrouzet et al., 2009).These phenomena offer significant insights into the intricate coupling processes among the solar wind, magnetosphere, and ionosphere.
Regarding the statistical research on the spatial distribution of ULF wave characteristics, numerous studies have utilized ground-based measurement data and satellite data (e.g., Anderson et al., 1990;Howard & Menk, 2005;Kokubun et al., 1989;Rae et al., 2012).The major conclusions drawn from these early observations include the dominance of poloidal modes on the dusk side, fundamental toroidal mode resonances occurring predominantly on the dawn side.Ascribed to the intricate drift-bounce resonance of the ions originating from the magnetotail and drifting westward, these poloidal Pc5 waves exhibit an antisymmetric mode structure and were discerned to be excited predominantly on the dusk side (Southwood et al., 1969;Southwood & Kivelson, 1982).Degeling et al. (2018) proposed that MHD fast mode waves would be trapped within the dusk-side plasmaspheric plume, which enables excitation of higher amplitudes by a cavity resonance.Meanwhile, simulations by Elsden and Wright (2022) show that 3-D FLRs (strong perturbations both in the poloidal and toroidal components) will occur prominently at the boundaries of the dusk-side plasmaspheric drainage plume where there are steep azimuthal gradients in the Alfvén frequency.Moreover, it was found that the occurrence rate, amplitude, and frequency of toroidal waves are markedly higher on the dawn side, utilizing data from the THEMIS satellites (Takahashi et al., 2015) and the IMAGE ground-based magnetometer network (Wharton et al., 2018).This distinctive dissimilarity was attributed to the greater prominence of the Kelvin-Helmholtz instability (KHI) on the dawn side (Lee & Olson, 1980;Mann et al., 1999;Nykyri, 2013), the local time dependence of the radial mass density variation and the associated Alfvén frequency gradient (Takahashi et al., 2016), and the larger and more coherent pressure gradients generated by asymmetric fast modes at dawn compared to dusk (Wright et al., 2018).
Despite previous research conducting statistical analyses on ULF waves, the spatial distribution and underlying excitation mechanisms of ULF waves in the plasmaspheric plume area remain inadequately understood, which have been found to play a key role in the propagation and coupling of ULF waves (e.g., Degeling et al., 2018;Elsden & Wright, 2022;Sandhu et al., 2023;Zhang et al., 2019).Our primary objective is to undertake a comprehensive investigation of the global distribution of ULF waves, with a specific focus on their statistical characteristics within the plume region.

Data and Methodology
In this investigation, we employed magnetic field data from the Fluxgate Magnetometer (FGM) and plasma data from Electrostatic Analyzer (ESA) with a temporal resolution of 3 s, both of which are installed on board the THEMIS satellite constellation.The observation period spanned from September 2012 to September 2017.The THEMIS mission comprises five satellites, denoted as THEMIS A, B, C, D, and E, each possessing an orbital inclination of approximately 10°.The apogee for THEMIS A, D, and E is at an approximate distance of 12RE, whereas THEMIS B and C orbit even farther away.Hence, for this study, we selectively focused on utilizing data from THEMIS A, D, and E.
First and foremost, as shown in Figure 1b, we present criteria for determining the structure of plasma plumes.Density enhancements are identified manually, following the same method as Hartley et al. (2022).Additionally, we reference the plasma density model proposed by Sheeley et al. (2001).Plasmaspheric plumes are defined as events lasting over 20 min.This differs from the approach of Hartley et al. (2022), who select events based on a width exceeding 0.2 R E .In order to establish a reliable comparison, we also selected observations without plasmaspheric plume for further analysis.After identifying and removing all the plasmaspheric plume events, we proceeded to eliminate data segments with electron density exceeding 5 cm 3 .This step was taken to exclude time intervals when the satellite was outside the magnetopause (e.g., Takahashi et al., 2015).Meanwhile, as shown in Figures 1c-1h, we focus on Pc-4 and Pc-5 ULF waves with the peak frequency of 2 ∼ 22 mHz.These ULF waves exhibit a sustained duration exceeding three complete oscillation cycles, persisting for a minimum of 7 min, and demonstrating quasi-monochromatic attributes (signified by the highest peak in the power spectrum surpassing fourfold the amplitude of the second-highest peak).We employ wavelet transform to identify the onset and cessation times of the wave events, defining these points as the instances when the amplitude in the time-frequency spectrum reaches half of its peak.For these meticulously selected waveforms, we harness the fast Fourier transform (FFT) technique to compute their power spectral density (PSD), thereby extracting vital parameters for comprehensive analysis.It's worth noting that the frequency resolution of the FFT is equal to the reciprocal of the observation time, results in an error upper limit of 1/3 of the waveform frequency.It's important to emphasize that not all waveforms last only 3 periods, making these errors acceptable.In addition, Electric field perturbation which is used for statistics of wave power is given by formula , where δV → is ion velocity perturbation obtained by subtracting the sliding average of 30 min from the ion velocity.
Figure 1 shows a plume crossing event observed by the THEMIS-a satellite.For this event, perturbations are discernible in all poloidal, toroidal, and compressional components, aligning with the characteristics of threedimensional Field Line Resonances (3D FLRs) in the presence of a plume on the dusk side.In addition to internal plasma instability, one potential driving source could originate externally from variations in solar wind dynamic pressure.These variations excite fast waveguide modes, subsequently driving FLRs characterized by a sharp azimuthal density gradient (Elsden & Wright, 2022;Sandhu et al., 2023).Based on the aforementioned methods, using data from September 2012 to September 2017, we identified a total of 437 Pc4-5 ULF wave events within 955 plasmaspheric plume-like structures.Additionally, outside the plasmaspheric plume regions, we identified a total of 13357 Pc4-5 ULF wave events.

Occurrence Rate
Figure 2a illustrates the global distribution of observation times outside the plasma plume.Due to frequent magnetopause crossing events on the dayside, observation times within this range are relatively low.Figure 2b presents the global distribution of observation times within the plasma plume.It is evident that nearly all identified plumes are located within the regions of L = 8 ∼ 11 and MLT = 13 ∼ 20.We only observed a minimal number of plumes on the dawn side, as plumes primarily form on the dusk side.Figure 2c depicts the occurrence rate of toroidal modes outside the plume.It is evident that the occurrence rates on the dayside and flank regions are significantly higher than on the nightside, displaying a clear dawn-dusk asymmetry.This observation aligns with previous findings (e.g., Hudson et al., 2004).This phenomenon may be related to activity on the flank side, such as KHI being stronger on the dawn side (e.g., Nykyri, 2013).Figure 2d presents the occurrence rate of toroidal modes within the plume, which is significantly lower compared to outside.The primary reason for this discrepancy is that toroidal modes are typically considered to be caused by external sources.The pronounced density gradient at the plume boundary reflects the compressional waves, hindering their propagation into the high-density plume region and thereby impeding the initiation of FLRs (Degeling et al., 2018).Consequently, there are almost no toroidal modes observed within the plume.
Figure 2e illustrates the occurrence rate of poloidal modes outside the plume.The occurrence rate on the flank side is notably higher than on the nightside, consistent with the observations by Liu et al. (2009).However, it is noteworthy that frequent poloidal waves were not observed in the noon sector.Furthermore, the occurrence rate of poloidal modes on the dusk side is lower than on the dawn side.This suggests that most poloidal modes on the dusk side occur in the plumes, corroborating the conclusion drawn from Figure 2f. Figure 2f provides the occurrence rate of poloidal modes within the plume.On the dusk side, the occurrence rate of poloidal modes within the plume is higher than that outside the plume.This indicates that most of the poloidal modes on the dusk side are confined within the plume structure.The lower intrinsic frequencies of magnetic field lines within the plume make it more conducive for the excitation of low-frequency waves.In summary, within the plume, ULF waves exhibit distinct global distribution characteristics compared to outside the plume structure.
To delineate the distribution characteristics of ULF waves across different frequency ranges, we partition ULF wave events without plumes into two segments based on peak frequency: one with a peak frequency between 2 and 5 mHz, and the other with a peak frequency greater than 5 mHz.As Figure 3a shows, in the 2-5 mHz range, toroidal modes predominantly manifest in the L = 8 ∼ 10 region, exhibiting a relatively high occurrence rate from the dawn side to the dusk side.Conversely, in Figure 3c, for peak frequencies greater than 5 mHz, toroidal modes primarily occur at smaller radial distances with L < 9, and the toroidal mode occurrence rate is higher exclusively on the dawn side.Figures 3b and 3d shows that the distribution pattern of poloidal modes closely mirrors that of toroidal modes in Figures 3a and 3c.For poloidal modes with peak frequencies greater than 5 mHz, the significant occurrence range is closer to the Earth than that of the poloidal modes with peak frequencies between 2 and 5 mHz.

Frequency Distribution
Figures 4a and 4c depict the peak frequency distribution of toroidal and poloidal modes outside the plasma plume.
It is evident that both modes exhibit a decreasing trend with increasing radial distance represented by L, consistent with the Alfvén continuum theory (e.g., Allan & Poulter, 1992;Waters et al., 2000;Zhang et al., 2018).The frequency of toroidal waves is slightly lower on the dusk side than on the dawn side, in agreement with the findings of Takahashi et al. (2015).They attributed this asymmetry to higher particle density from the ionosphere in the near-Earth dusk region.In contrast, the peak frequency of poloidal waves does not exhibit significant dawndusk asymmetry, which differs from previous results (Liu et al., 2009).We attribute this difference to Liu et al.'s observations not distinguishing between the inner and outer regions of the plasma plume.In addition, it can also be seen that the frequency of poloidal waves is significantly higher than that of toroidal waves.This may imply that the toroidal wave is dominated by fundamental mode, while the second harmonic mode may be the most common phenomenon for poloidal waves.Furthermore, we have observed that the frequencies on the nightside are higher than those on the dayside.The specific reasons require further investigation in the future.However, one possible explanation is that the distinct excitation mechanisms of the night-side waves and the near-noon waves may result in more harmonics on the night-side.For instance, the poloidal modes on the night-side may predominantly be the local second harmonics excited by the drift-bounce resonance, while the poloidal modes near noon may also include the FLRs generated by the cavity and waveguide modes induced by solar wind dynamic pressure variations (Elsden & Wright, 2022;Liu et al., 2009;Wright et al., 2018).Figures 4b and 4d illustrate the peak frequency distribution within the plume.Compared to the outside, both wave modes exhibit lower peak frequencies within the plume.This alignment with the FLR theory suggests that the higher plasma density within the plume, relative to its exterior, leads to lower eigenfrequency.Note that the red and yellow bins in the upper portion of Figure 4b are caused by second harmonic waves and there are less than three events in each bin.

Power Distribution
Figures 5a and 5c present the peak power distribution outside the plume structure.For the fundamental mode, there is a node of magnetic field perturbations at the magnetic equator (Takahashi et al., 2015), so we used the electric field to calculate the wave power.We used the square root of the integral of the power spectral density instead of the power directly (Liu et al., 2009).
where δE r and δE ϕ are in units of mV/m.Both modes exhibit an increase with the radial distance represented by L, consistent with previous findings (Liu et al., 2009;Takahashi et al., 2015).Specifically, toroidal modes are stronger on the flank side, with dawn-side power exceeding dusk-side power, attributed to stronger KHI on the dawn side.Poloidal modes also show a slight enhancement on the flank side, but their average amplitude is only half that of toroidal modes.We propose that these enhancements may arise from the FLR of the toroidal mode and the compressional fast wave with a poloidal component (Elsden & Wright, 2022).On the night side, an asymmetry in power distribution is observed, with greater wave power observed before midnight.This aligns with prior theories suggesting that magnetospheric convection is one of the sources of ULF waves during substorm intervals (e.g., McPherron, 2005).Figures 5b and 5d display the peak power distribution within the plume.Both wave modes exhibit higher power levels at noon compared to the dusk side.The enhancement of poloidal modes at noon may result from ULF generated by solar wind pressure variations at the magnetopause (e.g., Liu et al., 2009;Elsden & Wright, 2022).In addition, trapped MHD fast mode waves enable the excitation of higher amplitudes by a cavity resonance (Degeling et al., 2018).The possible reason of the increase in toroidal mode wave power on the flank side could be the FLRs with strong response to the solar wind dynamic pressure at the edge of the plume structure (Elsden & Wright, 2022).This also implies that the presence of plume prevents the fast modes from propagating to the dusk side (Degeling et al., 2018).

Conclusion
In this study, we conducted a statistical analysis of Pc4-5 ULF waves in the radial distance of L = 6 ∼ 12 utilizing five years of THEMIS data.We provide insights into the occurrence rate, frequency distribution, and power distribution of these ULF waves.Furthermore, we distinguish between ULF waves within and without the plume region.Our analysis yields the following results: 1.In the absence of plasmaspheric plume, the distribution of wave characteristics of poloidal modes and toroidal modes are very similar, suggesting that there should be a certain relationship between them, which can be explained by FLR theory, that is, the coupling between fast waves with radial component and toroidal modes.
The majority of waves are toroidal modes.2. In the presence of plasmaspheric plume, both the wave frequencies of poloidal modes and toroidal modes are significantly lower than that in the absence of plasmaspheric plume.The majority of waves are poloidal modes.3.In the presence of plasmaspheric plume, both the wave power of poloidal modes and toroidal modes are higher near noon, while the wave powers are higher at two flank sides in the absence of plasmaspheric plume.
These findings imply a connection between the generation and transmission of ULF waves and the density patterns within the plasmaspheric plume.The occurrence rate and power of wave modes within the plumes may also support the hypothesis that these modes are driven externally (Elsden & Wright, 2022).Notably, the distinct sources of poloidal modes on the dawn and dusk sides underscore a fundamental disparity.This insight serves as a valuable addition to the work of Liu et al. (2009), where the influence of the plasmaspheric plume was overlooked.Both studies collectively emphasize the significant impact of plumes on the spatial distribution of ULF waves, highlighting their indispensable role in comprehending the dawn-dusk asymmetry in wave dynamics.This work was supported by the National Natural Science Foundation of China (11905109, 11905080, and 11947238), Shenzhen Municipal Collaborative Innovation Technology Program -International Science and Technology (S&T) Cooperation Project (GJHZ20220913142609017), the "Fourteen Five-Year Plan" Basic Technological Research Project (Grant Nos.JSZL2022XXXX001), and the fellowship of China Postdoctoral Science Foundation (2023M731505).We would also like to thank Pro.f Liu Kaijun for helpful discussions.

Figure 1 .
Figure 1.The plasma plume penetration event that occurred on 30 January 2015 from 11:35 to 13:15 UT.(a) The THEMIS-a satellite orbit during the event (solid red line), with dashed black line representing the orbit from 10:30 to 14:30 UT.The green area represents the plasmasphere and plume structure region at 12:30 UT, obtained from particle simulations in Goldstein et al. (2014).(b) Total electron density during the event.The dashed red lines indicate the edges of plume.(c, d) Compression magnetic field perturbations and their wavelet transforms.(e, f) Velocity perturbations in the azimuthal direction and their wavelet transforms.(g, h) Velocity perturbations in the radial direction and their wavelet transforms.The dashed lines indicate the identified quasi-monochromatic waves.(i, j) The red and blue curves represent the power of velocity perturbations in the azimuthal and radial directions, respectively, corresponding to the dashed time interval.

Figure 2 .
Figure 2. Statistics of observation time and occurrence rate.The variables were averaged in bins of 1 R E from 6 to 12 R E in radial distance and 15°in azimuth.Bins of the total observation duration of less than 4 hr have been removed.(a, b) Global distribution of observation times without and within the plasma plume.(c, d) Occurrence rates of the toroidal modes without and within the plasma plume.(e, f) Occurrence rates of the poloidal modes without and within the plasma plume.

Figure 3 .
Figure 3. Occurrence rate of different frequency of waves without the plasma plume.(a, b) Toroidal mode and poloidal mode occurrence rate with peak frequency of 2-5 mHz.(c, d) Toroidal mode and poloidal mode occurrence rate with peak frequency greater than 5 mHz.

Figure 4 .
Figure 4. Global distribution of (a, b) wave peak frequency of toroidal mode without and within the plasma plume, (c, d) wave peak frequency of poloidal mode without and within the plasma plume.

Figure 5 .
Figure 5. Global distribution of (a, b) the averaged wave amplitude of toroidal mode without and within the plasma plume, (c, d) the averaged wave amplitude of poloidal mode without and within the plasma plume.