# Multiscale Solar Wind Turbulence Properties inside and near Switchbacks
measured by Parker Solar Probe

Mihailo M. Martinović Lunar and Planetary Laboratory, University of Arizona,
Tucson, AZ 85721, USA.  LESIA, Observatoire de Paris, Meudon, France.
Kristopher G. Klein Lunar and Planetary Laboratory, University of Arizona,
Tucson, AZ 85721, USA.  Jia Huang Climate and Space Sciences and Engineering,
University of Michigan, Ann Arbor, MI 48109, USA  Benjamin D. G. Chandran
Department of Physics & Astronomy, University of New Hampshire, Durham, NH
03824, USA  Space Science Center, University of New Hampshire, Durham, NH
03824, USA  Justin C. Kasper Climate and Space Sciences and Engineering,
University of Michigan, Ann Arbor, MI 48109, USA  Smithsonian Astrophysical
Observatory, Cambridge, MA 02138 USA.  Emily Lichko Lunar and Planetary
Laboratory, University of Arizona, Tucson, AZ 85721, USA.  Trevor Bowen Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
Christopher H. K. Chen School of Physics and Astronomy, Queen Mary University
of London, London E1 4NS, UK  Lorenzo Matteini Department of Physics, Imperial
College London, London SW7 2AZ, UK  Michael Stevens Smithsonian Astrophysical
Observatory, Cambridge, MA 02138 USA.  Anthony W. Case Smithsonian
Astrophysical Observatory, Cambridge, MA 02138 USA.  Stuart D. Bale Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
School of Physics and Astronomy, Queen Mary University of London, London E1
4NS, UK  Physics Department, University of California, Berkeley, CA
94720-7300, USA  The Blackett Laboratory, Imperial College London, London, SW7
2AZ, UK

###### Abstract

_Parker Solar Probe (PSP)_ routinely observes magnetic field deflections in
the solar wind at distances less than 0.3 au from the Sun. These deflections
are related to structures commonly called ’switchbacks’ (SBs), whose origins
and characteristic properties are currently debated. Here, we use a database
of visually selected SB intervals—and regions of solar wind plasma measured
just before and after each SB—to examine plasma parameters, turbulent spectra
from inertial to dissipation scales, and intermittency effects in these
intervals. We find that many features, such as perpendicular stochastic
heating rates and turbulence spectral slopes are fairly similar inside and
outside of SBs. However, important kinetic properties, such as the
characteristic break scale between the inertial to dissipation ranges differ
inside and outside these intervals, as does the level of intermittency, which
is notably enhanced inside SBs and in their close proximity, most likely due
to magnetic field and velocity shears observed at the edges. We conclude that
the plasma inside and outside of a SB, in most of the observed cases, belongs
to the same stream, and that the evolution of these structures is most likely
regulated by kinetic processes, which dominate small scale structures at the
SB edges.

space plasmas, interplanetary turbulence, solar wind

##  1 Introduction

_Parker Solar Probe (PSP)_ is the first mission to measure the inner
heliosphere closer than 0.3 au from the Sun’s surface, with perihelion
distances of $\sim$ 0.16 au during its first three encounters with the Sun.
One of the most compelling results from these first encounters is the ubiquity
of so-called ’switchback’ structures (SBs)111These structures are also called
’spikes’, ’jets’ or ’magnetic field reversals’ in the literature.,
characterized by a rotation of the magnetic field $\mathbf{B}$ observed by the
spacecraft (Kasper et al., 2019; Bale et al., 2019). Along with the magnetic
field rotation, the magnitude of the solar wind bulk velocity
$\mathbf{v_{sw}}$ increases by the order of the local Alfvén speed
$v_{A}=B/\sqrt{\mu_{0}n_{p}m_{p}}$ (Matteini et al., 2014), where $\mu_{0}$ is
magnetic permeability of vacuum, and $n_{p}$ and $m_{p}$ are proton density
and mass, respectively. Results from previous missions had already motivated
the community to invest significant effort into understanding the nature of
SBs, though they were not observed nearly as frequently as by _PSP_. Now, this
interest has increased, as new data sets from _PSP_ provide significantly
larger populations of SBs and more detailed information on the associated
plasma and electromagnetic field conditions due to increased measurement
precision and cadence.

###  1.1 Previous observations of Switchbacks in the solar wind

The first observations of sudden magnetic field rotations were reported in
_Pioneer 6_ observations (Burlaga, 1969, 1971), and later in _Helios_
measurements (Marsch et al., 1981). A decade later, measurements from the
_Ulysses_ mission confirmed that the probability distributions of the magnetic
field orientation are not uniform, but are rather skewed toward the polarity
opposite to the dominant polarity of the solar hemisphere the spacecraft was
observing (Forsyth et al., 1995). This result was followed by a statistical
analysis of Ulysses high latitude observations (McComas et al., 1998) that
demonstrated the opposite polarity measurements occurred in a non-negligible
($\sim 7-8\%$) fraction of observations, but only a minor fraction (of the
order of a percent) of these featured a complete inversion of the magnetic
field (Balogh et al., 1999). These events were classified as
discontinuities—either tangential discontinuities (TD), usually separating two
different plasma streams; or rotational discontinuities (RD), with a single
stream following a folded magnetic field line (Burlaga et al., 1977; Neubauer
& Barnstorf, 1981). Applying the standard Minimum Variance Analysis (MVA)
procedure (see e.g. Sonnerup & Scheible (1998)), to single-point measurements
(Burlaga & Ness, 1969; Mariani et al., 1973; Smith, 1973; Neugebauer et al.,
1984; Lepping & Behannon, 1986), RDs are more frequently observed than TDs. In
contrast, analysis of sets of manually selected events simultaneously observed
by three different spacecraft at 1 au (Tsurutani & Smith, 1979; Horbury et
al., 2001), and shortly after of four-point measurements from the _Cluster II_
mission (Knetter et al., 2003, 2004), found TDs to be dominant. These authors
argued that results derived from MVA might be misleading due to presence of
surface waves on TDs (Denskat & Burlaga, 1977; Hollweg, 1982), and therefore
observations of magnetic field folds manifest in RDs are not as common as
initial work suggested.

Although there is an evident ambiguity in the identification of SBs, they were
considered to be isolated events (Horbury et al., 2001; Yamauchi et al., 2004;
Horbury et al., 2018) that had been studied in detail with some basic
properties well established prior to the launch of _PSP_. It was understood
that the magnetic field deflection in SBs is followed by both the proton
(Neugebauer & Goldstein, 2013) and electron beam (also known as the strahl)
(Kahler et al., 1996). Similar behavior was observed for relative drift
between proton and $\alpha$ particle populations (Yamauchi et al., 2004). The
fluctuations of the magnetic and velocity field at large scales, meaning the
scales in the MHD domain with size comparable to the size of a structure, are
linearly related, describing a dominantly Alfvénic turbulence (Gosling et al.,
2009). Moreover, it can be shown that the SB magnetic and velocity
fluctuations are spherically polarized in the reference frame of zero electric
field (Gosling et al., 2011; Matteini et al., 2014), which can be well
approximated by the $\alpha$ particle’s velocity, due to their weak
interaction with Alfvénic fluctuations of protons (Matteini et al., 2015).
Some turbulent properties, e.g. power spectra and cascade rates, are found to
be very similar inside and outside of SBs, though discrete changes of velocity
and the magnetic field are contributing to intermittency of turbulence within
observed events (Horbury et al., 2018).

Starting with the first _PSP_ solar encounter in November 2018, a plethora of
new results regarding SBs were reported, as these structures were consistently
observed and resolved with high resolution measurements for radial distances
less than 0.25 au. These first results provided an overview of SB properties,
finding some differences compared to previous reports. Most notably, _PSP_ did
not observe SBs as isolated events, but rather ’clustered’ phenomena (Dudok de
Wit et al., 2020), with measured parallel temperature changing as the
spacecraft moves through the cluster (Woodham et al., 2020). This claim is
supported by two independent arguments: 1) normalized magnetic field
deflections from the Parker spiral (scaled to values between 0 and 1 depending
on cosine value) exhibit a power law distribution, which is a property related
to phenomena such as entangled filaments and flux tubes, rather than a
Gaussian distribution, which would represent randomized events; and 2)
Detrended Fluctuation Analysis (Kantelhardt et al., 2001) shows an increase in
long range correlations in time series for SBs compared to quiescent
intervals. These ’clusters’ are found to be separated by quiet periods with
few SBs (Horbury et al., 2020). The origin of this discrepancy between _PSP_
and previous measurements is not easy to understand due to _PSP_ ’s unique
trajectories compared to previous missions, nearly co-rotating with the Sun’s
surface during significant fractions of its first encounters (Bale et al.,
2019; Badman et al., 2020a; Rouillard et al., 2020). This specific trajectory
is likely causing the observed magnetic field deflections to have shorter
duration in measurements for larger radial distance, as the spacecraft could
be slicing through an elongated structure, possibly aligned along the Parker
spiral (Laker et al., 2020). Apart from these unexpected characteristics, many
similarities between _PSP_ and previous SB observations remain. First, SBs are
found to be dominantly Alfvénic structures, with direction of Poynting flux
and turbulent fluctuations changing along with the magnetic field (McManus et
al., 2020; Bourouaine et al., 2020; Mozer et al., 2020), maintaining the
correlation between bulk velocity and magnetic field when the field is
inverted (Matteini et al., 2014), while proton temperature does not vary
significantly through sharp field reversals (Wooley et al., 2020). Second,
Mozer et al. (2020) found that that the Poynting flux remains conserved before
and after a SB, but significantly increases inside the structure. Third, the
Partial Variance of Increments (PVI) of the magnetic field, a proxy for
intermittency, is also found to be increased in clustered intervals (Chhiber
et al., 2020), in accordance with previous results. Finally, it is worth
emphasizing that a very careful case study by Krasnoselskikh et al. (2020)
demonstrated that not all SBs appear to be Alfvénic in nature. Some of the
events exhibit significant wave activity, especially at the borders of SBs.
This is consistent with signatures of current sheets (Farrell et al., 2020)
and Whistler modes (Agapitov et al., 2020), detected at the edges of these
structures, where the magnetic field is rotating.

###  1.2 A Mystery of Switchback Origins

The characteristics of SBs listed in Section 1.1 do not conclusively
demonstrate how SB are formed or how they evolve throughout the inner
heliosphere. Proposed mechanisms for the creation of SBs can be roughly
classified in two groups - they either form on or very close to the solar
surface, or are generated in the propagating solar wind - with both
interpretations being represented in the literature.

A connection between SB characteristics and surface source structures could
lend credence to the theories suggest SBs form on or close to the surface of
the Sun. Toward this end, many authors have investigated the connection
between the plasma streams of increased bulk velocity and coronal jets
(McComas et al., 1995; Neugebauer et al., 1995; Culhane et al., 2007;
Patsourakos et al., 2008; Neugebauer, 2012; Tian et al., 2014)222For an
extensive review of the remote sensing observations of interest, see Raouafi
et al. (2016).. Since the Alfvénic fluctuations in the chromosphere contain a
significant amount of energy (De Pontieu et al., 2007), one possible
interpretation of our _in situ_ observations, also supported by extensive
simulation work (Uritsky et al., 2017; Wyper et al., 2017; Roberts et al.,
2018), is that these fluctuations will continue to propagate even if a jet
does not escape the Sun’s gravity (Horbury et al., 2020). Another candidate
mechanism is interchange reconnection, which arises from magnetic flux
transport close to the surface through structures such as coronal loops (van
Ballegooijen et al., 1998; Fisk & Schwadron, 2001; Fisk, 2005), interacting
with open magnetic field lines to create S-shaped structure observed as a SB
(Fisk & Kasper, 2020; Zank et al., 2020). Recent MHD simulations show that, if
a SB is in equilibrium with the surrounding plasma, it can preserve its shape
long enough to reach _PSP_ (Landi et al., 2005; Tenerani et al., 2020).
Further on, it is expected to gradually merge with the surrounding plasma,
most likely via parametric decay instability (Derby, 1978), a process
potentially at work in the solar wind, as observed on _Wind_ mission (Bowen et
al., 2018a), and predicted by turbulence models (Chandran, 2018), hybrid
(Matteini et al., 2010) and MHD simulations (Tenerani & Velli, 2013; Primavera
et al., 2019).

The complementary argument is that magnetic field folds are generated as the
solar wind propagates. The main observational support for this argument is
that the fraction of the solar wind with inverted magnetic field increases
with radial distance (Forsyth et al., 1995; McComas et al., 1998; Borovsky,
2016; Owens et al., 2018; Macneil et al., 2020; Badman et al., 2020b). Also,
simulations non-compressible three dimensional MHD turbulence (Zhdankin et
al., 2012) show that measured the distribution of magnetic field orientation
at 1 au can be caused solely by turbulent fluctuations. Although these studies
provide information on the statistics of the magnetic field and electron
strahl orientation, more detailed research of strictly defined SBs, their
duration, deflection angles and other parameters for purposes of effective
comparison with _PSP_ results has yet to be performed. Regarding possible
mechanisms for the generation of magnetic field reversals as the solar wind
propagates outwards, MHD simulations show that they can develop from velocity
shears (Landi et al., 2006; Ruffolo et al., 2020). SBs are predicted to appear
as a product of turbulent fluctuations evolving while maintaining constant
magnetic field intensity (Vasquez & Hollweg, 1996). The discontinuity, which
forms due to the fact that a three dimensional configuration for which the
field fluctuation is of the order of the field intensity is not expected to be
possible (Barnes, 1976; Valentini et al., 2019), creates the observational
signature such the one observed by _PSP_ (Squire et al., 2020). It is
important to note that, even if the role of the turbulent cascade turns out
not to be crucial for SB generation, the effects of turbulent fluctuations on
SB evolution needs to be investigated in detail (Chen et al., 2020).

###  1.3 Outline of the paper

Having reviewed previous observational results related to SBs and associated
physical interpretations, we will now move to our analysis of SBs observed by
_PSP_ during encounters 1 and 2. The intention of this work is not to verify
or disprove either school of thought regarding SB origins described in Section
1.2, but rather to analyze a set of observations that have the combined
benefit of both being visually selected and also containing a large number of
events, which will provide additional insight into both MHD and kinetic scale
properties of SBs. This way, the results presented here provide a
comprehensive set of observational constraints for current, as well as
developing future models of SB generation and evolution.

We use a database of visually selected SBs, with associated quiet and
transition periods before and after each structure. For each event, we combine
several methods to analyze plasma parameters, turbulent spectra, and
intermittency levels. Additionally, as a proxy for turbulence behavior at ion
scales, we compare levels of Stochastic Heating (SH), a non-linear plasma
heating mechanism based on the breaking of magnetic moment by low-frequency
turbulent fluctuations at ion scales (Chandran et al., 2010), inside and
outside of SBs. Details of these procedures are described in Section 2. In
Section 3, we review our results and discuss possible implications. Finally,
Section 4 summarizes the main points.

##  2 Method

Figure 1: Example of an event from Encounter 1, on Nov 5 2018, with five
characteristic regions of interest — from left to right: leading quiet region
(LQR), leading transition region (LTR), switchback (SB), trailing transition
region (TTR), and trailing quiet region (TQR), given in different colors.

Within the heliospheric community, the exact definition and typical properties
of SBs are still debated. Several different criteria to identify SBs in data
sets have been introduced throughout the last few decades (Yamauchi et al.,
2004; Horbury et al., 2018) and in particular in studies related to _PSP_ data
(Dudok de Wit et al., 2020; Horbury et al., 2020; McManus et al., 2020; Mozer
et al., 2020; Bourouaine et al., 2020). Here, we use the database of SBs from
_PSP_ Encounters 1 and 2 (E1 and E2) where, for each event, five separate
regions are identified: 1) Leading Quiet Region (LQR) with stable velocities
and magnetic fields before the SB; 2) Leading Transition Region (LTR), where
the magnetic field rotates from LQR toward its SB orientation; 3) the SB
itself with stable field orientation; 4) Trailing Transition Region (TTR); and
5) Trailing Quiet Region (TQR), with conditions which are, in general, not
very different from the ones in LQR. The identification of events was done
through visual inspections of $\mathbf{B}$ and $\mathbf{v_{sw}}$ time series.
Magnetic field rotations coincident with the bulk velocity enhancement are
selected, with the results being very consistent throughout multiple
independent manual inspections. A total of 1,074 events333Further on, the term
’event’ will be used for a sequence of the five different observed regions,
while the term ’SB’ will be used only for the period where the rotated
magnetic field is observed. were identified (662 in E1 and 412 in E2), 921 of
which have all five distinct regions resolved. A typical example of a selected
event is given on Figure 1. For each of these regions separately, we extracted
the average plasma parameters, including the proton core population density
$n_{p}$, thermal velocity $v_{t}$ and solar wind bulk velocity
$\mathbf{v_{sw}}$ from the Solar Probe Cup (SPC) measurements (Kasper et al.,
2016; Case et al., 2020), magnetic field $\mathbf{B}$ from the Fields
instrument suite magnetometers (Bale et al., 2016) and temperatures
perpendicular $T_{\perp}$ and parallel $T_{||}$ to $\mathbf{B}$ from analysis
of effective temperatures measured by SPC and the fluctuating magnetic field
(Huang et al., 2020a). Only measurements flagged with the highest degree of
confidence by the instrument teams are used in this study.

Figure 2: Example of a power spectrum for LQR (red), SB (blue), and TQR
(green) for a single event, with linear fits over the inertial and dissipation
ranges. All three regions were sampled for $\sim$ 70 s. Shaded areas on the
left panel represent the [$e^{-0.5}f_{p}$, $e^{0.5}f_{p}$] interval. The right
panel shows the same three spectra with frequency normalized to $f_{p}$.

We also calculate the Fourier magnetic field trace Power Spectral Density
(PSD) spectrum444For details on calculating PSD, see e.g. Appendix A of
Bourouaine & Chandran (2013). for each region, focusing on two characteristic
regimes: 1) the inertial range where the Alfvénic fluctuations cascade toward
smaller scales, characterized by a logarithmic scale slope usually measured in
the domain [-1.7, -1.45] (Bruno & Carbone, 2013) and, after the spectral break
frequency $f_{b}$, 2) the kinetic range where the fluctuations both become
dispersive and the energy in the electromagnetic field is expected to more
efficiently dissipate, eventually leading to plasma heating, causing
significant steepening of the spectral slope (Alexandrova et al., 2012;
Verscharen et al., 2019)555This is a very simplified description of a typical
turbulent spectrum. The border between the two described spectral regions is
often observed as a third range at which the spectral steepening is associated
with ion heating through kinetic processes (Sahraoui et al., 2010; Bowen et
al., 2020).. As multiple other phenomena, such as coherent structures likely
generated by turbulence (Lion et al., 2016; Howes, 2016), magnetic
reconnection (Mallet et al., 2017; Loureiro & Boldyrev, 2017; Vech et al.,
2018), or instrument noise (Martinović et al., 2019) can interfere with
measuring the background turbulent spectra, and as the event durations vary
significantly, we calculated the spectral indices and the PSD at the ion scale
break using a two step process. First, a Levenberg-Marquardt fit using a two
line function, with the fitting range frequencies set manually assuring that
both injection range and instruments noise are excluded, is performed for each
LQR, SB and TQR region for a given event. This method provided results for the
spectral break locations and PSD intensities with high accuracy and
independent of initial guess and input parameters, but introduced a subjective
bias on the values of inertial range logarithmic slopes through the manual
selection of the low frequency limit. Therefore, the second step of the
fitting algorithm was determining the inertial slopes by a least-square line
fit of data in an automatically selected range. To avoid statistical
uncertainties and the possible measurements of the injection range turbulence,
the low frequency limit was set at 150% of the minimum frequency of the power
spectrum of a given region, while the high frequency limit was set at
$0.75f_{b}$, to avoid sampling the fine structure of the spectral break. An
example of such a fit is given on Figure 2. We see both LQR and TQR spectra
encounter the noise floor at $\sim$ 6 Hz. The frequency range contaminated by
the noise is manually removed from the fitting procedure as to not affect the
resulting slopes. In total, spectra from 1,863 regions (LQRs, SBs and TQRs)
were fit successfully, and are included in this study.

It is worth emphasizing that a significant increase of $v_{sw}$ in SBs causes
a shift of the spectrum in frequency space, in accordance with the shift
prescribed by Taylor hypothesis (Taylor, 1938). The _Right panel_ of Fig. 2
illustrates this property, with the three spectra, fairly different in terms
of measured power, overlap as the observed frequency is normalized to $f_{p}$.
Here, $f_{p}=v_{sw}\sin{\theta}/2\pi\rho_{p}$ is the convected gyrofrequency
(Bourouaine & Chandran, 2013), $\theta$ is the angle between $\mathbf{v_{sw}}$
and B, $\rho_{p}=m_{p}v_{t\perp}/e_{c}B$ is the gyroradius, and $e_{c}$ is the
elementary charge with all values averaged over a measured region.

As the magnetic field PSD does not capture the intermittent properties of
turbulence, including phenomena such as localized coherent structures at the
borders of the regions within each event, we estimate these properties using
PVI, which is defined at time $t$ for some time increment $\tau$ as

$\mathrm{PVI}=\frac{|\Delta B(t,\tau)|}{\sqrt{\langle|\Delta
B(t,\tau)|^{2}\rangle}}$ (1)

where $\Delta B=B(t)-B(t+\tau)$, $\langle\rangle$ denotes ensemble average
over a region of interest (QR, TR or SB). Based on previous studies, times
with values of PVI $>$ 3 have been associated with non-Gaussian structures,
events with PVI $>$ 6 categorized as current sheets, and with PVI $>$ 8 as
reconnection sites (Matthaeus & Goldstein, 1982; Osman et al., 2012; Greco et
al., 2018; Chhiber et al., 2020).

Detailed analysis of the turbulent spectra around $f_{p}$ provides estimates
of ion heating delivered through non-linear mechanisms such as stochastic
heating (SH). An overview of estimates for SH rates for the first two _PSP_
encounters is provided elsewhere (Martinović et al., 2020), while in this
study we will only focus on the comparison of SH in SBs and their associated
quiet regions. The key quantity for estimation of SH is the normalized level
of turbulent magnetic fluctuations at the convective gyroscale $\delta=\delta
B/B$, leading to a total heating rate estimated as (Hoppock et al., 2018)

$Q_{\perp}=v_{A}^{2}\Omega\Bigg{(}\sigma_{1}\delta^{3}\exp\left[-\frac{\sigma_{2}}{\delta}\right]+\frac{\sigma^{3}c_{1}\delta^{3}}{\beta^{1/2}}\exp\left[-\frac{c_{2}\beta^{1/2}}{\sigma\delta}\right]\Bigg{)}$
(2)

where $c_{1}=0.75$, $c_{2}=0.34$, $\sigma_{1}=5$ and $\sigma_{2}=0.21$ are
order unity constants found from test particle simulations that are expected
to be fairly insensitive to plasma parameters (Chandran et al., 2010; Hoppock
et al., 2018). The relation between velocity and magnetic field fluctuations
is quantified by the parameter $\sigma=1.19$, proton cyclotron frequency is
given as $\Omega=e_{c}B/m_{p}$, and $\beta=v_{t}^{2}/v_{A}^{2}$. A detailed
discussion of these constants is beyond the scope of this work, but can be
found in previous literature on SH in the solar wind (Chandran et al., 2010;
Xia et al., 2013; Hoppock et al., 2018; Martinović et al., 2019). The magnetic
field fluctuation magnitude $P_{B}$ at the convective gyroscale is calculated
by integrating the PSD in the vicinity of $f_{p}$, as shown on Figure 2)

$\delta
B=\Bigg{[}\frac{\pi}{C_{0}(n_{s})}\int_{e^{-0.5}f_{p}}^{e^{0.5}f_{p}}P_{B}(f)df\Bigg{]}^{1/2}.$
(3)

In Equation 3, $n_{s}$ is the PSD logarithmic slope in the integration range,
determining the geometrical factor $C_{0}$, which is described in detail in
previous studies (Bourouaine & Chandran, 2013; Vech et al., 2017). We note
that SH rates determined this way have very large uncertainties (Martinović et
al., 2019), but can be confidently used for comparison between different
regions in a single event.

##  3 Results and Discussion

###  3.1 Overview of Plasma Parameters

Figure 3: Comparison of parameters in LQRs and SBs. The results for all the
parameters are quite similar, with $v_{sw}$ being systematically increased in
SBs by an average value of $\sim v_{A}/2$.

The macroscopic parameters of SBs and LQRs are shown in Figure 3. Magnetic
field strength, core proton density and temperature666The SPC data set does
not provide full values of temperature, but rather ones extracted from the
reduced distributions. The scalar $T_{c}$ used here is provided by method
described in (Huang et al., 2020a)., as well as other parameters which are not
shown, are fairly similar inside and outside of SBs for the vast majority of
events. The two peaks along the diagonal appearing in all three plots are due
to the combination of data sets from E1 and E2, which have slightly different
typical parameters. This is also the case for comparison between SBs and TQRs,
not shown. These observations agree with the interpretation that plasma inside
SBs is from the same folded plasma stream as the quiet regions outside the SB
(see, e.g. (Yamauchi et al., 2004; Tenerani et al., 2020; Fisk & Kasper, 2020;
Wooley et al., 2020)). However, it is not intuitively clear why the bulk
velocity should increase by $\sim v_{A}/2$ within SBs. Even though the
deflection angle777In this work, deflection angle is defined as the difference
in average $\theta$ between LQR and SB for each event. The differences between
TQR and SB are consistent to within several percent for large majority of
events. of the bulk velocity for SBs is very small (Kasper et al., 2019), we
find that the intensity of the $\Delta\mathbf{v_{sw}}$ has a Gaussian
distribution centered at $\sim 0.75v_{A}$ (not shown). This value does not
necessarily conflict with previous results, which argue that magnetic field
deflections in SBs behave like almost purely Alfvénic phenomenon (Yamauchi et
al., 2004; Gosling et al., 2011; Matteini et al., 2015; Horbury et al., 2018),
as these values are not comparing the states just before and after the
transition regions, but are rather averaged over longer periods. One possible
interpretation for such behavior is that the difference in bulk speeds,
normalized to Alfvén speed, is proportional to the Alfvénic ratio between
energy of magnetic field and velocity fluctuations, expected to be lower than
unity in the solar wind (Matteini et al., 2014), but this phenomena requires
additional detailed investigation.

###  3.2 Turbulence

Figure 4: Histograms for turbulent spectral properties measured in quiet
regions and SBs, obtained from manually performed fits. _Top panels_ show that
the ion scale break frequency in Hz $f_{b}$ remains unchanged in the shifted
spectra of SBs, while the PSD at $f_{b}$ is enhanced due to the spectrum being
shifted by the increase in $v_{sw}$. _Bottom left panel_ illustrates the
fitted inertial range logarithmic slopes being similarly distributed in all
three types of regions, and equally represented in the range between
theoretically predicted values of -5/3 and -3/2. Finally, the _bottom right
panel_ shows a notable shift in $f_{p}$ for SBs, affecting its ratio to
$f_{b}$.

Figure 4 shows the results from 1,863 manually fitted turbulent spectra in SBs
and quiet regions. The most notable feature is that, even though the turbulent
spectra are shifted in frequency due to increased $v_{sw}$, the spectral break
from inertial to ion dissipation region spans the same frequency range. This
implies the lowest wavenumber at which the dissipation is observed is lower in
SBs, producing larger power at the break point (_top panels_). _Bottom right
panel_ shows comparison of the spectral break point with the convective
gyroscale, also shifted toward higher frequencies (Figure 2), which is due to
the increase in both $v_{sw}$ and $\theta$, given that all the other
quantities comprising $f_{p}$ do not significantly change inside SBs (Figure
3). The relation of the convected proton inertial length $f_{d}$ to $f_{b}$
(not shown) is very similar, which is expected as our data set does not
contain any extremely high or low plasma $\beta$ measurements, as studied for
example in Chen et al. (2014). It is important to note the contrast with
recent results (Duan et al., 2020) showing different scalings of $f_{p}$ and
$f_{b}$ in PSP data; this difference is most likely masked in our results due
to similarity in MHD plasma parameters in QRs and SBs (Figure 3).

The fitted values of inertial range turbulence slopes shown in the _bottom
left panel_ are consistent with two theoretically predicted regimes—a standard
’critically balanced’ cascade with inertial slope centered around -5/3
(Goldreich & Sridhar, 1995), and a ’dynamically aligned’ cascade with the
inertial slope of $\sim-3/2$ (Boldyrev, 2006), describing a majority of the
measured spectra with similar abundances in histograms. There is no preference
to either of these regimes when comparing SBs to quiet regions. In order to
understand the inertial slope behavior, we investigated possible relations of
the fitted slope with multiple parameters, including plasma $\beta$, Velocity
Distribution Function (VDF) moments, magnetic field and characteristics of its
turbulent fluctuations, and radial distance. None of these parameters showed a
correlation with the inertial slope values. However, this result does not
necessarily contradict with previous studies where the inertial slope was
observed to depend on the residual energy and intermittency (Bowen et al.,
2018b), cross helicity (Chen et al., 2013), radial distance (Chen et al.,
2020), or proximity to the Heliospheric Current Sheet (Chen et al., 2021), due
to relatively small ranges for these parameters measured in our data set.
Similar studies to this work on the inertial slope dependencies will be
preformed in the future as the _PSP_ data set expands, producing larger
numbers of visually identified SB for analysis.

###  3.3 Comparison of SH Rates

To investigate properties of the ion scale turbulence and its interaction with
VDF moments, we estimate the level of non-linear SH. Here, it is important to
note that the SH values are very difficult to estimate for majority of events
due to their relatively short duration and occasional strong wave activity.
Therefore, these estimates should be observed only as a proxy for behavior of
turbulence at the proton gyroscale, and its comparison inside and outside of
SBs. The SH rates measured on this mission are considered in more detail and
with more strict constraints elsewhere (Martinović et al., 2020), while
discussion of other solar wind heating mechanisms, such as Landau damping
(Quataert, 1998; Chen et al., 2019) or cyclotron heating (Hollweg, 1999;
Kasper et al., 2013), is beyond the scope of this paper.

The turbulent PSD at the convective gyroscale is related to the level of SH
through the value of $\delta$, which is largely controlled by the amplitude of
the PSD in the integration range around $f_{p}$, defined in Equation 3. This
range is, for most cases, closer to $f_{b}$ in quiet regions than in SBs
(Figure 4). However, as the PSD at the break is significantly higher for SBs,
the integration range levels, and consequently $\delta B/B$ at the gyroscale,
remain very similar, as shown on Figure 5. Considering that the parameters
shown on Figure 3 are also very similar, the value of SH in SBs remains
approximately the same and thus the existence of SBs is not expected to
enhance the contribution of SH to solar wind heating. Observing Figure 2, we
could argue that the similarity in the estimated SH values originates from the
wave vector PSD, Doppler shifted to different frequencies. However, two
important characteristics must be taken into account when considering SH rates
— 1) the PSD at $f_{b}$ is increased for SBs and 2) the frequency of interest
$f_{p}$ does not represent a standard Doppler shift, but is also affected by
the angle $\theta$, pushing the integration range to slightly higher
frequencies. These two factors have effects that partially negate one another,
providing to the level of accuracy available, similar values of SH rates.

Figure 5: Comparison of SH values (_left_) and normalized magnetic field
fluctuations at the convective gyroscale (_right_) in LQRs and SBs. The values
are very similar, as well as for the comparison between TQRs and SBs (not
shown). The spread in the results originates from high uncertainties of
quantities in Equations 2 and 3. Figure 6: PVI histograms averaged for
different regions. Solid lines represent the measurements made more than
$\tau_{0}=20$ s from edge of each respective region, while dashed lines are
from the data sampled within $\tau_{0}$ time from each edge.

###  3.4 Effects of Borders and Intermittency

Similar characteristics of the turbulent spectra shown in Figure 4 do not
reflect the potentially different intermittency levels in different regions.
The distribution of PVI values, used as a proxy for intermittency, is shown on
Figure 6 organized by the five kinds of regions, clearly demonstrating that
PVI is the highest for the case of SBs, then TRs, and is the lowest for QRs
(Huang et al., 2020b). As recent results (Krasnoselskikh et al., 2020;
Agapitov et al., 2020; Farrell et al., 2020) have shown that the edges of SBs
can be populated by various structures and exhibit strong wave activity, we
separately observe the measurements close to the edges of each region. Solid
lines are calculated using only PVI values measured at least $\tau_{0}=20$ s
from the region’s edge. The histograms start to deviate for SB and transition
regions for PVI = 3-6, with the difference of $\sim 10\%$ at PVI $\approx 3$
and for about a factor of 2.4 at PVI $\approx 10$. On the other hand, the
magnetic field sampled close to the region edges are featuring significantly
more drastic trends, with spread of a factor of 2.3 for PVI $\approx 3$, and
over an order of magnitude for PVI $\approx 10$. This result is insensitive to
the $\tau_{0}$ used; recalculating Figure 6 with $\tau_{0}$ of 1, 2, 5, 10,
and 50 s (not shown) yields very similar results. We also reproduced Figure 6,
but only taking into account periods either larger or smaller than various
time durations between 2 and 200 s. The results are very similar to the ones
shown on the Figure, and are not displayed here. From here, we conclude that
properties of PVI are independent of the region duration. This result is fully
consistent with previous observations findings of SBs having increased
intermittency (Horbury et al., 2018), as well as the presence of waves and
enhanced currents at the edges of SBs. The presence of these structures at the
SB borders suggests that they might be responsible for increased ’lifetime’ of
SBs, stabilizing these structures against decay (Landi et al., 2005).

Figure 7: Effects of intermittency on SH rates. _Left panel_ illustrates the
major importance of the exponential factors in Equation 2 for weaker SH, and
dominance of $\delta^{3}$ term for strong SH. _Right panel_ shows that the
effects of intermittency—illustrated through average SH rate of 5 s intervals
inside a SB compared to SH rate calculated for the entire region—are notably
more important when the exponential factor is able to significantly modify the
fluctuation damping rate (Mallet et al., 2019).

Finally, it is interesting to consider the effects of intermittency on SH.
Figure 7 illustrates the SH values calculated for every 5 s interval and then
averaged over the entire region. It is notable that the intermittency strongly
enhances the level of SH whenever that level is low, but it is not the case
for intervals with strong SH, which are responsible for the majority of SH’s
contribution to solar wind heating (Martinović et al., 2019, 2020). This
feature, shown on _right panel_ of Figure 7, was verified to also hold for
quiet regions and any time averages in [5-100] s range. To understand this
effect, we refer to the work of Mallet et al. (2019), who calculated
fluctuations damping rates $\gamma\sim Q/\delta z^{2}$ for different solar
wind heating mechanisms. Here, $Q$ is the theoretical heating rate, and $z$ is
the Elsasser variable (Elsasser, 1950) at a given scale. For the case of
Landau damping, intermittency has no effect on the heating intensity due to
the damping rate increasing linearly with $\delta z$. On the other hand, the
SH was expected to be strongly affected by intermittency due to the
exponential factor in Equation 2. However, the _left panel_ shows that SH
dependence on $\delta$ transitions from an exponential to a power law
function. This implies that the damping rate for the case of SH is approaching
the linear function as the exponential factor either approaches unity,
becoming a very weak function of $\delta$. Therefore, when SH is a very
important, and possibly dominant, heating mechanism, its level and variance
within SBs and quiet regions, are significantly less affected by intermittency
compared to intervals when SH is low.

However, another level of analysis of this topic is related to the relation
between intermittency and order unity constants $c_{1}$, $c_{2}$, $\sigma_{1}$
and $\sigma_{2}$, given in Equation 2. Results of RMHD simulations (Xia et
al., 2013) indicate that their predicted values can be notably decreased
depending on the nature of the turbulent fluctuations, increasing the
predicted SH rate. This topic requires separate detailed investigation which
is outside of the scope of this paper.

##  4 Conclusion

In this work, a comprehensive analysis of manually selected magnetic field SB
events in _PSP_ encounters 1 and 2 is performed in order to determine the
characteristics of plasma inside and outside of the regions where the field is
deflected. The investigation shows that the plasma has very similar bulk
quantities and magnetic field values inside SBs, compared to the equivalent
values in the quiet regions before and after a SB, with the very well known
shift in the solar wind bulk speed being of the order of the Alfvén speed.
Additionally, estimates of the non-linear SH rates inside and outside SBs show
that the heating rate due to this mechanism remains fairly similar, which
suggests that the plasma sampled by the spacecraft in different regions of a
single event most likely belong to a single plasma stream.

However, several questions are raised related to the turbulent cascade and
kinetic processes in these events. First, the magnetic field power spectra
ion-scale breaking frequency $f_{b}$ remains similar inside and outside of
SBs, in spite of the spatial turbulent spectra in SBs being shifted toward
higher frequencies due to effects of Taylor hypothesis. Second, we observed
the inertial range PSD logarithmic slopes to have a distribution consistent
with both the values of -5/3 and -3/2, characteristic for the classic critical
balance and dynamic alignment turbulence theories, respectively. The inertial
logarithmic slope is not sensitive to typical physical quantities such as
density, pressures, or radial distance, and its phenomenology remains unclear,
but this could be due to relatively limited range of the plasma parameters
measured. Third, the increased values of PVI in the transition regions and SBs
compared to quiet regions, with this property being independent of the region
duration, reflects the existence of small scale structures at the region
borders. The origin of these structures is directly related to the SB origins
and will be one of the central questions regarding SB evolution in need of
resolution.

In reference to the two different schools of thought about SB origins
discussed in Section 1.2, the fact that we generally observe the same plasma
inside and outside of what is most likely an S-shaped structure (Dudok de Wit
et al., 2020; Horbury et al., 2020), implies that either 1) a parcel of solar
wind plasma expands outwards until the turbulent fluctuations become
sufficiently strong to fold magnetic field back on itself, forming a SB; or 2)
the magnetic field structure travels through the solar wind with the relative
speed of the order of $v_{A}$, folding the plasma without significantly
altering its internal structure. In the second scenario, complicated processes
near the solar surface, such as interchange reconnection (see e.g. Fisk &
Kasper (2020); Zank et al. (2020)), create a structure that persists as it
propagates through the solar wind, temporarily increasing the speed of the
part of the stream which is being pulled through the structure.

The transition regions and edges themselves are not investigated in detail in
this work. Although the basic phenomenology of these regions is well
understood, their comparison with quiet regions and SBs is not trivial, and
the magnetic field PSD from these regions require a more sophisticated
interpretation. The kinetic scale processes and ion-scale turbulence inside
SBs, and development and dissipation of current sheets at their edges, remain
an open topic for future effort. Finally, we note that all the remarks made in
this paper are meant to describe characteristics of the majority of observed
events, and therefore provide observational constraints for models aiming to
describe SBs. We do not expect future case studies, which would cover single
or few interesting events in more detail, to necessarily replicate all the
general conclusions given here.

The SWEAP Investigation and this publication are supported by the _PSP_
mission under NASA contract NNN06AA01C. The FIELDS experiment was developed
and is operated under NASA contract NNN06AA01C. M. M. Martinović and K. G.
Klein were financially supported by NASA grants 80NSSC19K1390 and
80NSSC19K0829. B. D. G. Chandran is partially supported from NASA grants
NNX17AI18G and 80NSSC19K0829 and NASA grant NNN06AA01C to the Parker Solar
Probe FIELDS Experiment. E. Lichko is supported by the National Science
Foundation under Award No. 1949802. C. H. K. Chen was supported by STFC Ernest
Rutherford Fellowship ST/N003748/2 and STFC Consolidated Grant ST/T00018X/1.

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