# Parker Solar Probe observations of near-$f_{\rm ce}$ harmonics emissions in
the near-Sun solar wind and their dependence on the magnetic field direction

Sabrina F. Tigik Andris Vaivads Space Plasma Physics, Electrical Engineering
and Computer Science,
KTH Royal Institute of Technology, Stockholm 11428, Sweden  David M. Malaspina
Astrophysical and Planetary Sciences Department, University of Colorado,
Boulder, CO, USA  Laboratory for Atmospheric and Space Physics, University of
Colorado, Boulder, CO, USA  Stuart D. Bale Physics Department, University of
California, Berkeley, CA 94720-7300, USA  Space Sciences Laboratory,
University of California, Berkeley, CA 94720-7450, USA

###### Abstract

Wave emissions at frequencies near electron gyrofrequency harmonics are
observed at small heliocentric distances below about 40 solar radii and are
known to occur in regions with quiescent magnetic fields. We show the close
connection of these waves with the large-scale properties of the magnetic
field. Near electron gyrofrequency harmonics emissions occur only when the
ambient magnetic field points to a narrow range of directions bounded by polar
and azimuthal angular ranges in the RTN coordinate system of correspondingly
$80^{\degree}\lesssim\theta_{\bf B}\lesssim 100^{\degree}$ and
$10^{\degree}\lesssim\phi_{\bf B}\lesssim 30^{\degree}$. We show that the
amplitudes of wave emissions are highest when both angles are close to the
center of their respective angular interval favorable to wave emissions. The
intensity of wave emissions correlates with the magnetic field angular changes
at both large and small time scales. Wave emissions intervals correlate with
intervals of decreases in the amplitudes of broadband magnetic fluctuations at
low frequencies of $10\,\mathrm{Hz}-100\,\mathrm{Hz}$. We discuss possible
generation mechanisms of the waves.

solar wind, plasma waves, Parker Solar Probe

##  1 Introduction

The NASA’s Parker Solar Probe (PSP) mission was launched in August 2018 with
the objective to study the near-Sun environment that has never been explored
before (Fox et al., 2016). Since its first encounter with the near-Sun
environment, Parker Solar Probe has been measuring intense wave activity
around the electron cyclotron frequency $f_{\rm ce}$ (Ma et al., 2021;
Malaspina et al., 2020; Shi et al., 2022). The study by Malaspina et al.
(2020) investigates the correlation between the emissions of near-$f_{\rm ce}$
waves and the angle $(\theta_{\rm Br})$ between the ambient magnetic field
vector and the instantaneous radial direction. They detect peaks in the
electric field spectra, at frequencies $0.7\,f_{\rm ce}<f<1.1\,f_{\rm ce}$, in
an interval of $15\,\mathrm{R}_{\odot}$ (solar radii) around the closest
approach to the Sun, during encounters 1 and 2. The authors find that the most
favorable condition for the near $f_{\rm ce}$ waves to occur is $\theta_{\rm
Br}<25^{\degree}$, i.e., a quasi-radial ambient magnetic field. However, in a
statistical analysis of the same interval, comparing the total hours of
electron cyclotron harmonic emissions and the total hours of $\theta_{\rm
Br}<25^{\degree}$, the authors find that the waves are observed only during
$10\%\sim 30\%$ of this time. As a result, Malaspina et al. (2020) concludes
that a near-radial ambient magnetic field, though necessary, is not a
sufficient condition for the growth of near-$f_{\rm ce}$ waves. Earlier
studies have suggested that the near-$f_{\rm ce}$ harmonics structures are
composed of electron Bernstein waves, and at least two other wave modes that
have not been completely identified so far (Malaspina et al., 2020, 2021; Ma
et al., 2021; Shi et al., 2022). The generation mechanism of these waves is
still an open question.

In this work, we extend the analysis in Malaspina et al. (2020), focusing on
the connection of the emission of near-$f_{\rm ce}$ harmonics with large-scale
properties of the magnetic field. We show that near-$f_{\rm ce}$ harmonics are
observed only when the ambient magnetic field points to a narrow range of
directions. In addition, we show that these waves are observed only in regions
where the broadband magnetic turbulence is low. The results reported in this
work are crucial for understanding the origin of the near-$f_{\rm ce}$
harmonics waves measured by Parker Solar Probe at small heliocentric
distances.

##  2 Data

This study uses data from PSP’s $5^{\rm th}$ perihelion passage, on June 7,
2020. The closest perihelion distance in this encounter was about $\sim
27.87\,\mathrm{R_{\odot}}\,(0.13\,\mathrm{AU})$ at 09:06 UTC. During this
perihelion passage intense near-$f_{\rm ce}$ waves are observed under
different solar wind conditions and thus it is well suited for deeper study of
the waves.

We use Parker Solar Probe measurements of magnetic and electric fields from
the FIELDS experiment (Bale et al., 2016) and ions from the SWEAP experiment
(Kasper et al., 2016). Magnetic field data from both the Fluxgate Magnetometer
(FGM) and Search Coil Magnetometer (SCM) instruments are used. As for electric
field data, we use two distinct differential electric potential data sets,
both measured by the FIELDS antenna pairs $V_{12}$ and $V_{34}$: AC-coupled
spectra in survey mode, with a cadence of $\sim 0.87\,\mathrm{s}$, and burst
waveform data, sampled at $\sim 150\,\mathrm{kS/s}$ during $\sim
3.5\,\mathrm{s}$ snapshots. To convert the differential voltage data into an
electric field we use $3.5\,\mathrm{m}$ effective antenna length (Mozer et
al., 2020). Search coil magnetometer data and both electric voltage data sets
are produced by the Digital Fields Board (DFB) (Malaspina et al., 2016).
Proton velocity data are from the Solar Probe ANalyzer-Ions (SPAN-I)
instrument (Livi et al., 2021), which is part of the SWEAP experiment suite.

We use two coordinate systems. In the large-scale analysis, magnetic field and
proton velocity data are shown in RTN coordinates. Regarding the proton
velocity, we use SPAN-I level 3 (L3), version 4 (v04) data, which includes,
among other data products, ion velocities measured in SPAN-I instrument
coordinates transformed to the RTN coordinate system. The ${\bf R}$ (radial)
unit vector points from the Sun to the center of the spacecraft. The ${\bf T}$
(tangential) unit vector lies in the ecliptic plane and is defined as ${\bf
T}=({\bm{\omega}}_{\odot}\times{\bf R})/|{\bm{\omega}}_{\odot}\times{\bf R}|$,
where ${\bm{\omega}}_{\odot}$ is the Sun’s rotation vector. The ${\bf T}$ unit
vector points in the solar rotation direction, which is close to the PSP’s ram
direction during the encounters. The ${\bf N}$ (normal) unit vector completes
the triad (${\bf N}={\bf R}\times{\bf T}$) pointing upwards from the ecliptic
plane. Figure 1 shows the representation of the RTN coordinate system. When
analyzing magnetic and electric field data at small scales, we use spacecraft
(SC) coordinates defined with respect to the spacecraft (Malaspina et al.,
2016). In the SC coordinate system, the $V_{12}$ and $V_{34}$ antennae pairs
are located in the $x\,-\,y$ plane, where ${\bf\hat{x}}$ lies roughly in the
PSP’s ram direction, ${\bf\hat{y}}$ is approximately downwards with respect to
the ecliptic plane. The third component, ${\bf\hat{z}}$, is perpendicular to
the thermal shield, pointing towards the Sun. The SC coordinate system is
optimal for making detailed comparisons between electric and magnetic fields.

Figure 1: RTN coordinate system and angles used in the study. The magnetic
field vector (red arrow) represents a characteristic situation when the
magnetic field is close to radial and points sunward, such as during PSP’s
$5^{\rm th}$ perihelion. The two angles shown are: $\theta_{\bf
B}=\arccos({\rm B_{N}}/{\bf|B|})$ (in blue), the polar angle of ${\bf B}$ with
respect to ${\bf N}$, and $\phi_{\bf B}=\arctan({\rm B_{T}}/{\rm B_{R}})$ (in
green), the azimuthal angle between the projection of the magnetic field
vector onto the RT plane and the line parallel to the ${\bf R}$ direction.

##  3 Results

We study in detail wave emissions with frequencies around the electron
gyrofrequency and its multiples. The study is divided into two parts. The
first one analyses the connection of the near-$f_{\rm ce}$ waves events with
large-scale properties of the ambient magnetic field. In the second part, we
analyse the small scale properties of the emission intervals and the effects
of low frequency waves with frequency around ion gyrofrequency on the
near-$f_{\rm ce}$ waves.

###  3.1 Large scale analysis

Figure 2 shows a full-day overview of PSP’s $5^{\rm th}$ perihelion, on 2020
June 7. Figure 2(a) depicts the plasma density from quasi-thermal noise
measurements (Moncuquet et al., 2020). Figure 2(b) shows proton velocity from
the SPAN-I instrument. During the entire day, there is a slow solar wind with
the proton radial velocity varying around $250\,\mathrm{km/s}$. Figure 2(c)
shows the magnetic field. Though the magnetic field varies considerably during
the day, it does not change its radial direction, with ${\rm B_{R}}$ pointing
towards the Sun (${\rm B_{R}}<0$) during the whole period. Figure 2(d) shows
the angles $\theta_{\bf B}$ and $\phi_{\bf B}$ as defined in Figure 1.
Throughout the interval $\theta_{\bf B}$ mainly varies around $90^{\degree}$
and $\phi_{\bf B}$ varies around $0^{\degree}$, corresponding to a near radial
magnetic field. Figure 2(e) shows the magnetic field spectrogram in the
frequency range from $2\,\mathrm{Hz}$ up to $292.7\,\mathrm{Hz}$. We can see
broadband wave emissions with varying amplitude throughout the interval. The
lower-hybrid frequency is highlighted by the white line. Finally, Figure 2(f)
shows the sum of the electric field spectra measured by the $V_{12}$ and
$V_{34}$ antennae pairs, in a frequency range from $366\,\mathrm{Hz}$ to
$72\,\mathrm{kHz}$, where the electron gyrofrequency is given by the white
line and the proton plasma frequency is represented by the black line. High
frequency wave emissions around the electron gyrofrequency and its multiples
can be seen throughout the day; those emissions are the focus of this study.

Figure 2: Overview of the day surrounding the $5^{\rm th}$ perihelion of PSP.
(a) Plasma density based on plasma frequency measurements. (b) Proton
velocity. (c) Magnetic field. (d) The direction of magnetic field, spherical
angles $\theta_{\bf B}$ and $\phi_{\bf B}$. The gray dashed lines mark the
boundaries of the intervals favorable to near-$f_{\rm ce}$ harmonics
emissions: $80^{\degree}<\theta_{\bf B}<100^{\degree}$ and
$10^{\degree}<\phi_{\bf B}<30^{\degree}$. (e) Magnetic fluctuations spectrum.
The white line represents the lower-hybrid frequency $f_{\rm LH}$. (f) Sum of
the high-frequency AC electric field spectrum calculated from differential
voltage data measured by ${\rm V}_{12}$ and ${\rm V}_{34}$ antennae pairs. The
black line shows the proton plasma frequency $f_{\rm p}$ and the white line
represents the electron cyclotron frequency $f_{\rm ce}$. The white markers
over the black background on the top of panel (f) mark the points in time
where both $\theta_{\bf B}$ and $\phi_{\bf B}$ are inside their respective
angular ranges favorable to near-$f_{\rm ce}$ harmonics emissions as seen in
panel (d).

A closer inspection of Figure 2 reveals that the emission of near-$f_{\rm ce}$
harmonics waves tends to occur when the background magnetic field assumes a
preferential direction. Most of near-$f_{\rm ce}$ harmonics emissions coincide
with instances where the magnetic field direction points to a direction within
the following angle intervals $80^{\degree}\lesssim\theta_{\bf B}\lesssim
100^{\degree}$ and $10^{\degree}\lesssim\phi_{\bf B}\lesssim 30^{\degree}$. In
Figure 2(d) the upper and lower limits of both angle intervals are indicated
by dotted gray lines. Furthermore, to illustrate this correlation, every
instance the magnetic field vector is inside the region limited by the angle
ranges described above is plotted in white over a black background on the top
of Figure 2(f). It can be noticed that near-$f_{\rm ce}$ emissions occur
mostly at the same time as the white markers on top, suggesting a correlation
between the waves and a particular spatial configuration of the magnetic
field.

Figure 3: The average intensity of near-$f_{\rm ce}$ harmonics emissions as a
function of the magnetic field direction. Data represent the whole day of
2020-06-07. Each dot corresponds to one spectral measurement averaged in the
frequency range of the near-$f_{\rm ce}$ harmonics waves,
$1.8\,\mathrm{kHz}<f<10\,\mathrm{kHz}$. The magnetic field direction is
characterized by $\theta_{\bf B}$ (vertical axis) and $\phi_{\bf B}$
(horizontal axis).

In Figure 3, we further analyze whether the emission of near-$f_{\rm ce}$
harmonics indeed depends on the magnetic field direction. Figure 3 shows the
average wave power of the near-$f_{\rm ce}$ emissions as a function of
$\theta_{\bf B}$ and $\phi_{\bf B}$. The average wave power is approximated by
integrating each wave spectrum in the frequency range
$1.8\,\mathrm{kHz}<f<10\,\mathrm{kHz}$. In terms of the electron
gyrofrequency, the chosen frequency range corresponds to $0.55<f/f_{\rm
ce}<3.5$, which is the optimal frequency interval for the near-$f_{\rm ce}$
harmonics during the day of the $5^{\rm th}$ perihelion, determined by visual
inspection. Looking at Figure 2(f), one can see that the selected frequency
band is dominated by the near-$f_{\rm ce}$ harmonic waves. During a few
intervals there are other type of emissions within the same frequency band,
for example, in the period between 07:30 $\sim$ 9:30 there are emissions that
are most probably ion-sound waves. These other waves do not seem to correlate
with the magnetic field direction and are, therefore, the most probable source
of the few high amplitude points randomly distributed around the scatter plot.
In Figure 3(a) the angle range, $80^{\degree}\leqslant\theta_{\bf B}\leqslant
100^{\degree}$ and $10^{\degree}\leqslant\phi_{\bf B}\leqslant 30^{\degree}$,
are marked by gray lines. One can notice that nearly all the wave power above
$10^{-5}\,\mathrm{(mV/m)^{2}Hz^{-1}}$ is bounded by lines identifying the
angle intervals discussed above. This confirms that the emissions of electron
cyclotron harmonics peak in a limited range of magnetic field polar and
azimuthal angles.

To illustrate the angular dependence of near-$f_{\rm ce}$ emissions in more
detail, we select for closer analysis a one-hour interval during $5^{\rm th}$
encounter that contains a short interval of about 15min containing
near-$f_{\rm ce}$ wave emissions. Figure 4 shows the one-hour interval
describing the plasma environment surrounding the wave emission event under
scrutiny. Due to the stable magnetic field and very low wave activity in the
electric field spectrum, where there are no other kinds of waves rather than
the near-$f_{\rm cs}$ harmonics, this event provides a close to ideal
situation for this analysis. Further, Figure 5 zooms to the interval of wave
emissions and we make a detailed analysis of how near-$f_{\rm ce}$ harmonic
waves correlate with changes in the magnetic field.

Figure 4: Near-$f_{\rm ce}$ harmonics event showing a strong correlation
between wave emission and variations in the magnetic field direction. Same
panels as in Figure 2, see detailed description there. (a) Plasma density. (b)
Solar wind velocity. (c) Magnetic field. (d) Spherical angles. (e) Magnetic
fluctuations. (f) AC electric field spectrum.

Figure 4(a) and (b) show the plasma density and velocity, respectively. These
two quantities do not have a determining influence on the emission of
near-$f_{\rm ce}$ harmonics, but provide the large scale properties of the
solar wind. Figure 4(c) shows the ambient magnetic field. During this time
interval, the magnetic field is quite stable and mostly radial. Figure 4(d)
depicts the spherical angles $\theta_{\bf B}$ and $\phi_{\bf B}$. The dashed
gray lines mark the upper and lower boundaries of the angle intervals where we
expect wave emissions to occur. Figure 4(e) shows the low frequency magnetic
fluctuation spectrum. The white line represents the lower-hybrid frequency,
and the straight line at $\sim 10\,\mathrm{Hz}$ is an instrumental artifact
caused by the magnetic signature of the reaction wheels and has no influence
on this analysis. Figure 4(f) depicts the high frequency electric field
spectrum. The sole wave activity at high frequencies during the whole one-hour
interval is the near-$f_{\rm ce}$ harmonic wave emissions. The emissions are
observed only during the time intervals when both angles in Figure 4(d) are
concurrently inside their respective angle interval. These occurrences are
also marked in white over a black background on the top of Figure 4(f). In
addition, we can see that there is a decrease in the amplitude of low
frequency magnetic fluctuations during the intervals of the near-$f_{\rm ce}$
harmonic wave emissions.

Figure 5: Zoomed in figure of near-$f_{\rm ce}$ harmonics event in Figure 4.
Shown are only the bottom four panels of Figure 4. (a) Magnetic field, (b)
spherical angles, (c) magnetic fluctuation spectrum, and (d) AC electric field
spectrum.

Figure 5 shows a further 15 min zoom-in on the highlighted interval in Figure
4. Figure 5(a) shows the magnetic field and Figure 5(b) shows the
corresponding polar and azimuthal angles. The boundaries that delimit the two
angular intervals favorable for wave emissions are marked with dotted lines.
Periods where both $\theta_{\bf B}$ and $\phi_{\bf B}$ are inside their
respective angular boundaries, satisfying the favorable conditions for
near-$f_{\rm ce}$ harmonics emission, are highlighted in yellow. Along these
15 minutes, the magnetic field is very close to the radial direction, as can
be seen, by the radial component ${\rm B_{R}}$ being dominant and having its
value very close to the negative of the magnitude of magnetic field
$-{\bf|B|}$. The largest deviations from the radial direction occur in regions
where ${\rm B_{T}}$ reach larger values. During most of the time, the normal
component ${\rm B_{N}}$ remains relatively close to ${\rm B_{N}}\approx
0\,\mathrm{nT}$, corresponding to $\theta_{\bf B}\approx 90^{\degree}$, and
throughout the whole period $\theta_{\bf B}$ stays within the interval
favorable for the emissions, including the few instances where ${\rm B_{N}}$’s
amplitude peaks at $|{\rm B_{N}}|\lesssim 20\,\mathrm{nT}$. On the other hand,
$\phi_{\bf B}$, which corresponds to the ${\rm B_{T}}$ variations, enters and
exits for several times the $\phi_{\bf B}$ interval favorable for the
emissions. Thus, for this particular event, the emission of near-$f_{\rm ce}$
harmonics waves depend only on ${\rm B_{T}}$ variations. Looking at Figure
5(d), it is clear that near-$f_{\rm ce}$ harmonics emission coincide with the
regions marked in yellow, where both $\phi_{\bf B}$ and $\theta_{\bf B}$ are
inside their respective angular boundaries. Furthermore, the variation in the
waves amplitude in Figure 5(d) relates to the variation in the angles within
their respective intervals in Figure 5(b). The closer both angles are to the
center of their interval favoring emissions, the stronger the wave emissions
are. Such correlation could already be inferred from Figure 3, where one can
see that the highest wave amplitudes lie close to the center of the bounded
interval. However, Figure 5 shows in detail how even a slight change in one of
the spherical angles immediately impacts the properties of near-$f_{\rm ce}$
harmonics waves.

Figure 5(c) shows the low frequency magnetic fluctuations spectrum. The
spectrum contains broadband fluctuations with varying intensity throughout the
interval and, in addition, there are localized emissions having frequency
peaks up to $f\approx 7\,\mathrm{Hz}$. Broadband magnetic fluctuations have
distinctively lower amplitudes during the intervals of the near-$f_{\rm ce}$
harmonics wave emissions. The amplitude decrease follows the same correlation
as described above for the near-$f_{\rm ce}$ harmonics, with the lowest
broadband amplitudes corresponding to instances where both $\phi_{\bf B}$ and
$\theta_{\bf B}$ are closer to the center of their respective intervals
favoring wave emissions. On the other hand, the localized emissions at a few
Hz do not show any direct correlation with the near-$f_{\rm ce}$ harmonics
waves or the spherical angles.

Figure 6: Near-$f_{\rm ce}$ harmonics event from 2021 January 16 (orbit 7),
showing the same correlations between the magnetic field direction and the
high frequency electric field wave emissions. (a) Plasma density. (b) Solar
wind velocity. (c) Magnetic field. (d) Spherical angles. (e) Magnetic
fluctuations. (f) AC electric field spectrum.

Figure 6 shows an interval from orbit 7, measured at a distance of $\sim
22.78\,\mathrm{R_{\odot}}\,(\sim 0.10\,\mathrm{AU})$ from the Sun. In this
event, a series of near-$f_{\rm ce}$ harmonics waves occur in a more active
plasma environment. The figure clearly shows that the correlations and
conditions for near-$f_{\rm ce}$ harmonics wave emissions shown in Figure 5
hold also for different solar wind conditions and different PSP distances from
the Sun. Figure 6(a) and Figure 6(b) show the plasma density and velocity,
respectively. While the plasma density is higher in this event, the slow solar
wind has a similar plasma velocity as seen in Figure 4(b). The plasma density
makes several step-wise variations, however, there is no clear correlation
between the emission of near-$f_{\rm ce}$ harmonics in Figure 6(f) and plasma
density variations. Figure 6(c) shows the magnetic field, the radial component
points outwards from the Sun $({\rm B_{R}}>0)$. The magnetic field shows a
higher degree of variations than the period shown in Figure 4(c). Figure 6(d)
shows the spherical angles $\phi_{\bf B}$ and $\theta_{\bf B}$. Though
$\phi_{\bf B}$ varies in a larger range than $\theta_{\bf B}$, both spherical
angles go in and out of their wave emission intervals. Instances where both
$\phi_{\bf B}$ and $\theta_{\bf B}$ are concurrently within their respective
angular boundaries correlate very well with near-$f_{\rm ce}$ harmonics
emissions in Figure 6(f). In Figure 6(e), the low frequency magnetic
fluctuations spectrum shows a clear correlation between decreased amplitudes
in the broadband fluctuations and time intervals where both $\phi_{\bf B}$ and
$\theta_{\bf B}$ are inside their respective boundaries. Throughout the 40 min
interval, there are high amplitude wave emissions with spectra peaks at a few
Hz that is close to the ion-cyclotron frequency (black line).

###  3.2 Small scale analysis

Figure 7 shows a $\sim 3.5\,\mathrm{s}$ near-$f_{\rm ce}$ harmonics event
recorded in burst mode at $18:28:09.459$ on 2020 June 07 (PSP’s $5^{\rm th}$
perihelion). Figures 7(a) and (b) show magnetic field and high amplitude
oscillations with frequencies close to the ion gyrofrequency can be seen.
These are circularly polarized waves as can be seen from the $90^{\degree}$
phase shift between ${\rm B_{x}}$ and ${\rm B_{y}}$ components. Figure 7(c)
shows the azimuthal angle $\phi_{\bf B}$ and the polar angle $\theta_{\bf B}$
with the dotted lines marking the corresponding boundary of the angular range
favoring wave emissions. While $\phi_{\bf B}$ stays all the time within the
angle range favorable for the wave emissions, $\theta_{\bf B}$ crosses the
boundary for short intervals multiple times. Figure 7(d) shows the electric
field waveform of the burst data. One can observe time-localized wave packets
with amplitudes that correlate very well with the ion cyclotron wave
oscillations in the magnetic field. On close inspection, we can see that
minimum in the amplitude correspond to the intervals when $\theta_{\bf B}$
crosses the boundary of the angle range favorable for the wave emissions and
the maximum is when $\theta_{\bf B}$ is furthest from the boundary. Figure
7(e) shows the total electric field power spectrum, where a series of time-
localized near-$f_{\rm ce}$ harmonics structures, regularly distributed
throughout the interval can be seen. They correspond to the largest amplitude
wave packets seen in Figure 7(e). In addition, besides the near-$f_{\rm ce}$
harmonics one can also distinguish localized wave emissions having peak
frequencies at about a few times the electron cyclotron frequency that do not
show harmonic structure at electron cyclotron frequencies. Those are most
probably ion sound waves and they do not show any correlation with the
magnetic field direction. In addition, there are during the gap intervals the
are narrow banded emissions that seem to connect the harmonics structures.

Figure 7: High-cadence $\sim 3.5\,\mathrm{s}$ burst data showing a series of
time-localized near-$f_{\rm ce}$ harmonics wave emissions correlated with low
frequency magnetic field oscillations. (a,b) Magnetic field in spacecraft
coordinates. (a) ${\rm B_{z}}$ and $|{\rm B_{z}}|$. (b) ${\rm B_{x}}$ (left
axis) and ${\rm B_{y}}$ (right axis). (c) Spherical angles: $\phi_{\bf B}$
(left axis) with the black dotted line marking the $\phi_{\bf B}=30^{\degree}$
boundary, and $\theta_{\bf B}$ (right axis) with the blue dotted line marking
the $\theta_{\bf B}=80^{\degree}$ boundary. (d) Electric field waveform of the
burst data. The two available components, ${\rm E_{x}}$ and ${\rm E_{y}}$, are
in spacecraft coordinates. (e) Total electric field power spectrum. The white
dotted line represents the electron cyclotron frequency.

##  4 Discussion

Near-$f_{\rm ce}$ harmonics waves are observed exclusively during time
intervals where the magnetic field direction is within (or very close to) the
following ranges: $80^{\degree}\lesssim\theta_{\bf B}\lesssim 100^{\degree}$
and $10^{\degree}\lesssim\phi_{\bf B}\lesssim 30^{\degree}$ (see Figure 3).
This corresponds to the magnetic field orientation being close to radial but
slightly tilted in the direction of positive $\phi_{\bf B}$ (${\rm B_{T}}>0$
when ${\rm B_{R}}<0$, or ${\rm B_{T}}<0$ when ${\rm B_{R}}>0$) with the lower
boundary of $\phi_{\bf B}=10^{\degree}$. While earlier studies have shown that
the emissions prefer near-radial magnetic field conditions (Malaspina et al.,
2020), the current study for the first time quantifies the angular range where
waves are observed. It is important to mention that the correlation between
the near-$f_{\rm ce}$ harmonics waves and the angular interval estimated in
this study is a consistent observation for nearly every such wave event
measured by PSP throughout the 11 close encounters with the Sun available
until the end of this study (not shown).

Figure 8: Probability distribution with respect to $\phi_{\bf B}$ for the day
around the $5^{\rm th}$ orbit perihelion. (red) all data points, (yellow) data
points with wave emission power being above (a) $1.0\times
10^{-4}\,\mathrm{(mV/m)^{2}Hz^{-1}}$ and (b) $2.0\times
10^{-4}\,\mathrm{(mV/m)^{2}Hz^{-1}}$ , (blue) estimated Parker spiral angle
for all data. Each histogram bin corresponds to $1^{\degree}$, and the sum of
the bar heights of each quantity in the total angle range ($\pm 90^{\degree}$)
is $\leqslant 1$.

To characterize further the properties of the wave emissions we analyze the
probability distribution of wave emission intervals with respect to $\phi_{\bf
B}$. Figure 8 shows three probability distributions for the day around the
$5^{\rm th}$ orbit perihelion: the distribution of measured $\phi_{\bf B}$
values for all data points (red) and for the data points with wave emissions
above a given threshold value (yellow), as well as the estimated values of
Parker spiral angle $\phi_{\rm PS}$ for all data points. The probability
distribution of data points with wave emissions is shown for two different
threshold values: $1.0\times 10^{-4}\,\mathrm{(mV/m)^{2}Hz^{-1}}$ (on the
left) and $2.0\times 10^{-4}\,\mathrm{(mV/m)^{2}Hz^{-1}}$ (on the right). The
vertical dotted lines mark the $\phi_{\bf B}$ boundaries that we identify as
favorable to wave emission. During this day the average magnetic field has a
$\phi_{\bf B}$ value slightly below $10^{\degree}$ and the distribution of
$\phi_{\bf B}$ values for all data points has a broad peak between
$-40^{\degree}$ and $+40^{\degree}$. This can be compared to the expected
Parker spiral angle values that are in the range of $+11^{\degree}$ to
$+17^{\degree}$. We see that on average the magnetic field is more radial than
the expected Parker spiral angle. The result is different for magnetic field
lines containing the wave emissions. As expected, most of the wave power
(yellow bars) is within the dotted lines that mark the $\phi_{\bf B}$ angle
range favorable to wave emission, with most of the wave power concentrated
between $15^{\degree}\lesssim\phi_{\bf B}\lesssim 25^{\degree}$. Thus, during
wave emissions, the magnetic field has $\phi_{\bf B}$ values that are above
the average for the same period. In addition, while the Parker spiral angle
distribution is inside the angular range favorable to wave emission, it is
right below the angle interval where the highest wave-power data are
distributed. To summarize, the magnetic field during wave emissions has the
average value of $\phi_{\bf B}$ larger than the average values of magnetic
field throughout the interval and larger than the average values of Parker
spiral angle.

Both $\theta_{\bf B}$ and $\phi_{\bf B}$ directions are important and have
equivalent influence on the emission and properties of near-$f_{\rm ce}$
harmonics waves. Wave emission occurs as soon as $\phi_{\bf B}$ and
$\theta_{\bf B}$ are simultaneously inside their respective ranges and
immediately ceases when at least one of the angles leaves its wave emission
interval. Furthermore, the amplitudes of near-$f_{\rm ce}$ harmonics waves
increase gradually as both spherical angles approach the center of their
respective wave emission intervals. From Figures 3 and 8, one can already
infer that the highest wave powers tend to occur in an even more restricted
angular range, closer to the center of the angle intervals where wave emission
occurs.

Another important finding is the correlation between the intensity of
near-$f_{\rm ce}$ harmonics and the intensity of broadband magnetic
fluctuations within the frequency range between about $2\,\mathrm{Hz}$ and
$200\,\mathrm{Hz}$. It is known that near-$f_{\rm ce}$ harmonics tend to occur
when the magnetic field turbulence at low frequencies is weak. Malaspina et
al. (2020) shows that statistically more wave emissions are observed in
regions of lower magnetic turbulence amplitude at frequencies below
$1\,\mathrm{Hz}$ (which is below proton gyrofrequency). The present study
shows, for the first time, that the amplitude decrease of broadband magnetic
fluctuations at higher frequencies, well above proton gyrofrequency, is well
correlated to the amplitude increase of the near-$f_{\rm ce}$ harmonics. This
observation is a consistent feature of these waves, which may suggest a causal
relationship between the drop in the magnetic field turbulent energy and the
amplitude increase of the high-frequency electrostatic waves.

Figure 9: High and low-frequency wave emissions around the $5^{\rm th}$
perihelion. (a) High-frequency electric field spectrum from AC differential
voltage data measured by ${\rm V}_{12}$ and ${\rm V}_{34}$ antennae pairs
(same as Figure 2(f)). (b) Low-frequency magnetic field spectrogram from
fluxgate magnetometer data. (c) Degree of polarization, magnetic field. (d)
Ellipticity of the magnetic field waves in the spacecraft frame. Ellipticity
is +1 for right-hand and -1 for left-hand polarization. In panels (c) and (d)
the gray area corresponds to points where either the degree of polarization is
below 0.7 or the wave power is below $1\times
10^{-3}\,\mathrm{nT^{2}Hz^{-1}}$. In panels (b), (c) and (d) the black line
represents the ion gyrofrequency $f_{\rm ci}$. (e) Magnetic field in RTN
coordinates, same as Figure 2(c).

To better understand the relationship between the near-$f_{\rm ce}$ harmonics
waves and low frequency waves, we analyze the high-frequency electric field
spectrum and the low frequency waves as observed in magnetic field during
encounter 5, see Figure 9. Magnetic field spectrum in Figure 9(b) shows that
in addition to broadband magnetic field fluctuations there are localized wave
emissions with frequencies close to and above the proton gyrofrequency (black
line). These waves have high degree of polarization, see Figure 9(c), and are
left- or right-hand polarized, see Figure 9(d). There is no clear correlation
between the intervals of the low-frequency waves and the intervals of the
near-$f_{\rm ce}$ harmonics emissions. For instance, in the interval from
07:00 to 10:00, there are no near-$f_{\rm ce}$ harmonics, but there are low-
frequency waves with the same polarization properties as the waves observed
during near-$f_{\rm ce}$ harmonics events. As an opposite example, there is a
near-$f_{\rm ce}$ harmonics event from 16:25 to 16:35 that has no low
frequency waves occurring at the same time. Analyzing PSP data from different
orbits (not shown here), we notice the same pattern as described above:
near-$f_{\rm ce}$ harmonics waves and low-frequency electromagnetic waves are
most of the time observed to occur simultaneously, but there are a few
instances where both waves are observed to occur independently of each other.
Considering these observations, we cannot rule out the possibility of a simple
coincidence. Near radial, quiescent magnetic field, with relatively low levels
of turbulent fluctuations is a favorable environment for the occurrence of
both the high-frequency electrostatic waves with frequencies around $f_{\rm
ce}$, and the low-frequency waves with frequencies close to $f_{\rm cp}$, so
they end up being observed during the same time intervals most of the time.
Unlike the emission of near-$f_{\rm ce}$ harmonics, the emission of low-
frequency waves does not have any constraints regarding the magnetic field
direction and are observed also at large heliocentric distances. For example,
Solar Orbiter has observed these waves at distances above
$>40\,\mathrm{R_{\odot}}$ (Khotyaintsev et al., 2021). This clearly shows that
the emission of ion-scale waves is independent of the occurrence near-$f_{\rm
ce}$ harmonics. However, there is no conclusive evidence showing the opposite
situation, i.e., that the emission of near-$f_{\rm ce}$ harmonics is
independent of the occurrence of ion-scale waves. The high rate of co-
occurrence between the waves and the fact that some of the ion-scale waves can
affect near-$f_{\rm ce}$ harmonics properties at small timescales, shows that
the waves are not completely uncorrelated. It is possible that the instability
exciting the ion-scale waves may be responsible to trigger a secondary
instability at electron scales that could lead to the emission of the
near-$f_{\rm ce}$ harmonics. In this case, the few instances where we see the
near-$f_{\rm ce}$ harmonics waves in the absence of the low-frequency waves
could be explained by the stabilization of the ion-scale instability, while
the triggered electron-scale instability remains ongoing.

In this study, similar to earlier studies, we could not identify the
generation mechanism of the near-$f_{\rm ce}$ harmonics waves. Different
mechanisms have been suggested: electron cyclotron drift instability, loss-
cone instability, electromagnetic pump wave, and wave-wave interaction
(Malaspina et al., 2021). The observations reported in this study, that the
waves favor a particular range of magnetic field directions, do not support
any of the previously proposed wave generation mechanisms. Rather the
opposite, it is not obvious why the presence of gradients or loss-cone would
favor a particular range of magnetic field directions. Similarly, there are no
observations that other types of waves would favor similar dependence on the
magnetic field. As a result, the possibility that wave-wave interaction is the
source of the waves is small. One possibility could be that the emissions are
caused by spacecraft interaction with the environment, and the direction bias
would be due to the geometry of the spacecraft and/or electric field
instrument. However, this hypothesis does not explain why the waves are
observed only when the broadband magnetic turbulence is low unless their
presence somehow affects measurements made by both MAG and SCM instruments. So
far, the only definitive conclusion we can take from the results presented in
this work is that the near-$f_{\rm ce}$ harmonics must be generated locally
since their emission and amplitude levels correlate with the local properties
of the ambient magnetic field. Any generation mechanism proposed for the
near-$f_{\rm ce}$ harmonics must take into account their dependence on the
magnetic field direction. Moreover, understanding how and why these waves have
such a close connection with the magnetic field, including their correlation
with low levels of broadband magnetic turbulence, is crucial to identifying
the source of these waves and determining whether or not they are spacecraft
generated.

##  5 Conclusion

We have used Parker Solar Probe data to investigate the connection between the
ambient magnetic field and the emission of near-$f_{\rm ce}$ harmonics waves
in the near-Sun solar wind. The results suggest that the magnetic field vector
direction is a determining factor for the emission of near-$f_{\rm ce}$
harmonics. Wave emission occurs only when the ambient magnetic field points to
a narrow region in space, bounded by polar and azimuthal angular ranges in the
RTN coordinate system of correspondingly $80^{\degree}\lesssim\theta_{\bf
B}\lesssim 100^{\degree}$ and $10^{\degree}\lesssim\phi_{\bf B}\lesssim
30^{\degree}$. This dependence on the magnetic field direction is consistent
for almost every near-$f_{\rm ce}$ harmonics event measured by PSP throughout
its eleven close orbits to the Sun. The few outliers do not deviate more than
$5^{\degree}$ from one of the angular boundaries described above. Wave
amplitudes are highest when both angles are close to the center of their
respective angular interval favorable to wave emissions. In addition, the data
shows that the angular range for wave emission, as well as the intensity
peaking when both spherical angles are closer to center of their respective
wave emission interval, is observed for variations in the magnetic field
direction at time scales that vary from large-scale variations on a minute
scale down to small-scale magnetic field oscillations at a sub-second scale.

Near-$f_{\rm ce}$ harmonics emissions are well correlated to a noticeable
decrease in the intensity of broadband magnetic field fluctuations in the
frequency range $10\,\mathrm{Hz}\sim 100\,\mathrm{Hz}$. The higher the
amplitude of the near-$f_{\rm ce}$ harmonics, the more pronounced the decrease
observed in the magnetic field turbulent spectrum. During the wave emissions,
there can be present low-frequency waves with frequency $2\,\mathrm{Hz}$ to
$10\,\mathrm{Hz}$, which corresponds to a few times the proton gyrofrequency.
The low-frequency waves show a high degree of polarization and can be left- or
right-handed polarized, while broadband fluctuations do not show a distinct
polarization pattern. We could not identify a clear correlation between the
near-$f_{\rm ce}$ harmonics and the low-frequency waves. However, due to the
high co-occurrence rate between the high and low frequency waves, and the fact
that the ion-scale waves may interact with the near-$f_{\rm ce}$ harmonics at
small time scales, we could not fully discard a possible correlation. We
speculate that the ion-scale waves could trigger a secondary, electron-scale
instability that would be responsible for the emission of the near-$f_{\rm
ce}$ harmonics. However, more studies are required to determine if the low-
frequency waves actually have any contribution to the emission of the high-
frequency harmonics.

Regarding the generation mechanism, the observations presented in this work
are inconclusive and do not favor any of the emission mechanisms proposed in
Malaspina et al. (2021). The present report on dependence on the magnetic
field direction for the emission of the near-$f_{\rm ce}$ harmonics is, to our
knowledge, an unprecedented observation in the free solar wind plasma, which
suggests that these waves may be connected to a distinctive, underlying
property of the near-Sun environment. On the other hand, we do not discard the
possibility that these waves may be spacecraft generated. More studies are
required to understand the source of these these waves and their role in the
early stages of the solar wind evolution.

The authors thank the _Parker Solar Probe_ , FIELDS and SWEAP teams. The
FIELDS experiment on _Parker Solar Probe_ spacecraft was designed and
developed under NASA contract NNN06AA01C. All data used in this work are
publicly available on the FIELDS
(http://research.ssl.berkeley.edu/data/spp/data/) and SWEAP
(http://sweap.cfa.harvard.edu/pub/data/sci/sweap/) data archives. Data
analysis was performed using the IRFU-Matlab analysis package available at
https://github.com/irfu/irfu-matlab/tree/PSPdevel. SFT and AV acknowledge
support from Swedish National Space Board Contract 163/19. SFT acknowledges
Henriette Trollvik for the helpful conversations regarding the statistical
analysis and data visualization and Vinícius Ferreira for the insightful
discussion about wave modulation.

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