# Whistler Wave Observations by Parker Solar Probe During Encounter 1: Counter-propagating Whistlers Collocated with Magnetic Field Inhomogeneities and their Application to Electric Field Measurement Calibration

S. Karbashewski, O. V. Agapitov, H. Y. Kim, F. S. Mozer, J. W. Bonnell, C. Froment, Thierry Dudok de Wit, Stuart D. Bale, D. Malaspina, N. E. Raouafi

► To cite this version:

S. Karbashewski, O. V. Agapitov, H. Y. Kim, F. S. Mozer, J. W. Bonnell, et al. Whistler Wave Observations by Parker Solar Probe During Encounter 1: Counter-propagating Whistlers Collocated with Magnetic Field Inhomogeneities and their Application to Electric Field Measurement Calibration. The Astrophysical Journal, 2023, 947, 10.3847/1538-4357/acc527. insu-04087586

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# Whistler Wave Observations by Parker Solar Probe During Encounter 1: Counter-propagating Whistlers Collocated with Magnetic Field Inhomogeneities and their Application to Electric Field Measurement Calibration

S. Karbashewski<fnref n="1" />, O. V. Agapitov<fnref n="1" />,<fnref n="2" />, H. Y. Kim<fnref n="1" />, F. S. Mozer<fnref n="1" />, J. W. Bonnell<fnref n="1" />, C. Froment<fnref n="3" />, T. Dudok de Wit<fnref n="3" />,<fnref n="4" />, Stuart D. Bale<fnref n="1" />,<fnref n="5" />, D. Malaspina<fnref n="6" />,<fnref n="7" />, and N. E. Raouafi<fnref n="8" />

Received 2023 January 3; revised 2023 March 14; accepted 2023 March 15; published 2023 April 24

## Abstract

Observations of the young solar wind by the Parker Solar Probe (PSP) mission reveal the existence of intense plasma wave bursts with frequencies between 0.05 and 0.20f<sub>ce</sub> (tens of hertz up to ~300 Hz) in the spacecraft frame. The wave bursts are often collocated with inhomogeneities in the solar wind magnetic field, such as local dips in magnitude or sudden directional changes. The observed waves are identified as electromagnetic whistler waves that propagate either sunward, anti-sunward, or in counter-propagating configurations during different burst events. Being generated in the solar wind flow, the waves experience significant Doppler downshift and upshift of wave frequency in the spacecraft frame for sunward and anti-sunward waves, respectively. Their peak amplitudes can be larger than 2 nT, where such values represent up to 10% of the background magnetic field during the interval of study. The amplitude is maximum for propagation parallel to the background magnetic field. We (i) evaluate the properties of these waves by reconstructing their parameters in the plasma frame, (ii) estimate the effective length of the PSP electric field antennas at whistler frequencies, and (iii) discuss the generation mechanism of these waves.

Unified Astronomy Thesaurus concepts: Solar wind (1534)

## 1. Introduction

NASA's Parker Solar Probe (PSP), launched in 2018 August, is the first mission to collect in situ measurements of solar wind parameters within the orbit of Mercury (Fox et al. 2016). Following an initial gravity assist with Venus, PSP was inserted into its first encounter with the Sun and reached a perihelion of $35.67R_\odot$ (~0.166 au) from the Sun in November of 2018. Through a subsequent series of six gravity assists with Venus between 2019 and 2024, the perihelion of PSP's orbit will progressively shrink through 24 encounters to $9.86R_\odot$ (~0.046 au) (Fox et al. 2016). The mission is designed to address three core science objectives: (i) trace the flow of energy that heats the solar corona and accelerates the solar wind, (ii) determine the structure and dynamics of the plasma and magnetic fields at the sources of the solar wind, and (iii) explore mechanisms that accelerate and transport energetic particles. Wave-particle interactions involving both magnetohydrodynamic (MHD) and kinetic-scale waves, such as Alfvén and whistler waves, respectively, are ubiquitous in the solar wind and known to play an important part in addressing all three science objectives of the mission. In particular, wave-particle interactions are central to understanding the evolution

of the core, halo, and strahl suprathermal electron populations (Feldman et al. 1975).

The first encounter phase of PSP lasted from 2018 October 31 to 2018 November 12; during which the near corotation of the probe with an equatorial coronal hole meant the probe was immersed in a slow (less than 500 km s⁻<fnref n="1" />) and highly Alfvénic solar wind (Bale et al. 2019; Kasper et al. 2019). Halekas et al. (2020) reported that, during the encounter, the density and temperature of the core electron population increased with heliocentric distance and agreed with previous empirical radial scaling. The strahl population was more narrowly distributed at perihelion while the halo population nearly disappeared, similar to reports at 0.3 au by Štverák et al. (2009); given this, the strahl population dominated the suprathermal fraction by nearly a factor of 10 over the halo population at the closest approach. The observed fractional halo population observed (less than 1%) is significantly less than that reported between 1 and 4 au where it typically represents between 4% and 10% of the total population depending on the fitting procedures used (McComas et al. 1992; Štverák et al. 2009).

The evolution of the suprathermal populations is such that the halo increases with distance from the Sun while the strahl population decreases (Maksimovic et al. 2005; Štverák et al. 2009). This naturally leads to the interpretation of an exchange between the populations, with strahl electrons transferred to the halo through scattering in wave-particle interactions. There is strong evidence that whistler waves are, at least partially, responsible for this conversion process. Whistler waves are

<figure><img src="image_3.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA; sgkarbo@berkeley.edu</footnote>

<footnote n="2">²Astronomy and Space Physics Department, National Taras Shevchenko University of Kyiv, Kyiv, Ukraine</footnote>

<footnote n="3">³LPC2E, CNRS, CNES, University of Orléans, Orléans, France</footnote>

<footnote n="4">⁴International Space Sciences Institute, Bern, Switzerland</footnote>

<footnote n="5">⁵Physics Department, University of California, Berkeley, CA 94720-7300, USA</footnote>

<footnote n="6">⁶Astrophysical and Planetary Sciences Department, University of Colorado, Colorado, Boulder, CO, USA</footnote>

<footnote n="7">⁷USA Laboratory for Atmospheric and Space Physics, University of Colorado, Colorado, Boulder, CO, USA</footnote>

<footnote n="8">⁸Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA</footnote>
known to strongly interact with high-energy electron popula-
tions and have been observed to scatter the strahl in the solar
wind at 1 au (Kajdič et al. 2016) and control the dynamics of
the relativistic electron populations in the Earth’s radiation
belts (Horne 2007; Thorne 2010).

There was strong whistler wave activity throughout much of PSP's Encounter 1 and several authors have reported on these observations (Agapitov et al. 2020; Cattell et al. 2021a, 2021b, 2022; Dudok de Wit et al. 2022; Froment et al. 2022). Recently, whistler waves observed during Encounter 1 have been linked to the scattering of strahl at distances less than $50R_⊙$ (less than 0.25 au) from the Sun (Cattell et al. 2021a, 2021b; Jagarlamudi et al. 2021) including some of the cases reported in this article. The statistical study by Cattell et al. (2021b) of whistlers observed by PSP concludes that the waves are more intermittent closer to the Sun (less than 0.3 au) than at 1 au, have frequencies on the order of approximately $0.2f_{ce}$ and are often associated with higher magnetic activity. These periods of whistler wave activity are interspersed with higher frequency electron Bernstein modes in the range of $f_{ce}$ and its harmonics (Malaspina et al. 2020). Recent work has demonstrated that the observed electron Bernstein modes are most likely generated by the interaction between the solar wind and the Parker Solar Probe spacecraft wake (Malaspina et al. 2022; Tigik et al. 2022). Nonetheless, electron Bernstein waves are found to preferentially occur in regions of exceptionally low-amplitude magnetic turbulence, termed quiescent regions (Short et al. 2022). The lack of whistler observations in quiescent regions is consistent with the connection between whistler activity and magnetic structures reported in the current study.

The local magnetic field has a strong impact on the propagation of whistler-mode waves, and any inhomogeneities in the field strength or direction can lead to the reflection, refraction, absorption, or mode conversion of such waves. An interesting feature of the magnetic field of the solar wind measured during the PSP encounters is the observation of so-called “magnetic switchbacks” (Krasnoselskikh et al. 2020; Agapitov et al. 2022); these are defined as rapid deflections of the magnetic field that last seconds to hours before “flipping” back to the original field state and are sometimes observed as complete reversals of the magnetic field direction. A feature of the leading and trailing boundaries of the switchback structures is the presence of dips in the magnitude of the magnetic field (by a few percent) and enhanced wave activity, including whistler waves (Krasnoselskikh et al. 2020). A recent discovery made by PSP and reported by Agapitov et al. (2020) is the existence of sunward-propagating whistler waves collocated with these magnetic dips. The waves represented fluctuation amplitudes up to 10% of the background magnetic field. The sunward-propagating feature makes these waves a prospective candidate for the efficient scattering of strahl electrons into the halo population (Saito & Gary 2007).

In this article, we focus on the observation of whistler waves by PSP at ~42.6R⊙ (~0.2 au) that are collocated with magnetic inhomogeneities (changes in magnetic field magnitude or magnetic field direction on the timescale of 1–2 s), including the edges of switchback structures, but also at local rotations of the magnetic field and magnetic dips not associated with switchbacks. We focus on how the relative (in the solar wind frame) velocity of the inhomogeneities, which are presumably

the source of the whistler waves, determines their spectral
properties and propagation direction. We consider five
localized whistler wave packets captured during a 1 hour
interval in the high sampling frequency “burst mode” of PSP.
These cases have similar plasma conditions, such as density,
magnetic field magnitude, suprathermal electron population,
and electron distribution anisotropy. The whistler waves are
observed to propagate predominantly radially in counter-
propagating, sunward, and anti-sunward directions. We
determine the fine spectral-temporal structure of these waves
in the plasma frame by using the high-frequency burst magnetic
and electric fields data and utilize these observations, along
with the cold homogeneous whistler-mode dispersion relation,
to estimate the effective length of the electric field antennas that
are part of the PSP FIELDS instrument.

The paper is organized as follows: Section 2 contains the data description and the analysis of PSP data highlighting bursts of counter-propagating whistler waves; Section 3 is an application of the whistler wave measurements to a determination of the effective length of the electric field antennas that are part of the FIELDS instrument; and Section 4 is a discussion of the results and proposed wave generation mechanism and highlights the main conclusions.

## 2. Parker Solar Probe Observations

In this section, we investigate whistler waves in the solar
wind using measurements from PSP on 2018 November 3,
during Encounter 1 at the heliocentric distance of $40R_⊙$. The
coordinate system used throughout this manuscript is the
inertial RTN (Radial-Tangential-Normal) system with the
radial component *R* oriented along the Sun-spacecraft line,
the transverse component *T* is defined to be orthogonal to the
rotational axis of the Sun and the radial component, i.e.,
*T* = *Ω*<sub>☉</sub> × *R*, while the normal component *N* completes the
orthogonal right-handed triad and, in this case, is aligned with
the normal of the ecliptic plane (Russell et al. 2016). The
electromagnetic fields are measured by the FIELDS instrument
suite on PSP (Bale et al. 2016). The electric field measurements
are made using the electric fields instrument (EFI) consisting of
two pairs of dipole electric field antennas oriented in the *TN*
plane and extending beyond the PSP heat shield, and a fifth
antenna located behind the heat shield on the instrument boom;
the location of antenna V5 in the wake of PSP means the *R*
component is susceptible to detrimental interference by the
wake electric field and cannot be reliably interpreted (Bale et al.
2016). Two three-component flux-gate magnetometers (MAG)
measure the magnetic field from DC to approximately 60 Hz
(up to 293 measurements per second during 2–4 days around
perihelion) (Bale et al. 2016), while the three-component
Search Coil Magnetometer (SCM) measures the magnetic field
perturbations above approximately 3 Hz up to approximately
1 MHz (Jannet et al. 2021; Dudok de Wit et al. 2022). The
Digital Fields Board (DFB) is responsible for the signal
processing and digitization of these signals and outputs a
variety of data products for analysis (Bale et al. 2016;
Malaspina et al. 2016). The DFB Burst Memory (DBM) data
product is an output of approximately 3.5 s of waveform data
sampled at 150 kS s<sup>-1</sup>, providing more than adequate coverage
of the whistler frequency range that was typically below 1 kHz
during the first encounter. The SWEAP instrument suite on
PSP is responsible for measurements of the solar wind
protons, α particles, and electrons including their densities,
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Solar wind parameters during the 1 hour interval where the whistlers were observed on 2018 November 3, between 10:00 and 11:00 UTC. (a) The magnetic field (RTN coordinates in red, green, and blue, respectively, and the magnetic field magnitude is in black) from the MAG. (b) SCM magnetic field spectra, the black line indicates the lower hybrid frequency, $f_{\text{lh}}$, and the white line is $0.1f_{\text{ce}}$. (c) DC coupled electric field spectra from the potentials difference of antennas 1 and 2. (d) Solar wind bulk flow velocity in RTN coordinates (same colors as (a) with $v_{\text{TN}}$ on the left axis and $v_{\text{R}}$ on the right axis) from SWEAP/SPC. (e) Plasma proton density obtained from SWEAP/SPC. The vertical lines in all panels indicate the location of burst waveform intervals from the DBM.</figcaption></figure>

temperatures, and bulk velocities (Kasper et al. 2016). In the present analysis, we focus primarily on DBM measurements of the electric and magnetic fields using the EFI and SCM, but we also make use of the background magnetic field measurements from the MAG, low-cadence spectrograms of the EFI and SCM covering the whistler frequency range, and we use the Solar Probe Cup (SPC) instrument on SWEAP to derive the proton density and velocity (Case et al. 2020).

The observations investigated here occurred on 2018 November 3, between the hours of 10:00 and 11:00 UTC. Figure 1(a) presents the solar wind magnetic field dynamics recorded by the MAG during this observation window. The magnitude of the background magnetic field is indicated by the black curves and the three RTN components by the red, blue, and green curves. The magnitude of the interplanetary magnetic field varies between 40 nT and 70 nT and the radial component of the magnetic field vector is primarily directed toward the Sun (negative *R* direction). This time period is characterized by a high level of magnetic field perturbations and we observe

switchback-like structures, rapid rotational discontinuities, and magnetic dips of varying magnitudes and duration.

Figures 1(b) and (c) present the frequency spectra captured by the SCM and EFI, respectively. The magnetic field spectra are represented by a quadrature summation of the three components while the electric spectra are only available at this time, in this frequency range, for the differential voltage signal between antennas 1 and 2, oriented in the *TN* plane. The upper black curve represents $0.1f_{\text{ce}}$ and the lower black curve is the local lower hybrid frequency $f_{\text{lh}}$, which is roughly the lower limit of the whistler frequency range. There is extensive wave activity in the lower (presumably magnetohydrodynamic, MHD) frequency range corresponding predominantly to Alfvénic fluctuations and in the whistler frequency range from approximately 35–500 Hz (0.02–0.3$f_{\text{ce}}$). We note that the large bursts of whistler waves are often collocated with periods of high magnetic activity corresponding to magnetic field inhomogeneities like magnetic field magnitude local minima
or rapid magnetic field direction changes; this correlation will be discussed further in Section 4.

The solar wind proton velocity determined by the SPC instrument is shown in Figure 1(e). The R-component of solar wind velocity dominates with a typical magnitude in the range of 250–320 km s⁻¹. The variation in the structure of the solar wind velocity for each of the coordinates is highly correlated with the corresponding magnetic component (switchbacks, dips, rotations); however, while the radial magnetic field often approaches zero and sometimes even flips sign, the radial velocity component maintains a large magnitude and is always flowing out from the Sun. The density from SPC shown in Figure 1(f) varies between roughly 250 and 400 cm⁻³ and the structural variation is similarly seen to be correlated with the magnetic activity.

The DBM sampling intervals during this 1 hour interval are highlighted by the five vertical lines between 10:15:00 UTC and 10:45:00 UTC. In this particular observation interval, all of these 3.5 s high sampling-rate periods coincided with significant whistler wave activity, with four of them collocated with magnetic dips and/or rotations. We use three components of the magnetic field transferred into the RTN coordinate system and two components of the electric field recorded in the spacecraft plane transverse to the direction of the Sun (the R-axis of the RTN coordinate system) transferred into T and N components. The third R component of the electric field is calculated in the whistler frequency range from $E_w \cdot B_w^* = 0$, where $E_w$ and $B_w$ are spectral components of the magnetic and electric field. This gives, for the $E_{wR}$:

$$E_{wR} = (E_{wT} \cdot B_{wT}^* + E_{wN} \cdot B_{wN}^*)^* B_{wR} / |B_{wR}|^2. \quad (1)$$

This estimation of $E_{wR}$ is used to process the spectral power of the electric field in the processing of the $E_w/B_w$ ratio and evaluating the effective length of the PSP electric field antennas in Section 3.

We first turn our attention to the case that begins at 10:43:47.29 UTC. Figure 2(a) shows the magnetic field captured by the MAG for a few minutes surrounding the event with the duration of the actual burst event highlighted by the gray-shaded region. The wave burst is collocated with a magnetic rotation associated with the trailing edge of a larger magnetic event that has the typical characterizations of a magnetic switchback (observed from approximately 10:43:00 UTC to 10:43:48 UTC). The magnetic rotation also results in a depletion of the magnetic field magnitude by about 10% from the background field of approximately 55 nT. Figure 2(b) displays the MAG measurement during the DBM event highlighted by the gray shading in Figure 2(a); the rapid rotation of the magnetic field, taking less than 1 s, is more clearly seen.

Figures 2(c) and (d) present the T component of the DBM magnetic and N component of the DBM electric field measurements recorded by the SCM and EFI, respectively. The wave packet, which will be shown to be in the whistler frequency range, reaches an amplitude greater than 2 nT peaking on the maximum of the gradient of the magnetic rotation and is reduced significantly toward the end of the DBM event. The frequency spectrogram of the three-component magnetic field DBM, $B_w$, is shown in Figure 2(e) with the upper and lower curves representing 0.1$f_{ce}$ and $f_{ih}$ as in Figure 1. The bulk of the wave power is contained between 100 and 250 Hz and belongs to the whistler wave frequency range; this leaves little doubt that these are indeed electromagnetic whistler waves. The ratio $E_w/B_w$ displayed in

Figure 2(f) in the spacecraft frame is calculated using the three-component magnetic spectra and three-component electric spectra (with the R component estimated from $E \cdot B = 0$). The white contour border outlines the regions of high wave spectral density from Figure 2(e) with the rest of the data omitted; the background color indicates the $E_w/B_w$ ratio expected for parallel propagating whistlers at each frequency. Importantly, the ratio is significantly larger than the expectation for whistler waves, as has been observed by others (Agapitov et al. 2020); this outcome is a by-product of the EFI antennas having a frequency-dependent effective length and will be addressed in Section 3.

The Poynting flux in the spacecraft frame is calculated using the complex $E_w$ and $B_w$ spectra shown in Figure 2(g) (Webber 2012). The most striking feature of this analysis is the existence of both sunward and anti-sunward waves, which we have elected to call counter-propagating whistler waves. The counter-propagating nature is profoundly strong evidence that the PSP spacecraft is traveling through, or very near to, the source region of the whistler waves; such a burst of sunward and anti-sunward waves occurring simultaneously with the symmetry observed here can presumably be explained by this scenario. Figure 2(h) shows the wave normal angle (WNA) between the wavevector and the background magnetic field, $\theta_{k \cdot B}$ (the R component of the background magnetic field from the MAG was negative during the event, i.e., directed toward the Sun) using the magnetic spectral matrices calculated from three-component SCM measurements (Santolík et al. 2003); this reinforces the counter-propagating feature and reveals the waves are propagating mostly quasi-parallel (<20°) to the magnetic field but do reach oblique angles of propagation up to 60° (but with much lower amplitude) because of curvature of the background magnetic field.

In the spacecraft frame, the frequencies of the sunward waves are less than the anti-sunward waves, however, there is a significant Doppler shift associated with both propagation directions due to the solar wind bulk velocity. The sunward-propagating waves will have experienced a Doppler downshift and thus have a higher frequency in the plasma frame; oppositely, the anti-sunward waves will have been Doppler up-shifted in the spacecraft frame and will have a lower frequency. The frequency of the whistler waves in the plasma frame, $f'$, can be mapped from the spacecraft frame frequency, $f$, using the cold plasma whistler dispersion, plasma density $n$, background magnetic field $B$ from the MAG, solar wind bulk velocity $v_{sw}$, wave normal angle $\theta_{k \cdot B}$ with respect to the background magnetic field, and the angle between the wavevector and velocity. A detailed description and illustration of this procedure are provided in Agapitov et al. (2020). The same frequency domain panels in the panels for Figures 2(e)–(h) are shown with the corrected (recalculated to the solar wind plasma frame) anti-sunward and sunward Doppler shifts in Figures 3(a)–(d) and Figures 3(e)–(h), respectively. The anti-sunward waves cover a frequency range of 100–180 Hz (0.065–0.12$f_{ce}$) and the sunward waves range from 150 to 225 Hz (0.1–0.15$f_{ce}$). The asymmetry in the counter-propagating frequencies is an interesting outcome regarding the possible wave generation mechanism and will be discussed further in Section 4.

The evolution of the distribution of wave normal $k$ relative to the background magnetic field direction of the counter-propagating whistlers is shown in Figure 4. Each point represents the wave’s normal direction with the unit vector $n = k/|k|$, of a single frequency and time from the Doppler-shifted spectrogram. The size of the point represents the wave
<figure><img src="image_5.png" /><figcaption>**Figure 2.** The burst interval collected starting at 10:43:47.29 UTC on 2018 November 3. (a) Dynamics of the magnetic field in RTN coordinates from the MAG over a short time interval around the burst. (b) The shaded region of (a) shows the magnetic field only during the burst. (c) Burst waveform of the magnetic field *T* component, δ*B*<sub>T</sub>. (d) Burst waveform of the electric field *N* component, δ*E*<sub>N</sub>. (e) Spectrogram of the magnetic field burst waveforms (the white contour indicates the spectral power threshold applied to the data in panels (f)–(h)). (f) The ratio of electric spectrogram to magnetic wave spectral power; the white contour denotes the data boundary with lower amplitude regions omitted for clarity and the background coloring represents the expected *E*<sub>w</sub>*B*<sub>w</sub> ratio for parallel propagating whistler waves. (g) The radial component of the Poynting vector, *S*<sub>R</sub>, is calculated from the spectrograms of each component and transformed to a symmetric log scale as discussed in the text. (h) Wave normal angle with respect to the background magnetic field, θ<sub>kB</sub>, ranging from 0° to 180° and indicating parallel and antiparallel propagation, respectively. The lower and upper solid black curves in (e)–(h) indicate *f*<sub>lh</sub> and 0.1*f*<sub>ce</sub>, respectively.</figcaption></figure>

spectral density, the color is the frequency in the plasma frame,
and the radius is time; i.e., the distributions of unit wavevectors
at each time are shown as concentric spheres. Panels ((a)–(c))
show the spherical projections of the 3D distribution in (d) onto
two-dimensional planes with the dashed circles indicating the
times corresponding to the major x-axis ticks in Figure 3. The
solid black curve indicates the direction of the magnetic field
while the dashed black curve is the antiparallel vector to the
background magnetic field. The counter-propagating whistler
waves are observed to propagate in cones surrounding the
magnetic field and the wave amplitude is largest during the
magnetic rotation. The anti-sunward waves maintain quasi-
parallel propagation throughout the burst at all frequencies. The
lower frequency sunward waves experience more oblique
propagation along the *N*-component as the magnetic field
rotates at the end of the interval, however, the wavevector
direction remains largely unchanged.

The case at 10:34:18.22 UTC is another example of counter-
propagating waves and is presented in Figure 5 in the same
format as Figure 2. During this interval, there is a rapid rotation
of the *N* component of the magnetic field and an associated
magnetic dip (approximately 10% variation) collocated with

the whistler wave burst. The wave burst amplitude is maximal
at the steepest part of the gradient of *B*<sub>N</sub> from the MAG. For the
*E*<sub>w</sub>*B*<sub>w</sub> ratio in Figure 5(f) there is an increasing gradient with
frequency for the measured ratio, which is again much larger
than the expectation. The Poynting and (WNA) in Figure 5(g)
and Figure 5(h), respectively, reveal the counter-propagating
nature of the burst with quasi-parallel propagation of both the
sunward and anti-sunward waves. In the spacecraft frame, the
anti-sunward waves occupy a frequency range of approxi-
mately 160–225 Hz (0.11–0.15*f*<sub>ce</sub>), but a Doppler correction
(not shown) reveals the anti-sunward waves experience a
Doppler upshift in the spacecraft frame from approximately
110–180 Hz (0.07–0.12*f*<sub>ce</sub>) in the plasma frame. Similarly, the
sunward waves are Doppler downshifted from approximately
150–240 Hz (0.1–0.16*f*<sub>ce</sub>) in the plasma frame to 90–160 Hz
(0.06–0.11*f*<sub>ce</sub>) in the spacecraft frame.

Figure 6 presents the burst at 10:19:15.56 UTC in the same
format as Figure 2. The burst interval is situated in a narrow
magnetic well with an approximately 30% decrease of the
background magnetic field magnitude at the well minimum that
occurs at the end of a period of MHD scale magnetic
fluctuations. The whistler wave fluctuations are broken into
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Doppler-shifted frequencies, *f*', of the burst waveform spectrogram starting at 10:43:47.29 UTC on 2018 November 3; only the largest amplitude whistles (above the threshold in Figure 2(e)) are retained. (a/e) The magnitude of magnetic field fluctuations. (b/f) *E*<sub>w</sub>/*B*<sub>w</sub> ratio with the expected ratio represented by the colored background. (c/g) *R* component of the Poynting vector. (d/h) Wave normal angle with respect to the background magnetic field. The left column (a–d) is for the anti-sunward-propagating whistlers and the right column (e–h) is for the sunward-propagating whistlers. The upper and lower solid black curves indicate 0.1*f*<sub>ce</sub> and *f*<sub>th</sub>, respectively.</figcaption></figure>

two clear wave packets that are maximal on the leading and trailing edges of the well. The Poynting flux and WNA reveal that the whistler waves are nearly all propagating sunward. The *E*<sub>w</sub>/*B*<sub>w</sub> ratio is greater than the theoretical expectation shown as the background in Figure 6(f), but we note the striking gradient in frequency that will be used in Section 3 to empirically calibrate the EFI antennas. In the spacecraft frame, the whistlers cover a frequency range of 60–160 Hz (0.06–0.16*f*<sub>ce</sub>) and are Doppler downshifted from a plasma frame frequency of 100–250 Hz (0.1–0.25*f*<sub>ce</sub>).

In Figure 7, we present the unit wavevector direction evolution for this sunward-propagating burst at 10:19:15:56 UTC. The three-dimensional distribution in Figure 7(d) shows the propagation of the wave burst is quasi-parallel along the magnetic field directed toward the Sun. In the *RT* plane and *RN* plane, the wavevectors are highly collimated along the magnetic field; there is also a weak dependence on the frequency where the lowest frequency waves are more oblique. In the *TN* plane the distribution is more widely dispersed around the parallel field and has very little frequency dependence; this more scattered distribution can largely be attributed to the dominance of the *R* component of the background magnetic field seen in Figure 6(b) that is not captured in Figure 7(c).

Another example of exclusively sunward propagation is given by the case at 10:30:30.15 UTC shown in Figure 8. A

rotation of the magnetic field and a corresponding dip in the magnitude of the magnetic field by nearly 50% is collocated with a very short burst of whistler waves. The magnetic dip observed here has more of a form of a step decline in magnetic field magnitude that takes much longer to recover. It is observed that the whistler wave burst is localized to the initial sharp gradient, similar to the gradient localization in the previous case, but this time only on the leading edge as the trailing edge is a gradual recovery. The Poynting flux reveals that this wave packet contains only sunward-propagating waves; similar to the other cases, the waves are again nearly parallel to the background magnetic field as shown in the WNA plot. This case has a lower frequency than the rest of the cases and the *E*<sub>w</sub>/*B*<sub>w</sub> ratio at low frequencies is observed to approach the ratio expected from a cold plasma dispersion (see Figure 8(f)). This indicates that the effective antenna length is close to the geometrical antenna length at these lower frequencies; this is consistent with observations reported by Mozer et al. (2020).

The case at 10:33:31.03 is presented in Figure 9; in this case, we observe the waves in a calm magnetic field during the DBM event. The Poynting flux shows that the packets of whistler waves are all propagating primarily away from the Sun and the WNA shows they are nearly parallel to the background magnetic field. Given that anti-sunward waves do not strongly interact with the solar wind particle populations, and that there
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Distribution of unit wavevectors for the burst waveform starting at 10:43:47.29 UTC on 2018 November 3. In each of the plots, the distance from the origin is time, the size of the dot is the wave spectral density, and the color is the frequency in the plasma frame, $f'$. The solid black curve indicates the direction of the magnetic field and the dashed black curve is the antiparallel vector to the background magnetic field direction. (a) RT plane spherical projection. (b) RN plane spherical projection. (c) TN plane spherical projection. (d) Three-dimensional distribution in RTN coordinates.</figcaption></figure>

is a lack of a strong inhomogeneity in the magnetic field, it is possible that the waves observed here have traveled from another source to the spacecraft; this interpretation is supported by the lack of sunward waves collocated with this event and the lack of a magnetic inhomogeneity to indicate some source mechanism. The amplitude of these waves is also significantly less than the other four cases, this is further evidence they may have originated elsewhere in the solar wind and experienced some light damping as they have propagated to the location of PSP. The hatched regions in the frequency-dependent plots serve only to remove signals introduced by voltage perturbations due to plasma clouds generated by dust impacts on the spacecraft (Malaspina et al. 2020; Page et al. 2020). These waves are again Doppler shifted to a frequency of 160 Hz to 220 Hz (0.1–0.14$f_{ce}$) in the spacecraft frame from a frequency of 100 Hz to 160 Hz (0.06–0.1$f_{ce}$) in the plasma frame.

### 3. Application to the FIELDS Instrument

The five cases presented in the previous section, and specifically the two counter-propagating cases, offer an opportunity to calibrate the electric field antennas on the FIELDS instrument in the whistler waves frequency range. The

methodology presented here is similar to that used for the wave cases presented by Mozer et al. (2020). There are two frames of interest in this problem, the spacecraft frame (denoted by unprimed quantities) and the plasma frame (denoted by primed quantities), with the plasma frame moving at a relative velocity, $v$, with respect to the spacecraft frame. The electromagnetic fields in the spacecraft frame are related to the plasma frame fields by the Lorentz transformations for special relativity (Feynman 1964),

$$E_{\parallel} = E'_{\parallel} \qquad (2)$$

$$B_{\parallel} = B'_{\parallel} \qquad (3)$$

$$E_{\perp} = \frac{(E' - v \times B')_{\perp}}{\sqrt{1 - v^2/c^2}} \qquad (4)$$

$$B_{\perp} = \frac{(B' + \frac{v \times E'}{c^2})_{\perp}}{\sqrt{1 - v^2/c^2}} \qquad (5)$$

where here the transformation is in the format of the so-called “inverse” transformation due to our choice of the spacecraft frame as the unprimed frame. In the case where $v < < c$, as is the case for the PSP observations presented herein,
<figure><img src="image_8.png" /><figcaption>**Figure 5.** The burst interval starting at 10:34:18.22 UTC on 2018 November 3. (a) Dynamics of the magnetic field in RTN coordinates from the MAG over a short time interval around the burst. (b) The shaded region of (a) shows the magnetic field only during the burst. (c) Burst waveform of the magnetic field T component, δB_T. (d) Burst waveform of the electric field N component, δE_N. (e) Spectrogram of the magnetic field burst waveforms (the white contour indicates the spectral power threshold applied to the data in panels (f)–(h)). (f) The ratio of electric spectrogram to magnetic spectrogram; the white contour denotes the data boundary with lower amplitude regions omitted for clarity and the background coloring represents the expected E_w/B_w ratio for parallel propagating whistler waves. (g) The radial component of the Poynting vector, S_R, is calculated from the spectrograms of each component and transformed to a symmetric log scale as discussed in the text. (h) Wave normal angle with respect to the background magnetic field, θ_kB, ranging from 0° to 180° and indicating parallel and antiparallel propagation, respectively. The lower and upper solid black curves in (e)–(h) indicate f_lh and 0.1f-ce, respectively.</figcaption></figure>

Equations (4) and (5) are reduced to,

$$E_{\perp} = (E' - \mathbf{v} \times \mathbf{B}')_{\perp} \qquad (6)$$

$$B_{\perp} = B'_{\perp}. \qquad (7)$$

The effect of special relativity on the whistler wave measurements is an additional electric field proportional to $\mathbf{v} \times \mathbf{B}'_w$ associated with the whistler wave that is measured by the FIELDS electric antennas, where $\mathbf{B}'_w$ refers to the magnetic field waveform in the plasma frame. It is well documented that electric field antenna response is further modified by an undetermined transfer function due to coupling between the spacecraft and plasma (Gurnett 1998). This coupling can be determined by the spacecraft geometry, frequency and wavelength of the electromagnetic or electrostatic fields being measured, and the plasma parameters. A simple way to correct for this interaction is to use an effective length, $l_{eff}$, calibration (Hartley et al. 2015, 2016; Mozer et al. 2020),

$$\frac{l_{\text{ant}}}{l_{\text{eff}}} \mathbf{E}_{\text{exp}} = \mathbf{E}'_{\text{w}} - \mathbf{v} \times \mathbf{B}'_{\text{w}} \qquad (8)$$

where $\mathbf{E}_{\text{exp}}$ represents the measured electric field using the nominal antenna length, $l_{\text{ant}} = 3.5$ m. The effective antenna

length for a given frequency can be estimated from whistler waves measured by PSP using the apparent velocity from the ratio of the measured $E_w$ and $B_w$, the plasma flow speed $V_{SW}$, the unit wavevector $\mathbf{k}/|\mathbf{k}|$, and the expected whistler phase velocity from the cold plasma dispersion for the frequency in the plasma frame, $v_{ph}$,

$$l_{\text{eff}}(\omega) = l_{\text{ant}} \frac{|\mathbf{E}_{\text{w}}(\omega)|/|\mathbf{B}_{\text{w}}(\omega)|}{v_{\text{ph}}(\omega') + V_{\text{SW}} \cdot \hat{\mathbf{k}}(\omega)} \qquad (9)$$

where $|\mathbf{E}_{\text{w}}(\omega)|$ denotes the amplitude of a wave at a specific frequency in the spacecraft frame; $\omega$ is the wave frequency in the spacecraft frame; $\omega'$ is the wave frequency in the plasma frame.

Using Equation (9) and the Doppler corrected wave spectra in the plasma frame, like those presented in Figure 3, we estimate the effective length $l_{eff}$ for each of the five DBM cases. Figure 10 displays the result of this analysis with $l_{eff}$ normalized to the plasma density. Panels (a) through (d) present the bivariate PDF of the results for each of the individual cases overplotted on the aggregate of all the cases with sunward and anti-sunward waves indicated by blue and
<figure><img src="image_9.png" /><figcaption>**Figure 6.** The burst interval staring at 10:19:15.56 UTC on 2018 November 3. (a) Dynamics of the magnetic field in RTN coordinates from the MAG over a short time interval around the burst. (b) The shaded region of (a) shows the magnetic field only during the burst. (c) Burst waveform of the magnetic field *T* component, δ*B*<sub>*T*</sub>. (d) Burst waveform of the electric field *N* component, δ*E*<sub>*N*</sub>. (e) Spectrogram of the magnetic field burst waveforms (the white contour indicates the spectral power threshold applied to the data in panels ((f)–(h))). (f) The ratio of electric spectrogram to magnetic spectrogram; the white contour denotes the data boundary with lower amplitude regions omitted for clarity and the background coloring represents the expected *E*<sub>*w*</sub>*B*<sub>*w*</sub> ratio for parallel propagating whistler waves. (g) The radial component of the Poynting vector, *S*<sub>*R*</sub>, is calculated from the spectrograms of each component and transformed to a symmetric log scale as discussed in the text. (h) Wave normal angle with respect to the background magnetic field, θ<sub>*kB*</sub>, ranging from 0° to 180° and indicating parallel and antiparallel propagation, respectively. The lower and upper solid black curves in (e)–(h) indicate *f*<sub>lh</sub> and 0.1*f*<sub>ce</sub>, respectively.</figcaption></figure>

red, respectively. The overlap of the common frequencies in
both the counter-propagating cases represents a strong valida-
tion of the result as the Lorentz-transformed electric field and
Doppler shift are opposite for each wave propagation direction.
Figure 10(e) displays the aggregate of all the cases along with a
comparison of the result from Mozer et al. (2020) with
frequencies of 1, 3, 4, 10, 20, 35, and 60 Hz. There is excellent
agreement among all five cases and with the Mozer et al.
(2020) results at low frequencies. The right axis shows an
estimate of the l<sub>eff</sub>. It is observed that at low frequencies up to
about 60 Hz, l<sub>eff</sub> is less than or equal to the antenna geometrical
length of 3.5 m. Above 60 Hz, l<sub>eff</sub> increases to 22–28, i.e.,
nearly a factor of 6–8 greater than expected, and peaks locally
at a frequency of 140–150 Hz before decreasing and appearing
to have a plateau at and above 160–170 Hz (over to the
frequency range investigated) at around 13–17 m. Overall,
l<sub>eff</sub>(ω) ranges from ~5 m to ~28 m for the frequency range
35–300 Hz. The obtained l<sub>eff</sub> has a comparably large spread of
values (10%–15%), which is a result of the measurements
regime and approximations used for processing: the PSP
plasma data sampling rate of 1 s is a result of plasma
parameters averaging during the processing intervals when

the actual solar wind bulk velocity and plasma density can vary
in the range of 5%–10% (Mozer et al. 2023) affecting the
evaluated wave properties; the approximation of field-aligned
whistler waves propagation (we filtered out waves with WNA
above 20° for the estimations) can lead to less than 5% error of
estimation since the maximum of WNA distribution is at ~5°;
the possible dependence on the background plasma density
(reported by Hartley et al. (2017) for the Van Allen Probes
measurements). The slightly higher values of l<sub>eff</sub> obtained at
higher density (Figure 10(a))will need further investigations.
Such a large range of effective lengths over a small frequency
range of the FIELDS EFI instrument (which covers DC to
60 kHz) is concerning regarding the reliability and potential for
misinterpretation of electric field measurements made by PSP
but is not wholly unexpected. The effective length l<sub>eff</sub> of the
electric field sensor for the spacecraft measurements in very
variable plasma conditions cannot be accurately obtained in the
ground test (Imachi et al. 2007; Hartley et al. 2017). Thus the
in-flight calibrations, which can be based on the suitable
plasma models and the properties of plasma wave modes, can
be applied. The wide range of plasma density in the solar wind
(or magnetosphere plasma (Hartley et al. 2017)) can lead to
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Distribution of unit wavevectors for the burst waveform starting at 10:19:15.56 UTC on 2018 November 3. In each of the plots, the distance from the origin is time, the size of the dot is the wave spectral density, and the color is the frequency in the plasma frame, *f*'. The solid black curve indicates the direction of the magnetic field and the dashed black curve is the antiparallel vector to the magnetic field. (a) *RT* plane spherical projection. (b) *RN* plane spherical projection. (c) *TN* plane spherical projection. (d) Three-dimensional distribution in RTN coordinates.</figcaption></figure>

different approximations for the antenna effective length, for
example, when the geometrical antenna scale is less about the
local Debye length in plasma, and or the antenna lengths are of
the same order as the spacecraft size. This case is regular for the
PSP EFI measurements in the solar wind with Debye length of
about units or tens of meters and the relatively short antennas
(3.5 m), which are less about the PSP linear sizes. Here we used
the waves processed in the cold plasma approximation to
estimate the frequency dependence of l\_eff(\*\omega) for the cases
collected during the 1 hour interval with similar plasma
conditions. The overestimation of the electric field from the
measurements on board the two Van Allen Probes in the
Earth's magnetosphere with the short boom antenna (dependent
on the background plasma density and wave frequency) was
reported by Hartley et al. (2017). Hartley et al. (2017)
discussed a possible source of this anomalous gain to be in
the geometry of the voltage-biased surfaces around the sensors.
The extra gain was significant for the shorter spin-axis sensors
(the length was 14 m versus 40 and 50 m for the spin-plane
sensors): the short boom axial antennas provide 1.5–3 times
overestimated electric field measurements (Hartley et al. 2017).
We presume that the geometry of the voltage-biased surfaces of

the PSP spacecraft can be a source of the observed over-
estimation of electric field measurements in the frequency
range from 50 to 160 Hz observed in plasma with a density of
300–380 cm⁻³. The estimates presented here could be used in
conjunction with those made by Mozer et al. (2020) to assist
with the interpretation of the FIELDS antennas and guide
future attempts at empirical calibration. It should be noted that
these results are for cases with similar densities and magnetic
field strengths and caution should be used in applying these
effective lengths to other regimes.

4. Discussion

High sampling-rate SCM and EFI PSP observations of electromagnetic fields in the burst mode (150 kS s⁻¹) have revealed the occurrence of localized bursts of low-frequency whistler wave packets: the typical frequency range is 100–300 Hz (0.05–0.2f<sub>ce</sub>) in the spacecraft frame (in an agreement with statistical studies by Jagarlamudi et al. (2021) and Cattell et al. (2022)). These waves often coincide with local minima of the magnetic field magnitude or with edges of magnetic switchbacks containing sharp rotations of the
<figure><img src="image_11.png" /><figcaption>**Figure 8.** Analysis of the burst waveform starting at 10:30:30.15 UTC on 2018 November 3. (a) Dynamics of the magnetic field in RTN coordinates from the MAG over a short time interval around the burst. (b) The shaded region of (a) shows the magnetic field only during the burst. (c) Burst waveform of the magnetic field *T* component, *δB*<sub>*T*</sub>. (d) Burst waveform of the electric field *N* component, *δE*<sub>*N*</sub>. (e) Spectrogram of the magnetic field burst waveforms. (f) The ratio of electric spectrogram to magnetic spectrogram; the white contour denotes the data boundary with lower amplitude regions omitted for clarity and the background coloring represents the expected *E*<sub>w</sub>/<i>B</i><sub>w</sub> ratio for parallel propagating whistler waves. (g) The radial component of the Poynting vector is calculated from the spectrograms of each component and transformed to a symmetric log scale as discussed in the text. (h) Wave normal angle with respect to the background magnetic field, *θ*<sub>*k*,B</sub>, ranging from 0° to 180° and indicating parallel and antiparallel propagation, respectively. The lower and upper solid black curves in (e)–(h) indicate *f*<sub>th</sub> and 0.1/*f*<sub>ce</sub>, respectively.</figcaption></figure>

background magnetic field direction: the distribution of the
whistler waves amplitudes in their relation to the magnetic field
magnitude perturbations (Δ|B|) and magnetic field perturbation
(|ΔB|) is presented in Figure 11 indicating collocation of wave
sources with magnetic dips for highest amplitudes of sunward-
propagating waves (Figure 11(a)), and collocation of wave
sources (the counter-propagating waves) with changes of
magnetic field direction (Figure 11(b)). The high amplitude
predominantly sunward waves appear to be strongly associated
with the local dips of the background magnetic field (high Δ|B|
in Figure 11(a)). The counter-streaming cases were observed
simultaneously with sharp changes in magnetic field direction
(|ΔB|) but do not contain a dip of background magnetic field |B|.
The anti-sunward case was found to be unrelated to any
inhomogeneity in the magnetic field (we presume that the
waves observed with lower values of |ΔB| are anti-sunward
waves). The sunward and counter-streaming cases observations
suggest that we observe them in the vicinity of the source
region the sunward whistlers can interact efficiently with the
strahl, so are rapidly decaying interacting with the strahl
electrons in the source region (possibly by the gyrosurfing
mechanism (Artemyev et al. 2013; Kis et al. 2013)). This
presumably determines the high correlation of sunward

whistlers with magnetic field inhomogeneities. However, anti-
sunward waves, even generated in the vicinity of magnetic field
inhomogeneities, can propagate long distances without s
significant decay. Wave amplitudes peaked at 2 nT representing
up to 0.05 of the background magnetic field magnitude. The
polarization of these waves was found to be predominantly
quasi-parallel to the background magnetic field confirming the
statistics reported by Cattell et al. (2020) and Froment et al.
(2022). The quasi-parallel propagation appears to be the
predominant mode of the observed whistler waves in the solar
wind in all the range of heliocentric distances covered by
observations by PSP and Solar Orbiter: the probability to
observe waves with wave normal angle above 50° is less than
0.01 (Tong et al. 2019; Kretzschmar et al. 2021; Cattell et al.
2022). This makes propagation direction to be the key factor
for wave-particle interaction efficiency: the scattering of the
strahl electrons by the counter-propagating (sunward-propagat-
ing) whistler wave is more than an order of magnitude more
efficient than scattering by anti-sunward-propagating quasi-
parallel waves. All whistler waves observed by Solar Orbiter at
0.5–1.0 au were found to propagate from the Sun (Kretzschmar
et al. 2021). Thus, the sunward-propagating whistler population
in the young solar wind (observed by PSP at 25–45 R⊙) can be
<figure><img src="image_12.png" /><figcaption>**Figure 9.** The burst interval staring at 10:33:31.03 UTC on 2018 November 3. (a) Dynamics of the magnetic field in RTN coordinates from the MAG over a short time interval around the burst. (b) The shaded region of (a) shows the magnetic field only during the burst. (c) Burst waveform of the magnetic field *T* component, δ*B<sub>T</sub>*. (d) Burst waveform of the electric field *N* component, δ*E<sub>N</sub>*. (e) Spectrogram of the magnetic field burst waveforms (the white contour indicates the spectral power threshold applied to the data in panels ((f)–(h))). (f) The ratio of electric spectrogram to magnetic spectrogram; the white contour denotes the data boundary with lower amplitude regions omitted for clarity and the background coloring represents the expected *E*<sub>w</sub>*B*<sub>w</sub> ratio for parallel propagating whistler waves. (g) The radial component of the Poynting vector, *S*<sub>R</sub>, is calculated from the spectrograms of each component and transformed to a symmetric log scale as discussed in the text. (h) Wave normal angle with respect to the background magnetic field, θ<sub>kB</sub>, ranging from 0° to 180° and indicating parallel and antiparallel propagation, respectively. The lower and upper solid black curves in (e)–(h) indicate *f*<sub>Ih</sub> and 0.1*f*<sub>ce</sub>, respectively.</figcaption></figure>

a potentially key factor for the scattering of the strahl electron
population. It raises a question about the mechanism of their
local generation process (to explain the heliocentric distances
range of generation of these waves reported by Cattell et al.
2022) and requires a statistical study of whistler wave
occurrence rates and their parameters in the young solar wind
to include into the consideration both the wave amplitude
distribution and wave polarization parameters including their
propagation direction.

The lower sampling rate of the electron distribution function evaluated by the SWEAP instrument suite on board PSP (one per 27 s during Encounter 1, and 14 s during later Whittlesey et al. 2020) does not allow direct processing of the electron instabilities inside the discussed magnetic holes (their duration is ~1 s), so we use the available plasma density measurements and the averaged electron distribution functions for a model for the electron distribution inside the magnetic field minimum. The plasma density did not have any peculiarities inside the magnetic field magnitude depression region (Agapitov et al. 2020). This suggests existing of the hot trapped population with the loose cones and respectively with the transverse anisotropy favorable for the generation of quasi-parallel whistlers. Similar structures with

magnetic field magnitude depletion were observed in the vicinity
of the Earth bow shock by the four MMS spacecraft with high-
time resolution electron measurements aboard (Ahmadi et al.
2018), and these observations nicely confirm our assumptions
highlighting the trapped hot electron population. Such distribution
(with the proper plasma temperature) can guide the generation of
whistler waves propagating along and opposite the background
magnetic field through the cyclotron instability (Drake et al. 2015;
Agapitov et al. 2018; Ahmadi et al. 2018; Malaspina et al. 2022;
Tigik et al. 2022). For the magnetic hole, which is in the rest with
respect to the background core plasma, i.e., propagating with the
bulk solar wind velocity, such a distribution is shown
schematically in Figure 12(e). However, the wave sources are
connected with the magnetic field inhomogeneities (like switch-
back boundaries) propagating in the solar wind plasma with
significant relative velocity δV in the range of 50–150 km s⁻¹.
This relative velocity leads to the shift of distribution function of
the trapped electron population with δV (schematically shown in
Figure 12(f)) and the corresponding generation of sunward and
anti-sunward whistler waves with the difference in the wave
frequency (Figures 12(a), (b) present whistler spectra in the
plasma frame) and amplitude (Figure 12(a)). Numerical estimation
<figure><img src="image_13.png" /><figcaption>**Figure 10.** (a)–(e) The bivariate PDFs of the density normalized effective antenna length and wave frequency for the five cases of whistler bursts in the 1 hour interval presented herein: (a) sunward case from Figure 6; (b) sunward case at 10:30:30.15 and anti-sunward case from Figure 9; (c) counter-propagating case at 10:34:18.22; (d) counter-propagating case from Figure 2; and (e) the aggregate of all the cases. In (a) through (d), the sunward waves are shown by blue markers and anti-sunward by red markers, with the aggregate from (e) shown in the background for clarity. The hollow black triangles in (e) represent the result from Mozer et al. (2020) for comparison. The right axis is an estimate of the effective length in meters using the density of $n = 300 \text{ cm}^{-3}$ from Mozer et al. (2020) and is similar to the density for the five cases presented here.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Distribution of whistler bursts observed during 10–11 UT (the interval presented in Figure 1) in the wave amplitude and the background magnetic field perturbation domain: (a) On the vertical axis is the maximum amplitude of each temporal bin of the spectral matrices within the frequency range of the whistler waves and on the horizontal axis the normalized perturbation of the magnetic field magnitude of $|\Delta\mathbf{B}|/|\mathbf{B}|$ during the same time window. (b) The same vertical axis with the horizontal axis represents the perturbation of the magnetic field vector of $|\Delta\mathbf{B}|/|\mathbf{B}|$ during the same time window. The black circles indicate each of the temporal bins, while the five cases are highlighted by the colored markers in the legend of (a) and are indicated as either sunward (SW), anti-sunward (ASW), or counter-propagating (CS).</figcaption></figure>

of the growth rate reveals this difference in Figures 12(c), (d): propagation with half of the local Alfvén velocity separates the sunward and anti-sunward waves in frequency with a maximum growth rate at a higher frequency for the sunward wave and at lower for anti-sunward wave (Figure 12(b)); propagation of magnetic hole with the local Alfvén velocity changes both the frequency and value of the maximum wave growth rate

(Figure 12(c)): sunward waves have several times higher growth rate than the anti-sunward ones (Figure 12(d)). Because of the lack of high sampling-rate measurements of the electron distribution function, we can provide the qualitative reliance on the observed whistlers properties: the faster propagating magnetic hole can lead to a greater difference in the frequency of growth rates for sunward and anti-sunward waves (Cases in Figures 2 and
<figure><img src="image_15.png" /><figcaption>**Figure 12.** (a)–(d) Comparison of the Doppler-shifted spectrum with the growth rates estimate under the quasi-linear approximation. (a) Predominantly sunward waves from the case at 10:19:15 UTC. (b) Counter-propagating waves with shifted frequency peaks from the case at 10:43:47. (c) Predicted growth rates using $T_{\perp}$ = 100 eV and $T_{\parallel}$ = 20 eV, and magnetic hole relative velocity $v_{rel}$ = $v_A$ (solid curves) and $v_{rel}$ = $2v_A$ (dashed curves). (d) Predicted growth rates using $T_{\perp}$ = 100 eV and $T_{\parallel}$ = 20 eV, and magnetic hole relative velocity $v_{rel}$ = $v_A$. (e) and (f) Schematic descriptions of the wave generation mechanism by trapped particle populations. (e) A symmetric loss cone distribution is equally unstable to whistler waves of the same frequency propagating both sunward (SW) and anti-sunward (ASW). (f) A loss cone distribution with a positive relative drift velocity, $v_{rel}$, is unstable to counter-propagating whistler waves where the peak frequency of the SW waves is larger than that of the ASW waves; a larger $v_{rel}$ results in lower growth rates of the ASW waves.</figcaption></figure>

5 of counter-propagating whistlers in the current paper) up to full suppression of anti-sunward waves generation (Cases in Figures 6 and 8 in the current paper and the cases reported by Agapitov et al. 2020). Additionally, observations of sunward whistlers only at their generation regions and observations of anti-sunward-propagating whistler waves unrelated to the source regions support the difference in the wave–particle interaction efficiency: sunward waves are rapidly decaying on the edges of the source region but anti-sunward quasi-parallel waves propagate almost without decay.

## 5. Conclusions

1. We have identified the presence of intense whistler waves in their generation region from Parker Solar Probe observations at a heliocentric distance of approximately $35.7R_\odot$ and evaluated their parameters:

(a) the counter-propagating nature of the sunward and anti-sunward whistler waves indicates that observations were captured in the wave source region;

(b) the occurrence of sunward-propagating whistler waves is related to local inhomogeneities of the magnetic field often associated with switchback boundaries. Such close relation indicates that sunward-propagating whistlers are observed in their generation region and rapidly decay because of efficient wave-particles interaction outside the source;

(c) the anti-sunward waves experience slower decay and can be observed far from their source regions;

(d) observed waves had mostly quasi-parallel to the background magnetic field wave normal angles (less than 20°). Some deviations from the quasi-parallel propagation (with wave normal angle up to 60°) were found to be connected with the inhomogeneous geometry of the background magnetic field.

2. The generation mechanism for these sunward and anti-sunward-propagating quasi-parallel whistler waves is presumably the cyclotron instability (suggested by simultaneous generation of counter-propagating wave

bursts) driven by suprathermal electron population (of the order of 100 eV) trapped in magnetic field dips and/or the strahl population with perturbed geometry at local inhomogeneities of the magnetic field (sharp rotations) at switchback boundaries. The derived difference in frequency of sunward and anti-sunward counter-propagating waves confirmed their simultaneous generation in the source region moving in the solar wind frame.

3. The effective length of the PSP antennas applied to electric field measurements being carried out by the Electric Field instrument (EFI) has been processed in the frequency domain of whistler waves making use of the Search Coil Magnetometer (SCM) measurements of magnetic field perturbations and the whistler wave dispersion relation. The simultaneous observations of counter-propagating whistler waves were used to eliminate the moving media effects of solar wind flow. The EFI effective length ($l_{eff}$) is about 1 m for the near DC measurements, grows up to 3.5–4.5 m for frequencies of 10–50 Hz (Mozer et al. 2020), and peaks from 4.5 m to ~25 m when the frequency increases from 50 Hz to 130–150 Hz for the plasma density range of 280–350 cm⁻³. Above 150 Hz $l_{eff}$ saturates at approximately 15 m (we covered the frequency range below 300 Hz in this work). From an experimentalist’s perspective, perhaps the most consequential outcome of this analysis is that these values of $l_{eff}$ can cause significant overestimation of the measured electric field and require proper calibration before processing the efficiency of wave–particle interactions at frequencies above 50 Hz.

S.K., O.V.A., and H.Y.K were supported by NASA grant Nos. 80NNSC19K0848, 80NSSC22K0433, and NSF grant No. 1914670. O.V.A. was partially supported by NASA grant Nos. 80NSSC20K0218, 80NSSC20K0697, and 80NSSC21K1770. The PSP/FIELDS experiment was developed and is operated under NASA contract No. NNN06AA01C.
ORCID iDs

S. Karbashewski https://orcid.org/0000-0003-4988-492X

O. V. Agapitov https://orcid.org/0000-0001-6427-1596

H. Y. Kim https://orcid.org/0000-0003-3304-0722

F. S. Mozer https://orcid.org/0000-0002-2011-8140

J. W. Bonnell https://orcid.org/0000-0002-0675-7907

C. Froment https://orcid.org/0000-0001-5315-2890

T. Dudok de Wit https://orcid.org/0000-0002-4401-0943

Stuart D. Bale https://orcid.org/0000-0002-1989-3596

D. Malaspina https://orcid.org/0000-0003-1191-1558

N. E. Raouafi https://orcid.org/0000-0003-2409-3742

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