Parker Solar Probe evidence for scattering of electrons in the young solar wind by narrowband whistler-mode waves

Observations of plasma waves by the Fields Suite and of electrons by the Solar Wind Electrons Alphas and Protons Investigation (SWEAP) on Parker Solar Probe provide strong evidence for pitch angle scattering of strahl-energy electrons by narrowband whistler-mode waves at radial distances less than ~0.3 AU. We present two example intervals of a few hours that include 8 waveform captures with whistler-mode waves and 26 representative electron distributions that are examined in detail. Two were narrow; 17 were clearly broadened, and 8 were very broad. The two with narrow strahl occurred when there were either no whistlers or very intermittent low amplitude waves. Six of the eight broadest distributions were associated with intense, long duration waves. Approximately half of the observed electron distributions have features consistent with an energy dependent scattering mechanism, as would be expected from interactions with narrowband waves. A comparison of the wave power in the whistler-mode frequency band to pitch angle width and a measure of anisotropy provides additional evidence for the electron scattering by whistler-mode waves. The pitch angle broadening occurs in over an energy range comparable to that obtained for the n=1 (co-streaming) resonance for the observed wave and plasma parameters. The additional observation that the heat flux is lower in the interval with multiple switchbacks may provide clues to the nature of switchbacks. These results provide strong evidence that the heat flux is reduced by narroweband whistler-mode waves scattering of strahl-energy electrons.


Introduction
Whistler-mode waves have long been proposed as a potential mechanism for scattering of solar wind electrons. Many previous theoretical arguments indicated that the waves must propagate sunward to match the resonance condition for waves propagating parallel to the interplanetary magnetic field (Vocks et al., 2005;Saito and Gary, 2007a;Roberg-Clark et al., 2016). Several studies have presented evidence for parallel propagating whistler-mode waves (Lacombe et al., 2014;Stansby et al.,2016;Tong et al., 2019); however, only few examples of waves propagating sunward have been found (Agapitov et al., 2020;Cattell et al., 2021a). Although most theoretical and simulation studies have assumed parallel propagation, Pistinner and Eichler (1998) examined the role of obliquely propagating whistlers, and concluded that they can control the electron heat flux. Utilizing particle-in-cell (PIC) simulations with an imposed electron heat flux, Roberg-Clark et al. (2016;2018) showed that large amplitude oblique whistlers controlled the heat flux in high beta plasmas.
Electric field waveform captures from STEREO at 1 AU provided evidence that large amplitude narrowband whistlers were obliquely propagating, enabling resonant interactions with electrons without the need for sunward propagation (Breneman et al., 2010;Cattell et al., 2020). Similar obliquely propagating whistler-mode waves are observed by Parker Solar Probe (PSP) inside ~0.3 AU (Cattell et al., 2021a;Agapitov et al., 2020), in addition to waves propagating close to the magnetic field, which are sometimes sunward propagating.
Many possible instability mechanisms have been proposed, including temperature anisotropy (Gary and Wang, 1996), heat flux instabilities (Forslund, 1970;Feldman et al., 1975;Gary, 1978;Gary et al.,1975;Shaaban et al.,2018;Roberg-Clarke et al., 2018), heat flux fan instability (Bošková et al., 1992;Krafft and Volokitin, 2003;Vasko et al., 2019; oblique heat flux whistler/fast magnetosonic instability (Verscharen et al., 2019); and electron beam instabilities (Sauer and Sydora, 2010). A number of studies have examined electron properties, to determine instability mechanisms for the observed whistlers (Lacombe et al., 2014;Stansby et al., 2016;Tong et al., 2019). One likely mechanism for generating the oblique waves observed in the STEREO (Cattell et al., 2020), and Parker Solar Probe (Cattell et al. 2021a) waveform capture data is the heat flux fan instability. The electron beam mechanism might operate also, but the necessary beams have not been observed. Agapitov et al. (2020) concluded that temperature anisotropy in concert with magnetic field gradients might destabilize whistlers observed by PSP. A few authors have made direct comparisons of whistlers and electron distributions for evidence of scattering by the waves. Kajdič et al. (2016) found that strahl electron distributions were broader when parallel propagating whistlers were observed when compared to intervals without waves. Gurgiolo et al. (2012) showed that halo electrons were scattered from the strahl, but did not see any evidence of the type of waves needed to provide the scattering.
Evolution of the electron distributions has been examined using data from satellites at radial distances from the Sun ranging from Parker Solar Probe inside ~.2 AU (Halekas et al., 2020a(Halekas et al., , 2020b to Ulysses at ~4 AU (Štverák et al., 2009). These observations are consistent with the existence of significant wave scattering in addition to collisional effects (Maksimovic et al., 2005;Štverák et al., 2009;Wilson III et al.,2000).
In this letter, we present the first direct evidence, using waveform capture data and electron distributions, for scattering of solar wind electrons by narrowband whistler-mode waves at distances less than 0.3 AU from the Sun.
Statistical properties of these waves were presented by Cattell et al. (2021a).
Data sets are presented in section 2. Overviews of two intervals showing wave spectra, electric and magnetic field waveforms, and electron pitch angle data, statistical comparison of wave power to broadening of electron pitch angle distributions, and a set of shorter intervals with electron distributions are presented in section 3. Discussion and conclusions are given in section 4.

Data sets and overview
We utilize data from the Parker Solar Probe Fields (Bale et al., 2016) and Solar Wind Electrons Alphas and Protons Investigation (SWEAP) (Kasper et al., 2016) instrument suites. From FIELDS (Bale et al., 2016), we utilize the Level 2 waveform capture data for the three electric field components and the three search coil magnetic field components obtained during the first solar encounter. The waveform data utilized in this study were obtained for ~3.5 s intervals at ~150 ksamples/s. Storage and transmission of these data was controlled by a quality flag, and in the first encounter dust impacts often triggered the quality flag. Therefore, the waves observed in this data set are often not the largest that occur. To further examine the occurrence, duration and amplitudes of the whistler-mode wave activity, we show one electric field and one magnetic field channel in the DC coupled bandpass filter (BPF) data which is obtained at a cadence of 1 spectrum every ~1.7 s, over a frequency range from 0.4 Hz to 4000 Hz. We also utilized one electric field and one magnetic field channel in the higher frequency resolution DC coupled spectral data, which is obtained at a rate of 1 spectra every ~28 seconds, over a frequency range of ~10 Hz to 4.8 kHz (Malaspina et al., 2016). The Level 2 quasi-static magnetic field data in RTN (radial-tangential-normal) coordinates at ~4 samples per second are used for determining pitch angles and the background magnetic field structure.
The electron parameters were obtained from the SWEAP Solar Wind Electrons Alphas and Protons Investigation (Kasper et al., 2016) Solar Probe Analyzers (SPAN-A-E and SPAN-B-E)(Whittlesey et al. 2020). We show pitch angle distributions for energies from ~20 to 2000 eV, covering core, halo and strahl (Halekas et al., 2020a;Halekas et al., 2020b). A complete energy-angle scan is obtained in .256 Cy (~.87 s), which are summed on board to obtain a distribution every ~28 s. The solar wind velocity, used to transform distributions into the plasma frame, was obtained from the Level 2 Solar Probe Cup (SPC) moments (Case et al., 2020). For some intervals, we also show heat flux. The solar wind density, and electron core and nonthermal temperatures were obtained from the Fields Quasi-thermal Noise (QTN) data (Moncuquet et al., 2020). Figure 1 presents an overview of one interval that contains large amplitude narrrowband whistler-mode waves (2018 November 2 from 11:10 to 13:10 UT) that we will discuss in detail. Two different waves types are clearly distinguishable in the electric field (panel a) and magnetic field (panel b).

Observations of whistlers and electron scattering
The waves at harmonics of the electron cyclotron frequency (fce), intermittently from ~11:10 to 11:50 UT and again at ~12:16 and 12:24 UT in panel a, have been identified as electrostatic whistler/Bernstein waves seen primarily in regions of quiet radial magnetic field (Malaspina et al., 2020).
The electron pitch angles are narrowly field-aligned as expected for strahl, indicating that these waves do not strongly scatter electrons at these energies (panels d, e and f). Note that both the electric field spectrum and the waveform capture 2 indicate that much weaker electron Bernstein waves occurred simultaneously with the whistlers around 1229 UT. The whistler waves, observable in panels a and b from ~12:25 to ~12:37 at <~0.1 fce, occur when the magnetic field is more variable and usually smaller. Strong scattering at energies from ~200 to ~800 eV is seen in association with the most intense whistler waves from ~1227 to ~1231 UT. Scattering at the lower energies continues in concert with the weaker waves. This association provides strong evidence for scattering of strahl-energy electrons by narrowband whistlers.   Figure 2, panel 1, which plots the entire 3.5 s waveform capture, illustrates the wave packet structure and variability often seen in the narrowband whistlers; the waveform for one packet is shown in panel 2. The waveforms in panels 2, 4, 5 and 6 are coherent; the example in panel 3 is less monochromatic, and, in addition, has the signature of higher frequency waves (also seen in Figure 1h). In both Figures 1 and 2 The waveform capture data show that the whistler wave packets vary on sub-second time scales. The ~28s averaged spectral data often under-estimate the wave amplitudes and ,as expected, miss the highly variable nature of the waves that is clear in waveforms and in the BPF data. The waveforms in Figure   2 show that individual wave packets can have durations of <~0.1 s. The electron distributions have the same ~28 s cadence as the spectral data, and are, therefore, averages over regions that could include strong waves interspersed with weak or no waves. For this reason, we would not expect a one-to-one correspondence between the pitch angle broadening and wave amplitudes. In addition, electrons may have interacted with waves upstream of the observations.
Utilizing the complementary information provided by high time resolution BPF data and the better frequency resolution spectral data, and a technique developed to study whistler-mode waves in the radiation belts (Tyler et al., 2019a(Tyler et al., , 2019b, both the frequency and peak power of the whistler-mode waves can be more accurately determined at the higher time resolution of the BBF data. We compare these power values to two methods to assess the electron broadening. Because there are cases of very broad distributions(see examples in Figure 4, panels g and h)when the peak flux is not at 180° (away from the sun), the pitch angle width was defined to be the width at half height between the maximum flux (using the actual pitch angle measurement for non-180° peaks or the splined value at 180° pitch angle if 180° is the peak flux) and the minimum flux. Note that the plotted values are full-widths. The second method is an 'anisotropy,' defined to be the measured maximum flux over the minimum flux between 90° and 180° degrees. When the strahl is narrowly peaked, the 'anisotropy' is large; when the distribution is very broad, the anisotropy is very small. Figure 3 plots the pitch angle width and 'anisotropy' for the energy band centered at 314 eV versus the magnetic power in the whistler-mode frequency band for the two days containing the shorter intervals shown in UT and 10:43:58 UT. The occurrence of peaks that are not at 180° in some energy bands (seen most clearly in panels g and h, but also observable at high energies in d and e) is consistent with the results of our particle tracing code (Vo et al., 2020;Cattell et al., 2021b). Table 1 presents a qualitative relationship between the whistler-mode wave amplitudes and durations and the twenty-six representative energy-pitch angle distributions (categorized as narrow, broadened and very broad, as discussed above) that were obtained when the distributions were not affected by the rapid changes in the magnetic field (13 on 2018 November 2, and 13 on 2018 November 3). As discussed above, a perfect correlation would not be expected because the obliquity of the waves varies, the electrons may have interacted with waves upstream of the observation location, and due to the mismatch between wave packet and distribution function timescales. Waves at the time of each distribution were categorized as: (1) very intense (BPF electric field amplitude >4 x10 -4 V, BPF magnetic field >.05 nT, spectral electric field >1x10 -8 V 2 /Hz and spectral magnetic field >5.0x10 -4 nT 2 /Hz, in the appropriate frequency band); (2) intense (meeting criteria for very intense for only part of the interval, or only for the BPF or the spectra, or occurring in all channels at reduced intensity); (3) moderate (occurring for part of interval with reduced intensities(BPF electric field amplitude >1.5 x10 -4 V, BPF magnetic field >.015 nT, spectral electric field >1x10 -9 V 2 /Hz and spectral magnetic field >1.0x10 -5 nT 2 /Hz, in the appropriate frequency band); (4) waves detected only in the electric field; and (5) very weak intermittent waves or no detectable waves. Table 1 summarizes the comparison of distribution characteristics and wave amplitudes, which can also be inferred from Figure   3. Of the two narrow distributions (blue arrows), the narrowest, Figure 4b, occurred when the waves were below the threshold, and Figure 4a

Discussion and conclusions
The intervals shown above provide strong evidence for scattering of solar wind electrons at energies of ~100 eV to ~1 keV by narrowband whistlermode waves to produce the halo and reduce the electron heat flux. This is consistent with the results of a particle tracing code (Vo et al., 2020;Breneman et al., 2010) for wave and plasma parameters based on 1 AU measurements. Strong scattering by oblique whistlers has been observed in PIC simulations of the solar wind (Micera et al., 2020), and in other contexts including solar flares (Roberg-Clark et al., 2019), the radiation belts (Katoh and Omura, 2007;Allanson et al., 2020), and high beta astrophysical plasmas (Roberg-Clark et al., 2018;Komarov et al., 2018). PIC simulations have also provided evidence for scattering by parallel whistlers in the radiation belts (Camporeale and Zimbardo, 2015) and for the case of sunward propagating waves in the solar wind (Saito and Gary, 2007b).
The intense narrowband whistlers occur primarily in regions where the solar wind magnetic field is variable and often weaker, and in association with magnetic switchbacks, as also reported by Agapitov et al. (2020).
Statistics on narrowband whistlers inside ~.3 AU (Cattell et al., 2021a) confirmed this association. In several of the events we showed above, although the whistlers are strongest near the switchback edges, intermittent weaker waves fill the entire switchback. The occurrence of intense whistlers and electron scattering within switchbacks may aid in understanding their origin.
For the set of waveform captures obtained on 11/2/2018 and 11/3/2018, we calculated the range of resonant energies for the observed wave frequencies, wave angles, and wave vector magnitude, using the resonance condition, − $⃗ • ( $$$⃗ = Ω ( , where is the wave frequency, $⃗ is the wave vector, ( $$$⃗ is the electron velocity and Ω ( is the electron gyrofrequency. We include the Doppler shift of the wave frequency and the electron velocity relative to the solar wind flow. For these events, the ratio of our measured frequencies to the electron gyrofrequency is ~0.09 to ~.3. The phase velocities ranged from ~650 km/s to ~1100 km/s, larger than the solar wind speeds of ~250 to 350 km/s. The wave angles varied from ~3° to ~45°, and the magnitude of the wave vector varied from ~.6 to 2.5 km -1 . Resonant energies ranged from ~100 eV to 700 eV for the n=-1 resonance, and ~90 eV to ~400 eV for the n=1 resonance. For the distributions when waveform captures were not obtained, we can estimate the wave angle range and the phase velocity from the measured ∂E ∂B 0 ratios. These estimates increase the range of wave angles to ~70°, thus increasing the highest resonant energies to ~1 keV. The range of energies over which the electron distributions are broadened is consistent with the inferred resonant energies. For the Landau resonance, 1 2 ∥ = (∥ , the resonant energies range from a few to 10s of eV, which might be associated with the increase in core electron temperature seen in association with the largest whistlers. This core heating was also observed in a statistical study of whistler waves observed by PSP (Cattell et al., 2021).
For the three cases (11/2/2018 12:27:51 and 12:30:11, and 11/3/2018 10:43:58) where a waveform capture was obtained within the 28 s interval of one of the 26 electron distributions, we can compare the energies over which broadening occurred (from ~100 eV to ~1 keV) and determine the range of resonances involved. The waves propagated sunward and within 15° of the solar wind magnetic field; the energy associated with n=-1 was ~100 eV, and the n=-3 resonance reached energies of 900 eV to ~1 keV. The observed electron scattering was consistent interaction with resonances from n=-3 to n=-1 for these three events.
With the assumption that the waves are parallel propagating, many authors have shown that scattering rates are proportional to Ω ( 6 6 0 ∝ 6 0 (Brice, 1964;Kennel and Petschek, 1966;Lyons et al., 1972, andAlbert, 2017, for whistlers in the context of radiation belt; Kulsrud, 2005,  Our results are consistent with the conclusions of Halekas et al. (2020b) that the electron heat flux is regulated by the whistler fan heat flux instability and/or oblique fast magnetosonic/whistler mode instability. Given the wide range of observed wave angles, we cannot rule out the operation of other instability mechanisms at times, such as the temperature anisotropy Wang, 1996, Agapitov et al., 2020) or the heat flux instability (Gary, 1975;Shaaban et al., 2018). We show directly, for the first time, that the strahl energy electrons, which carry the heat flux, are strongly scattered by the narrowband whistler-mode waves, reducing the heat flux. The highly oblique wave mode is observed at 1 AU by STEREO (Cattell et al., 2020;Breneman et al., 2010), and, at this distance from the Sun, the heat flux fan instability is the most likely wave generation mechanism. These narrowband large amplitude whistler-mode waves are, therefore, the most likely candidates for regulating the electron heat flux and scattering of strahl electrons into the halo, as confirmed by PIC simulations (Micera et al., 2020) and the particle tracing results (Vo et al., 2020 Table 1. Qualitative comparison of electron distribution and wave properties. Narrow, broad and very broad refer to electron pitch angle widths, as described in the text. 1,2,3, 4 and 5 refer to wave characteristics, with 1 being the most intense, 4 having only electric field signatures, and 5 having very weak and intermittent or no waves. See text for details.