Multiband Electrostatic Waves below and above the Electron Cyclotron Frequency in the Near-Sun Solar Wind

Using the Parker Solar Probe measurements, this Letter reports two new types of multiband electrostatic waves in and near the heliospheric current sheet. They are classified into the f < f ce and f > f ce multiband electrostatic waves, in which most (or all) of the bands in the former type are lower than f ce, and all of the bands in the latter type are higher than f ce, where f and f ce denotes the wave frequency and the electron cyclotron frequency, respectively. This Letter also exhibits observational evidence of the existence of nonlinear wave–wave interactions of both types of electrostatic waves. In particular, the f > f ce multiband electrostatic waves are found to be modulated in the presence of low-frequency oblique ion-scale waves. According to the observed frequency distribution, this Letter proposes that the mode nature of the f < f ce multiband electrostatic waves could be the oblique ion acoustic wave or the lower-hybrid wave, and the f > f ce multiband electrostatic waves are the electron Bernstein mode wave. These findings provide a challenge to understand the complex electron and ion dynamical processes in and near the heliospheric current sheet.


Introduction
One mystery in the solar atmosphere and the near-Sun solar wind is how the energetic particle population and the associated high-frequency waves interact with each other and evolve. When these energetic particles contain free energy behaving (possibly, but not the only possibility) as the beam, ring, ring beam, or loss cone in the particle velocity distribution function, they can radiate different kinds of high-frequency plasma waves (e.g., whistler waves, ion acoustic waves, electron acoustic waves, electron Bernstein waves, and high-frequency electromagnetic waves) under different plasma parameters (e.g., Ashour-Abdalla & Kennel 1978;Umeda et al. 2007;Chen et al. 2020;Shi et al. 2020;Wu et al. 2020;Zhou et al. 2020;Sun et al. 2021). Before the launch of the Parker Solar Probe (PSP; Fox et al. 2016), only remote radio observations could enable indirect conjectures regarding the dynamical process between high-frequency waves and high-energy electrons in the solar atmosphere and its neighborhood (e.g., Bastian et al. 1998). Nowadays, in situ measurements by PSP provide a unique opportunity to directly investigate both high-frequency waves and high-energy particles in the inner heliosphere. This Letter focuses on high-frequency waves between the lower-hybrid frequency f LH and the electron plasma frequency f pe detected by PSP.
From PSP observations, Malaspina et al. (2020) first reported the existence of multiband high-frequency electrostatic waves in the near-Sun solar wind below 50 solar radii, in which the wave power densities are peaked at the frequency f ∼ 0.7f ce and f ∼ f ce and their harmonics (also see Ma et al. 2021), where f ce is the electron cyclotron frequency. Malaspina et al. (2020) also found that these waves preferentially are located in the region where the magnetic field is quiet and nearly following the Parker spiral lines. The detected waves at f ∼ f ce are proposed to be the electron Bernstein wave; however, the nature of the waves at f ∼ 0.7f ce is unknown (Malaspina et al. 2021). Malaspina et al. (2021) further found two new types (Type B and Type C) of narrowband electrostatic waves near f ce in a steady magnetic field environment; however, their mode nature is still unknown. Moreover, ion acoustic waves extending to the ion plasma frequency f pi were observed in the near-Sun solar wind (Mozer et al. 2020(Mozer et al. , 2021a. Mozer et al. (2021b) recently found a new continuous narrowband electrostatic wave below f pi , which is interpreted as the ion acoustic mode wave.
Based on PSP measurements, this Letter will report two new types of multiband electrostatic waves observed in and near the heliospheric current sheet. Their frequency distributions are obviously different from the electrostatic waves found by Malaspina et al. (2020Malaspina et al. ( , 2021 and Mozer et al. (2021b). According to their frequency distribution features, they are classified into the f < f ce and f > f ce multiband electrostatic waves. For the f < f ce type waves, most (sometimes all) of the frequency bands are below f ce , and for the f > f ce type waves all of the frequency bands are above f ce . Moreover, this Letter will provide direct observational evidence for nonlinear wave −wave interactions relating to both the f < f ce and f > f ce multiband electrostatic waves.

Data and Event Overview
We analyze the data from 20:30:00 UT on 2020 September 24 to 00:00:00 UT on 2020 September 26 measured by PSP, which is in Solar Encounter 6 and at ∼28 solar radii (∼0.13 au). The plasma parameters including the solar wind velocity, the proton number density, the proton temperature, and the electron pitch angle distribution are measured by the SWEAP instrument suite (Kasper et al. 2016). The magnetic and electric field data come from the FIELDS . The magnetic field from the FGM has a resolution of ∼292 Hz. The electric field spectra we use include AC-coupled survey spectra with a ∼0.9 s cadence and DC-coupled survey spectra with a ∼28 s cadence. The electric field waveform burst data are sampled at 35,000 per second in E6 (Malaspina et al. 2016). In addition, we use the FIELDS RFS electric field spectrum to estimate the electron number density by using the QTN method (Moncuquet et al. 2020). Figure 1 presents an overview of our high-frequency wave event. Spectrograms of the AC and DC differential voltage V 12 power spectra are presented in Figures 1(a) and (b), respectively, where the white solid line denotes f ce . We see that multiband electrostatic waves near f ce appear from ∼22:00:00 UT on 2020 September 24 to ∼05:30:00 UT on 2020 September 25 (labeled by Regime I in Figures 1(a)). The frequency distribution of these observed waves is similar to that of the waves found by Malaspina et al. (2020Malaspina et al. ( , 2021, i.e., in the wave power centering at ∼ 0.7f ce and ∼ f ce , as well as the harmonics. We also find two more new types of multiband electrostatic waves during ∼08:30:00−19:30:00 UT on 2020 September 25 (labeled by Regime II in Figures 1(a)), in which the strongest wave power is either lower than f ce or higher than f ce . Different from electrostatic waves near f ce that mainly occupy a steadily radial magnetic field environment (Regime I; |B R |  2|B T,N |), the new types of electrostatic waves arise in a relatively disturbed magnetic field environment (Regime II), as shown by the magnetic field distribution in Figure 1(c). We also filter the three magnetic field components in the lowfrequency range of 0.001-1 Hz to quantify the disturbed magnetic field in Regime I and Regime II, and we find that the averaged peak of the disturbed magnetic field is about 0.02 (0.1) times the background magnetic field in Regime I (II). Figure 1(c) further shows that the direction of the magnetic field is radially outward (inward) before (after) the PSP crosses Regime II, and the magnetic field direction roughly changes five times in Regime II. Moreover, from the distributions of the solar wind speed (Figure 1(d)), the proton and electron density (Figure 1(e)), and the proton temperature (Figure 1(f)), we see that the averaged solar wind speed and the proton temperature in Regime II are nearly the same as those in Regime I; however, the proton density in Regime II is nearly 4-8 times larger than that in Regime I. Furthermore, to identify the magnetic configuration, Figure 1(g) presents the pitch angle θ distribution of electrons with an energy of 315 eV. The θ ∼ 0°a nd 180°outside Regime II indicate the existence of the steady open magnetic field, because the strahl electrons stream outward from the Sun in the open magnetic field. The pitch angle of these strahl electrons is broader in Regime II. According to the observed distributions of the magnetic field and the pitch angle distribution of the strahl electrons, we conclude that the two new types of electrostatic waves in Figure 1 occur in and nearby the heliospheric current sheet.
In order to illustrate the difference between these new types of multiband electrostatic waves and the multiband waves found by Malaspina et al. (2020), we will exhibit their electric field spectra in four typical events (occurring at ∼10:24:07, 15:08:56, 15:37:39, and 18:24:13 UT on 2020 September 25) labeled by the dashed-dotted lines in Figure 1.

The f < f ce and f > f ce Multiband Electrostatic Waves
We plot all spectral power densities (PSDs) of the burst mode data from the electric field measurement in Regime II. Of all 49 burst mode events on 2020 September 25, 29 events contain only the f < f ce multiband electrostatic waves, 5 events contain the f > f ce multiband electrostatic waves, and 3 events contain both types of waves. We choose four typical events to illustrate the main features of the observed waves (see Figure 1): Event #1 and Event #2 for the f < f ce multiband electrostatic waves, Event #4 for the f > f ce multiband electrostatic waves, and Event #3 for the mix of two types of waves.
First, we analyze the f < f ce type multiband electrostatic waves in Events # 1 and #2, in which the strongest wave power occurs below the f ce .
Figures 2(a1)-(f1) present the PSDs of the electric fields E x and E y and summarize the information of the peak, bandwidth, and amplitude in each frequency band in Event #1. The solid and dotted curves in Figures 2(a1) and (d1) denote, respectively, f ce ; 801 Hz and f LH ; 19 Hz under the averaged magnetic field strength B 0 ; 28.6 nT. There exist six frequency bands in both the E x and E y PSDs. The first frequency band where the frequency is lowest contains the strongest wave power peaking at f 1 ; 200 Hz ; 0.25f ce (or ; 11f LH ). The other frequency bands center at the harmonics of f 1 , and the highest frequency band is higher than the f ce . The wave amplitudes δE x ( f n ) and δE y ( f n ), shown in Figures 2(c1) and (f1), are obtained by filtering the electric field data in the frequency range [f n − Δf n /2, f n + Δf n /2], where n = 1, ..., 6 denotes the number of the frequency band. We can see that both of the wave amplitudes decrease with increasing n, which implies the possibility that the harmonic waves (n > 1) are excited by the nonlinear interaction of the fundamental band wave (n = 1).
Figures 2(a2)−(f2) present the PSDs of the two electric field components and their key parameters in Event #2.
The lowest frequency f 1 ; 260 Hz ; 0.08f ce (or ; 3f LH Consequently, these two band waves could be emitted independently. According to the wave amplitudes in Event #2, the bands centered at f 2j+1 can be classified as one subgroup, and the bands at f 2j+2 correspond to another subgroup, where j = 1, ..., 5. The amplitude of the f 2j+2 band wave is normally larger than that of the f 2j+1 band wave at the same j. Moreover, f 2j+1 approximates the sum of f 2j and f 1 . Therefore, we conclude that one source of the f 2j+1 band wave could be the nonlinear interaction between the f 2j and f 1 band waves. We note that the nonlinear interactions among the second band waves can generate the f 2j band waves. Here, we summarize the basic features of the f < f ce multiband electrostatic waves: (1) the strongest wave power arises at a frequency much smaller than f ce ; (2) the waves can extend to the frequency larger than f ce ; and (3) the nonlinear wave−wave interactions are evident in the presence of the two band waves with the finite amplitude. Now, we proceed with the analysis of the f > f ce multiband electrostatic waves in Events #3 and #4. These waves have distinct frequency distributions, as shown in Figures 3 and 4. Figure 3 shows the f > f ce multiband electrostatic waves in Event #3. These waves have four frequency bands, in which the powers center at ∼1.7f ce , 2.5f ce , 3.3f ce , and 4.2f ce . The f < f ce multiband electrostatic waves also arise in Event #3, in which the observed features are similar to those in Event #2. Moreover, Event #3 contains the Type B narrowband electrostatic wave found by Malaspina et al. (2021). A new finding of the Type B narrowband electrostatic wave is the coexistence of fundamental and second harmonic waves (see Figure 3(a)).  Figure 4 shows the f > f ce multiband electrostatic waves in Event #4. The four bands are nearly centered at 3.9f ce , 4.9f ce , 5.6f ce , and 6.8f ce . Actually, the number of the frequency bands is more than four (the band higher than 17,500 Hz cannot be measured due to the burst data sample rate used on 2020 September 25). Moreover, these four bands are higher than the ion plasma frequency f pi . 10 -7 10 -5 10 -3 10 -1 10 1 10 -7 10 -5 10 -3 10 -1 10 1 Figure 2. The PSDs of the electric field components E x and E y and the wave parameters in Events #1 and #2. Panels (a1)−(c1): spectrogram of the E x PSD, the averaged E x PSD, and wave parameters including the peak, bandwidth, and amplitude in each E x frequency band in Event #1. Panels (d1)−(g1): spectrogram of the E y PSD, the averaged E y PSD, and wave parameters including the peak, bandwidth, and amplitude in each E y frequency band in Event #1. Panels (a2)−(g2) for Event #2: the description is the same as for panels (a1)−(g1). The solid and dotted white lines denote the electron cyclotron frequency f ce and the lower-hybrid frequency f LH , respectively.
Another important finding in Figure 4 is the modulation of the f > f ce multiband electrostatic waves by low-frequency waves. Figures 4(a) and (d) exhibit the periodical change of the wave power at a timescale of ∼1/3 s. This timescale is coincident with the period of oblique ion-scale waves appearing in Event #4 (not shown). Figures 4(g)-(j) present the wavelet crosscoherence spectra between the strength of the f > f ce multiband electrostatic waves and the magnetic field perturbations of oblique ion-scale waves to exhibit their correlation. Here, we use the following procedures to obtain the strength of the f > f ce multiband electrostatic waves: (1) both δE x and δE y in burst data are filtered in the range [f pi , 17,500 Hz]; and (2) the filtered field strength, |δE x ( f > f pi )| and |δE y ( f > f pi )|, are resampled at the FGM resolution. For the magnetic field perturbations δB x and δB y of low-frequency waves, they are filtered in the range 1-10 Hz. Figures 4(g)-(j) illustrate that the high correlation coefficients (0.75) arise at f ∼ 3 Hz as the f > f ce multiband electrostatic waves have a finite amplitude. This is strong evidence of the modulation of the f > f ce multiband electrostatic waves by the low-frequency oblique ion-scale waves.
The basic features of the f > f ce multiband electrostatic waves are summarized as: (1) the frequency bands are located either between nf ce and (n + 1)f ce (n = 1, 2, 3, and 4) or near mf ce (m = 4, 5, 6, and 7); (2) the frequency difference between two neighboring bands is about f ce ; and (3) the waves can be modulated by low-frequency electromagnetic waves.

Summary and Discussion
This Letter provides observational evidence of two new types of multiband electrostatic waves beyond the multiband electrostatic waves near f ce found by Malaspina et al. (2020Malaspina et al. ( , 2021: one is most (or all) of the frequency bands below f ce , and the other is all of the frequency bands above f ce . Their observational features are explored in Section 3. Here, we proceed with further discussions on their mode nature and excitation mechanism.
The wave frequency and wavevector k in the plasma frame are the key parameters in determining the mode nature. Due to the Doppler frequency shift induced by the bulk flow of the solar wind particles, the relation between the wave frequency in the plasma frame f pl and the frequency in the spacecraft frame f sc is given as There are three possibilities associated with the wave frequency: (1) f pl = f sc ; (2) f pl ∼ f sc ; and (3) f pl ? f sc . The first possibility corresponds to 2πf sc ; k · V 0 , which results in two predictions, i.e., the wavenumber along V 0 approximates as k f V 2 V sc 0 0 p , and the wave phase speed v p = 2πf pl /k is much smaller than V 0 . In the second possibility, the wavenumber can be still approximated by the expression k f V 2 V sc 0 0 p . The third possibility occurs as the wave propagates against V 0 , in For the mode nature of the f < f ce multiband electrostatic waves, since there is no eigenmode of the plasma wave existing at multiple frequencies shown in Events #1 and #2, we can exclude the second possibility of the wave frequency in the plasma frame, i.e., f pl ∼ f sc . According to f sc and V 0 in Events #1 and #2, the wave scale is approximated as =~km in Event #1 and λ = 1.0 km in Event #2 in the first frequency band. Due to the ion (electron) inertial length being λ p ; 6.0 km (λ e ; 0.14 km) in Event #1 and λ p ; 6.7 km (λ e ; 0.16 km) in Event #2, the first frequency band wave is located on a sub-ion scale. For higher frequency bands with f n , the wave scale is about λ( f 1 /f n ), and the wave scale can even extend down to the electron scale, e.g., ∼0.08 km for the f 12 band wave in Event #2. Therefore, the candidate for the observed (quasi-)electrostatic mode between the ion and electron scales could be the ion Bernstein wave, the ion acoustic (or beam) wave, or the lowerhybrid wave. For the ion Bernstein wave, the frequency difference between two neighboring bands is on the order of the proton cyclotron frequency, which is inconsistent with the frequency feature, i.e., f 2 − f 1 ; 0.25f ce in Event #1 and f 2 − f 1 ; 0.08f ce in Event #2. As a consequence, the mode nature of the f < f ce multiband electrostatic waves could be the ion acoustic (or beam) wave or the lower-hybrid wave.
In order to further identify the wave mode, we analyze the correlation between two electric field components for the first frequency band wave in Event #1 and the first and second band x y | | | | d d ¢~¢ for the second frequency band wave. The former exhibits right-handed elliptical polarization, and the latter is linearly polarized in the x-y plane. We note that due to |B x | = |B y | in the spacecraft frame in Event #2, we rotate the magnetic field data in a frame in which B 0 x ¢ = . Although the values of E E x y Y d d of the first and second band waves in Event #2 are similar to feature (1) of ion acoustic waves, both the waves have a large electric component perpendicular to the local magnetic field, and this is different from feature (2). Consequently, we can exclude the possibility of an ion acoustic wave propagating along the magnetic field. However, we cannot exclude the possibility of an oblique ion acoustic wave. We also analyze the electric field distribution of one f < f ce electrostatic wave event (∼11:31:49−11:32:02 UTC) in which the magnetic field is nearly located in the spacecraft spin plane (e.g., B B B 0.3 z x y 2 2 +~), and we find the electric field component perpendicular to the magnetic field E ⊥ is much stronger than in the parallel electric field component E ∥ .