# Density And Velocity Fluctuations of Alpha Particles in Magnetic Switchbacks

Michael D. McManus Physics Department, University of California, Berkeley, CA
94720-7300, USA  Space Sciences Laboratory, University of California,
Berkeley, CA 94720-7450, USA Michael D. McManus<EMAIL_ADDRESS>Jaye
Verniero Heliophysics Science Division, NASA, Goddard Space Flight Center,
Greenbelt, MD 20771, 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  Trevor A. Bowen Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
Davin Larson Space Sciences Laboratory, University of California, Berkeley, CA
94720-7450, USA  J. C. Kasper Climate and Space Sciences and Engineering,
University of Michigan, Ann Arbor, MI 48109, USA  Smithsonian Astrophysical
Observatory, Cambridge, MA 02138 USA  Roberto Livi Space Sciences Laboratory,
University of California, Berkeley, CA 94720-7450, USA  Lorenzo Matteini The
Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK  Ali Rahmati
Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450,
USA  Orlando Romeo Space Sciences Laboratory, University of California,
Berkeley, CA 94720-7450, USA  Phyllis Whittlesey Space Sciences Laboratory,
University of California, Berkeley, CA 94720-7450, USA  Thomas Woolley The
Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK

###### Abstract

Magnetic switchbacks, or sudden reversals in the magnetic field’s radial
direction, are one of the more striking observations of Parker Solar Probe
(PSP) thus far in its mission. While their precise production mechanisms are
still unknown, the two main theories are via interchange reconnection events
and in-situ generation. In this work density and abundance variations of alpha
particles are studied inside and outside individual switchbacks. We find no
consistent compositional differences in the alpha particle abundance ratio,
$n_{\alpha p}$, inside vs outside, nor do we observe any signature when
separating the switchbacks according to $V_{\alpha p}/V_{pw}$, the ratio of
alpha-proton differential speed to the wave phase speed (speed the switchback
is travelling). We argue these measurements cannot be used to rule in favour
of one production mechanism over the other, due to the distance between PSP
and the postulated interchange reconnection events. In addition we examine the
3D velocity fluctuations of protons and alpha particles within individual
switchbacks. While switchbacks are always associated with increases in proton
velocity, alpha velocities may be enhanced, unchanged, or decrease. This is
due to the interplay between $V_{pw}$ and $V_{\alpha p}$, with the Alfvénic
motion of the alpha particles vanishing as the difference $|V_{pw}-V_{\alpha
p}|$ decreases. We show how the Alfvénic motion of both the alphas and the
protons through switchbacks can be understood as approximately rigid arm
rotation about the location of the wave frame, and illustrate that the wave
frame can therefore be estimated using particle measurements alone, via sphere
fitting.

##  1 Introduction

One of the more striking results of Parker Solar Probe’s (PSP, Fox et al.
(2016)) mission thus far is the ubiquity, in the near-Sun solar wind, of
magnetic switchbacks - large, sudden rotations of the magnetic field,
accompanied by spikes in the radial solar wind velocity. While switchbacks
have previously been observed both in the inner heliosphere using Helios
measurements (Borovsky, 2016; Horbury et al., 2018), and at 1AU and beyond
(Kahler et al., 1996; Neugebauer & Goldstein, 2013), these recent PSP
observations have sparked renewed interest in their nature and origins.

###  1.1 Properties

Switchbacks (hereafter SBs) are long, thin (Laker et al., 2021; Horbury et
al., 2020), S-shaped (McManus et al., 2020) magnetic structures, most likely
oriented along the magnetic field direction (Laker et al., 2021). They are
mostly Alfvénic in nature, with constant magnitude $|\mathbf{B}|$ field
corresponding to the condition of spherical polarisation. The Alfvénic
correlations between $\mathbf{B}$ and $\mathbf{v}$ mean that the field
rotations of SBs are accompanied by large positive spikes in the proton
velocity, regardless of the underlying polarity of the magnetic field
(Matteini et al., 2014). They don’t occur continuously but rather appear in
“patches” (Bale et al., 2019; de Wit et al., 2020), separated by periods of
quiet, steady flow and radial magnetic field. The proton core temperature
appears unchanged within individual SBs (Woolley et al., 2020; Martinović et
al., 2021), however the patches themselves appear to be overall hotter than
the quiet interstitial periods (Woodham et al., 2021; Bale et al., 2021).

###  1.2 SB Formation Theories

The question of what mechanisms are responsible for switchback generation is
still an open one. Several ideas have been put forward, generally coming in
two main flavours; the first involves generation via magnetic reconnection.
Fisk & Kasper (2020) postulate that due to large scale equatorial circulation
of the photospheric magnetic field, open magnetic field lines are dragged
across closed loops at lower latitudes, causing interchange reconnection
events which launch S-shaped kinks into the corona. Zank et al. (2020)
describe a similar idea but with the reconnection occurring significantly
higher up in the corona and launching fast magnetosonic type modes both up and
down the open field lines.

An alternative idea is that switchbacks naturally form in the solar wind as it
expands and travels outwards. Magnetic field fluctuations decay more slowly
with radial distance $R$ than the mean magnetic field does, resulting in
normalised amplitudes of Alfvénic fluctuations increasing as a function of
$R$. This means that out of the bath of initially small amplitude, linear
Alfvén waves known to be present at the base of the corona, the normalised
fluctuation amplitudes grow as the plasma travels outwards until they
eventually become large enough to cause the field to switch back on itself.
Mallet et al. (2021) develop an analytical model of such large-amplitude
Alfvén waves in an expanding solar wind and make several testable predictions
for the properties of the SBs produced, and similar results have been found
via MHD simulations (Squire et al., 2020; Shoda et al., 2021). In-situ
generation of SBs then very naturally explains the observation that the SB
filling fraction increases as a function of radius (Badman, Samuel T. et al.,
2021; Mozer et al., 2020; Macneil et al., 2020), something that is difficult
to explain for theories involving a purely low coronal origin. It would also
explain switchback “patches” as corresponding to wind that has undergone
greater expansion in transit; Bale et al. (2021) provide strong evidence that
at least some of the patches observed by PSP so far are due to superradially
expanded wind originating from the boundaries of supergranules at the solar
surface.

Recent analysis of Ulysses, Helios and PSP data by Tenerani et al. (2021)
however suggests that the scaling of SB occurrence as a function of radial
distance $R$ in fact depends on the size or duration of the switchback, with
shorter duration SBs decaying with $R$ and longer ones persisting. This, along
with the non-uniform properties of SBs (wide range of durations (de Wit et
al., 2020), some exhibiting compressibility while most do not (Krasnoselskikh
et al., 2020), different types of discontinuity at the boundaries (Larosa, A.
et al., 2021), etc) could be evidence that both types of generation mechanism
are occurring, and we are seeing a combination of short-duration SBs naturally
decaying via processes like parametric decay within a few tens of solar radii
(Tenerani et al., 2020), while in-situ generation is replenishing the
population of longer duration SBs. At this stage this is still speculative and
there are many open questions regarding the formation, evolution and eventual
decay of SBs.

For completeness we mention that there are other potential SB generation
mechanisms unrelated to the two just described. Ruffolo et al. (2020) suggest
they may be associated with the onset of shear-driven turbulence at or above
the Alfvén critical surface. Velocity shears between adjacent flux tubes can
then potentially be large enough to trigger the onset of Kelvin-Helmholtz type
instabilities and their associated vorticity roll-ups, producing the large
deflections in $\mathbf{B}$ that we observe as SBs. Schwadron & McComas (2021)
also postulate that SBs are produced by shear interactions between fast and
slow streams above the Alfvén surface (when ram pressure becomes dominant), in
particular pointing out that this should occur in the super-Parker spiral type
magnetic fields produced by footpoint motion across the leading edges of
coronal holes.

In this paper we focus on one small piece of the picture, namely the behaviour
of alpha particles inside vs outside individual switchbacks, and whether or
not this can help distinguish between any potential generation mechanisms.

##  2 Methods

###  2.1 Data

For this study we focus on PSP’s third and fourth encounters (E3 and E4) from
Aug 27th to Sep 8th, and 2019 and Jan 23rd to Feb 3rd, 2020, respectively. We
use data from the FIELDS magnetometers (Bale et al., 2016) for high resolution
magnetic field $\mathbf{B}$ measurements and down-sample to match particle
measurement cadences as needed. 3D ion velocity distribution function (VDF)
measurements are taken from the SPAN-Ion electrostatic analyser (Livi et al.,
2021; Kasper et al., 2016), with proton and alpha counts spectra produced at
cadences of 7s and 14s respectively. To the proton channel spectra we fit a
bi-Maxwellian to both the core and beam populations, with the proton beam
constrained to lie along the magnetic field relative to the core velocity. The
alpha channel contains a small ($2\%$) contamination from the proton channel,
which manifests as scaled down proton core and beam VDFs in the alpha channel.
This was accounted for by taking the previously fitted proton parameters and
reducing the density down to fit the extraneous protons. An additional single
bi-Maxwellian was then fit to the alpha part of the spectrum and the core
alpha particle VDF parameters extracted. The $\sim 2\%$ scaling factor is a
free parameter in the fit; it was checked that there was no energy or angle
dependence in the contaminant protons so that an overall scaling was
sufficient. The uncertainties on the fitted alpha densities are approximately
$10\%$.

We will on occasion require proton density measurements. For this we use
quasi-thermal noise (QTN) estimates derived from extraction of the plasma line
from FIELDS RFS spectra (Romeo et al., 2021), and approximate
$n_{e}=n_{p}+2n_{\alpha}\approx n_{p}$, as the alpha abundance in PSP’s early
encounters is very low (Woolley et al., 2021).

###  2.2 Switchbacks

Our dataset of SBs consists of 92 examples chosen by visual inspection from E3
and E4. Using SPAN-Ion data as the source of our ion measurements means we are
constrained by the alpha particle 14s measurement cadence to selecting
relatively longer SBs, and so cannot use quite as large an event database as
in some previous studies (Martinović et al., 2021). Following Martinović et
al. (2021) we split each SB into five distinct regions: Leading Quiet (LQ),
the relatively quiescent period immediately preceding the switchback; Leading
Transition (LT), the transition corresponding to the rotation of the magnetic
field; Switchback (SB), the interior of the switchback structure; Trailing
Transition (TT), the second transition, and finally the Trailing Quiet (TQ)
region, the quiescent field immediately following the passing of the SB. In
this work we are mainly interested in comparing the quiescent “background”
conditions to the interior of the switchback, rather than the transition
regions which represent the edges of the magnetic structure (and display a
host of interesting physics, including signatures of reconnection (Froment et
al., 2021) and wave activity (Agapitov et al., 2020; Krasnoselskikh et al.,
2020)). Figure 1 shows a prototypical example SB, with the five regions
indicated with vertical dashed lines.

Figure 1: Z-component (in PSP spacecraft coordinates) of the magnetic field
for a typical switchback, showing the demarcation of different regions:
Leading Quiet (LQ), Leading Transition Region (LTR), Switchback interior (SB),
Trailing Transition Region (TTR), and Trailing Quiet (TQ).

##  3 Results and Discussion

###  3.1 Density and Abundance Changes

The left plot in figure 2 shows a histogram of the fractional change in alpha
number density between SB interiors and their leading quiet (LQ) regions,
$(n_{\alpha}^{\mathrm{SB}}-n_{\alpha}^{\mathrm{LQ}})/n_{\alpha}^{\mathrm{LQ}}$.
While the spread in fractional density changes is quite large, the mean (and
median) of $\Delta n_{\alpha}/n_{\alpha}^{\mathrm{LQ}}$ are both very close to
zero (0.05 and 0.02 respectively). This is qualitatively very similar to the
proton fractional density changes in SBs reported both in observations and
simulations (see Figure 4 of Larosa, A. et al. (2021) and Figure 10 of Shoda
et al. (2021) respectively).

Figure 2: Histograms of fractional change in alpha density (left) and alpha
abundance (right) between switchback interiors and their leading quiet
regions.

The histogram on the right in figure 2 shows the change in alpha abundance
$\Delta n_{\alpha
p}=n_{\alpha}^{\mathrm{SB}}/n_{p}^{\mathrm{SB}}-n_{\alpha}^{\mathrm{LQ}}/n_{p}^{\mathrm{LQ}}$
between the same two regions (note that although one might have $\langle\Delta
n_{p}\rangle\approx 0$ and $\langle\Delta n_{\alpha}\rangle\approx 0$, a
priori they need not be statistically independent). For proton densities we do
not use SPAN-Ion measurements of $n_{p}$, but rather estimates of $n_{p}$ from
FIELDS QTN measurements as detailed in section 2. The large $\delta\mathbf{V}$
associated with SBs often moves the proton VDF significantly out of SPAN-Ion’s
field of view (FOV), which results in a large (unphysical) proton density
decrease as measured by SPAN. While fitting does mitigate the problem
somewhat, it is often not enough to completely eliminate these instrumental
density decreases. Using SPAN-Ion measurements only would then appear to show
large spikes in the alpha abundance inside SBs compared to outside (not
plotted here), which as we have shown is not the case. As we will explain in
the next section, the alpha particle VDFs tend to move much less in velocity
space during SBs, and so the problem is much less significant and any motion
that does occur can be properly captured by the fitting routines. Again, while
the spread in the right histogram of figure 2 is relatively large, the
distribution is clearly peaked about $\Delta n_{\alpha p}/n_{\alpha p}\approx
0$. We interpret these two figures then as showing there is no statistically
significant change in either the alpha density or the alpha abundance inside
SBs vs outside.

Figure 3: Scatter plot of fractional change in alpha abundance in switchbacks
vs alpha proton drift as a fraction of the local Alfvén wave phase speed. The
distribution is symmetric showing no strong dependence.

Lack of a compositional signature difference between the SB and LQ regions
strongly suggests we are measuring the same plasma inside vs outside, in
agreement with previous interpretations of SBs (Yamauchi et al., 2004; McManus
et al., 2020; Martinović et al., 2021; Woolley et al., 2020). We would
certainly expect SBs generated in-situ to not display any compositional
differences in the plasma inside the SB compared to outside. However, these
observations do not rule out coronal origins of SBs. SBs generated by
interchange reconnection events further down in the corona may very well be
expected to display compositional differences at the time they are generated.
This is because the properties of plasma confined in closed magnetic loops is
known to change (relative to open field lines) over the confinement time, due
to processes like gravitational settling and the first ionization potential
(FIP) effect (Laming et al., 2019; Rakowski & Laming, 2012). However the only
way this would be measurable at PSP is if the alpha particles and the SB
travel outwards together at exactly the same speed from their point of origin,
preserving the compositional signature difference. While it has long been
generally understood that alpha particles _do_ travel faster than the protons
at approximately the local wave speed (Thieme et al., 1990; Steinberg et al.,
1996; Matteini et al., 2015), giving rise to the phenomenon of alpha particle
“surfing” whereby alpha particles are less affected by the Alfvénic
fluctuations, we now show that this isn’t always the case, and that expecting
a compositional signature to persist to PSP distances would require rather
unphysical fine-tuning.

In fig. 3 we plot the change in alpha abundance $\Delta n_{\alpha p}$ vs the
ratio of alpha-proton drift speed to wave speed, $V_{\alpha p}/V_{W}$.
$V_{\alpha p}$ is calculated as $|V_{\alpha}-V_{pc}|$ where $V_{\alpha}$ and
$V_{pc}$ are the alpha and proton core velocities respectively, and $V_{W}$ is
computed by taking the normal N-component of the equation

$\delta\mathbf{V}=\pm V_{W}\frac{\delta\mathbf{B}}{|\mathbf{B}|},$ (1)

which serves to define the wave speed (Goldstein et al., 1995). Plotting
$V_{N}$ vs $B_{N}/|\mathbf{B}|$ over the LQ region associated with each SB and
taking the gradient of a line of best fit then yields an estimate of the local
wave phase speed. (Note that equation 1 is effectively an empirical
measurement of the speed of Alfvénic fluctuations - it is not yet fully
understood why $V_{W}$ is usually less than $V_{A}$ in the solar wind
(Goldstein et al., 1995; Neugebauer et al., 1996).)

From figure 3 we clearly see that alpha particles do not always travel at the
local wave speed; when considering short intervals such as these, there is a
very wide range of $V_{\alpha p}/V_{W}$ values. There also does not appear to
be any trend in $\Delta n_{\alpha p}$ with $V_{\alpha p}/V_{W}$. In
particular, there is no signature around $V_{\alpha p}/V_{W}\approx 1$, where
one might expect such a compositional signature to be were it present when the
SB was generated; the spread in points around $V_{\alpha p}/V_{W}\approx 1$
appears no different than the spread at other values. In retrospect however
this is not too surprising, for two reasons. First, even in the model of Zank
et al. (2020) where the interchange reconnection is occurring relatively high
up (compared to the photospheric reconnection models of Fisk & Kasper (2020)
and Drake et al. (2021)), in coronal loops with scale height $\sim
6R_{\odot}$, the local Alfvén speed is very high ($V_{A}\gtrsim 1000$ km/s),
and the alpha particles are not expected to ever drift at such high speeds
ahead of the protons. Rather, the phenomenon of alpha particles surfing at the
Alfvén speed is only expected to kick in at greater radial distances, once the
Alfvén speed has decayed enough to be comparable to $V_{\alpha p}$ (and after
which it may act as an instability threshold preventing $V_{\alpha p}\gg
V_{A}$ (Verscharen et al., 2013)). Thus, we wouldn’t expect $V_{\alpha
p}/V_{W}\approx 1$ to be possible at the site of interchange reconnection, and
the alphas would not be able to carry a compositional signature with the SB to
be observed at PSP. Secondly, even if the alpha particles could leave the
interchange reconnection event at the same speed as the SB, a PSP encounter
with perihelion distance $\sim 30R_{\odot}$ still represents a travel distance
of several hundred Alfvén crossing times (using a typical SB length scale
$l\sim 5\times 10^{4}$ km (Laker et al., 2021; Tenerani et al., 2020)).
Therefore, barring some rather unphysical fine-tuning, any compositional
signature would have long since decayed away by the time the alpha particles
reached PSP, and we would expect to observe something like figure 3. In
conclusion then, our results are all consistent with in-situ generation
mechanisms of SBs, but cannot be used to rule out origin mechanisms occurring
further down in the corona or at the surface of the Sun.

###  3.2 Alfvénic Motion of the Alphas

Figure 4: Cartoon showing the idealised expected alpha particle motion in
velocity space during a SB in the three scenarios (i) $V_{\alpha p}<V_{w}$,
(ii) $V_{\alpha p}\sim V_{w}$, (iii) $V_{\alpha p}>V_{w}$, corresponding to
the rows of figure 5. $R$ and $B$ denote the radial and magnetic field
directions respectively.

SBs are known to be highly Alfvénic and spherically polarised
($|\mathbf{B}|=\text{constant}$), and we therefore expect the particle motion
to be spherically polarised too. To see why, consider a particle at rest in
the frame co-moving with the Alfvén wave. The magnetic field, being Galilean
invariant, is still spherically polarised, and the wave being stationary means
that energy is conserved in this frame (and that the electric field should
almost vanish). A particle with perturbed velocity $\delta\mathbf{v}$ relative
to this frame must therefore trace out a sphere in velocity space in order to
conserve energy. Boosting back into the spacecraft frame we infer that the
observed motion should be spherically polarised, centred at the wave frame
velocity, with radius equal to the wave speed relative to whichever particle
population we are considering. (For a more in-depth discussion of this see
Matteini et al. (2015).) With this picture in mind, one can potentially expect
three different types of alpha particle motion, depending on the relative
magnitudes of $V_{\alpha p}$ and $V_{pw}$; these are sketched out in the
cartoon in figure 4. In scenario (i) we have $V_{\alpha p}<V_{pw}$, and would
expect to observe spherical polarisation of both the protons and alpha
particles, with the alpha particles tracing out a sphere of smaller radius
than the protons, approximately given by $V_{\alpha w}\approx V_{pw}-V_{\alpha
p}$. In case (ii), the position of the alphas in velocity space roughly
coincides with the wave frame, $V_{pw}\approx V_{\alpha p}$, and one would
expect the protons to be spherically polarised but the alphas to be roughly
stationary. In case (iii) we have $V_{\alpha p}>V_{pw}$, and so would again
expect the protons and alphas to be spherically polarised, but importantly the
alphas should move in anti-phase with the protons. This potential for alphas
to move either in-phase or in anti-phase with protons during Alfvénic
fluctuations depending on the relative values of $V_{\alpha p}$ and $V_{pw}$
was first pointed out by Goldstein et al. (1995) using Ulysses data.
Understanding this in terms of spherical motion of each species in velocity
space is exactly the model laid out in Matteini et al. (2015), the only
difference here is that the cadence and quality of the SPAN-Ion measurements
allow us to distinguish between the three cases over short timescales, and
directly observe and measure the spherical polarisation of the alphas.

Figure 5: Three example SBs showing the different types of alpha-particle
Alfvénic motion. First column is the 3D proton and alpha velocity measurements
in instrument coordinates, in blue and red respectively, through the
switchback. Middle column are these particle velocities projected onto the
minimum variance magnetic field plane. Yellow circles indicate the start of
the SB interval, yellow triangles the point of maximum deflection during the
SB. Third column is the magnetic field measurements projected onto the same
plane, with circles of best fit in grey (blue crosses mark the circles’
centres).

In each row of figure 5 we show an example SB, illustrating the three main
types of alpha particle motion just described. The left column shows the 3D
proton and alpha velocity measurements (in blue and red respectively), in
instrument coordinates. In all three cases the proton motion (in blue) is
spherically polarised as expected. Regarded as single large-amplitude, low-
frequency Alfvén waves, these switchbacks are not just spherically polarised
but to a good approximation arc-polarised as well (Fisk & Kasper, 2020), as
first theoretically predicted by Barnes & Hollweg (1974) and observed many
times in the solar wind since (Lichtenstein & Sonett, 1980; Tsurutani et al.,
1994; Riley et al., 1996). Their maximum and intermediate principal component
axes define a plane that is almost constant through the SB interval, and the
tip of the $\mathbf{B}$ field roughly traces out an arc on the sphere of
$|\mathbf{B}|=\text{constant}$. If
$\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}$ are the orthonormal principal
components of the magnetic field measurements $\mathbf{B}_{i}$ for a single SB
interval with eigenvalues $\lambda_{1}\lesssim\lambda_{2}\ll\lambda_{3}$, we
can project the measurements onto the plane defined by $e_{2}$ and $e_{3}$,
which should be the plane in which they appear most circular; this is shown in
the third column of fig. 5. The fluctuations being Alfvénic means we can
project the velocity measurements onto the same plane; this is plotted in the
second column of fig. 5.

From the middle plot of the first SB example (taken from 2019-08-30/22:50:25
to 2019-08-30/23:12:47), we can see that the alpha particle velocities in red
are spherically polarised as well, albeit with a smaller amplitude - they move
on the surface of a smaller sphere. To a good approximation, the alpha and
proton velocities both subtend the same angle $\theta$ that the magnetic field
does, and appear to be rotating about a similar point in velocity space. The
yellow circle represents the start of the entire SB interval, and the yellow
triangle the point of maximum $\mathbf{B}$ field deflection during the SB.
From these we can see that the protons and alphas are moving in phase with
each other. This corresponds to scenario (i), $V_{\alpha p}<V_{pw}$, in figure
4. Circles of best fit to the proton and alpha motion are overlaid in blue and
red respectively, their centres marked with crosses. The proximity of the
centres of the alpha and proton circles shows good agreement between these two
components of the wave frame velocity. The third component can be estimated by
calculating the $(e_{2},e_{3})$ plane that minimises the least square distance
to the measured velocities for each species separately (this is then the plane
of arc-polarisation). For protons the sphere centre is
$\mathbf{v}_{pw}=(-543,147,16)\text{ km/s}$ and for the alphas it is
$\mathbf{v}_{\alpha w}=(-547,152,28)\text{ km/s}$ (in instrument coordinates),
showing very good agreement in all three components. Because the alpha fits
are independent of the proton fits (they are not constrained to lie along the
magnetic field relative to the proton VDFs), fitting spheres in this way
represents two independent estimates of the wave frame velocity.

Using equation 1 during the LQ region of this SB, we estimate the local alpha
and proton wave phase speeds as $V_{pw}\approx 149$ km/s and $V_{\alpha
w}\approx 45$ km/s. This is in excellent agreement with the radii of the
spheres of best fit in column 2, which have radii of 143 and 44 km/s
respectively. With $V_{\alpha p}\approx 114$ km/s in the LQ region we also
have $V_{\alpha p}<V_{pw}$ and $V_{\alpha p}+V_{\alpha w}\approx V_{pw}$ as
expected. Comparing to the measured Alfvén speed $V_{A}\approx 147$ km/s, for
this SB we have that the proton phase speed and Alfvén speed are almost equal,
$V_{pw}\approx 1.01V_{A}$. In time series of the particle velocities for SBs
like this, the large spikes in the proton velocity would also be seen in the
alphas, albeit smaller in magnitude.

In the second example SB (middle row), (taken from 2019-08-29/21:08:04 to
2019-08-29/21:20:46), the proton velocities are still spherically polarised,
but the alpha’s are not - they appear relatively stationary in velocity space
through the SB and do not trace out an arc (the yellow markers for initial and
maximum $\mathbf{B}$ deflection lie almost on top of each other). This
corresponds to scenario (ii) in figure 4, where $V_{\alpha w}\approx 0$, and
for the LQ interval preceding this SB, we have $V_{\alpha p}\approx 158$ km/s,
$V_{pw}\approx 162$ km/s, and $V_{\alpha w}\approx 6$ km/s, with $V_{A}\approx
161$ km/s so that $V_{pw}\approx 1.01V_{A}$. The alphas are therefore roughly
comoving with the wave, and their location in velocity space serves as an
estimate of the wave frame. The centre of the proton circle of best fit in
blue lies reasonably close to the alpha velocities, but we note there is a
fair amount of scatter in the proton measurements for this SB.

Finally, in the third example SB in the bottom row (from 2019-08-29/08:37:52
to 2019-08-29/08:51:09), the alphas are again spherically polarised, but
moving in anti-phase with the protons, as can be seen by the relative
locations of the points of maximum SB deflection (yellow triangles). This
corresponds to scenario (iii) in figure 4. For this LQ region we have
$V_{\alpha p}\approx 167$ km/s, $V_{pw}\approx 101$ km/s, and $V_{\alpha
w}\approx-26$ km/s, with $V_{A}\approx 105$ km/s so that $V_{pw}\approx
0.96V_{A}$. The quantitative agreement is not quite as good as previous,
qualitatively however $V_{\alpha p}>V_{pw}$, and $V_{\alpha w}$ and $V_{pw}$
have opposite signs, as expected. For SBs such as these, a time series of
particle velocities would see spikes in proton velocity coinciding with dips
in alpha velocity. We note that this anti-phase motion would also be expected
to be observed in proton beams, since they typically travel at or slightly
above the Alfvén speed relative to the core (Alterman et al., 2018).

###  3.3 Relation to the de Hoffman-Teller Frame

Finally, for completeness we compare the wave frames determined using the
methods described above with the direct computation of the de Hoffman-Teller
(DHT) frame. First introduced by De Hoffmann & Teller (1950) in the context of
MHD shocks, it is defined to be the frame in which the plasma’s electric field
vanishes, and is usually computed by finding the velocity $\mathbf{V}$ that
minimizes the quantity (Khrabrov & Sonnerup, 1998)

$D(\mathbf{V})=\frac{1}{M}\sum_{m=1}^{M}\left|(\mathbf{v}^{(m)}-\mathbf{V})\times\mathbf{B}^{(m)}\right|^{2},$
(2)

where $\mathbf{v}^{(m)},\mathbf{B}^{(m)}$ denote velocity and magnetic field
values over a series of measurements indexed by $m=1,\dots,M$. By definition
we expect the DHT frame and the wave frames computed above to be one and the
same. To see geometrically why this is so for the SBs being considered here,
consider the ideal case of a perfectly spherically polarised Alfvén wave. The
minimum value of $D(\mathbf{V})=0$ will be achieved only if each term in eq. 2
vanishes, which requires $\mathbf{V}_{\text{DHT}}$ to lie on the line through
$\mathbf{v}^{(m)}$ parallel to $\mathbf{B}^{(m)}$, for each measurement $m$.
The point that uniquely satisfies this is the centre of the sphere in velocity
space (as it is the point of intersection of each of these lines through
$\mathbf{v}^{(m)}$). Thus, regarding SBs as essentially single, large-
amplitude spherically polarised Alfvén waves, we expect
$\mathbf{V}_{\text{DHT}}$ and $\mathbf{v}_{pw}$ to agree to good
approximation, and this is indeed the case for our three example SBs, as
summarised in Table 1.

Table 1: Wave speeds determined by sphere fitting vs $V_{DHT}$ SB | $\mathbf{v}_{pw}$ | $\mathbf{V}_{DHT}$
---|---|---
| (km/s) | (km/s)
(i) | (-543, 147, 16) | (-548, 116, 9)
(ii) | (-598, 155, 33) | (-614, 152, 12)
(iii) | (-483, 128, 1) | (-473, 123, 9)

##  4 Conclusions

In this work the density and abundance variations of alpha particles were
examined in a database of 92 switchbacks from PSP’s encounter 3 and 4. No
consistent compositional signature difference was observed in the alpha
abundance $n_{\alpha p}$ inside SBs vs outside, suggesting that PSP is
measuring the same plasma in both cases, in agreement with previous
interpretations of SBs (Yamauchi et al., 2004; Martinović et al., 2021;
Woolley et al., 2020). We argued that even if SBs are the results of
interchange reconnection events lower down in the corona, compositional
signatures are not likely to exist and be measurable at PSP for two reasons:
1) the local Alfvén speed at the postulated interchange reconnection sites is
very high and most likely precludes alphas being able to travel with the SBs
that are launched upwards along the field lines (thus preventing compositional
information being carried with the SB), and 2) even if the alphas are able to
travel with the SB, a small difference between $V_{\alpha p}$ and $V_{pw}$
would cause a compositional signature to have long decayed away due to the
distance (in Alfvén crossing times) to PSP’s perihelia. Thus, our observation
of there being no dependence of $\Delta n_{\alpha p}$ on $V_{\alpha p}/V_{pw}$
is to be expected and does not help distinguish between in-situ generation and
interchange reconnection as potential SB formation mechanisms.

In addition, we examined the three-dimensional nature of the velocity
fluctuations of both protons and alphas within individual SBs. We observed
spherical polarisation of both the proton and alpha velocities, which can be
understood as a consequence of energy conservation in the wave frame. Three
example SBs showed the alphas moving in-phase, stationary relative to, and in
anti-phase with, the protons. This corresponds to the three cases $V_{\alpha
p}<V_{pw}$, $V_{\alpha p}\approx V_{pw}$, and $V_{\alpha p}>V_{pw}$. Thus
while SBs are always associated with spikes in the proton velocity, alpha
velocities may be enhanced, unchanged, or decrease, depending on the relative
values of $V_{\alpha p}$ and $V_{pw}$. For the case $V_{\alpha p}<V_{pw}$,
where the alphas move in phase on a sphere of smaller radius than the protons,
the centres of the proton and alpha velocity spheres were in excellent
agreement, illustrating how one can make two independent particle measurements
to uniquely identify the wave frame. One can in principle use these methods to
estimate the wave frame over short time scales using purely particle
measurements, and we showed that this agreed well with the the usual method of
computing the de Hoffman-Teller frame via minimisation of the motional
electric field, $\mathbf{E}=-\mathbf{v}\times\mathbf{B}$. Intuitively then the
Alfvenic motion of both the alphas and the protons through SBs can be
understood as approximately rigid arm rotation about the location of the wave
frame in velocity space, as illustrated in 4 and discussed in Matteini et al.
(2014, 2015), and the length of the lever arms are to a good approximation
given by $V_{pw}$ and $V_{\alpha w}$ for protons and alphas respectively.

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