11institutetext: Institut de Recherche en Astrophysique et Planétologie, CNRS,
UPS, CNES, 9 Ave. du Colonel Roche 31028 Toulouse, France 11email:
<EMAIL_ADDRESS>
Laboratoire d’Astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allee
Georoy Saint-Hilaire, 33615 Pessac, France
Department of Mechanics and Aerospace Engineering, Southern University of
Science and Technology, Shenzhen 518055, People’s Republic of China
Department of Physics and Astronomy and Bartol Research Institute, University
of Delaware, Newark, DE 19716, USA
Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400,
Thailand
Space and Atmospheric Physics, Department of Physics, Blackett Laboratory,
Imperial College London, London, UK SW7 2AZ
CGAFD, Mathematics, CEMPS, University of Exeter, Exeter, UK
LDE3, DAp/AIM, CEA Saclay, 91191 Gif-sur-Yvette, France
Department of Space and Climate Physics, University College London, Mullard
Space Science Laboratory, Holmbury St. Mary, RH5 6NT, UK.

# Solar Orbiter Observations of the Kelvin-Helmholtz Instability in the Solar
Wind

R. Kieokaew 11 B. Lavraud 1122 Y. Yang 33 W. H. Matthaeus 44 D. Ruffolo 55 J.
E. Stawarz 66 S. Aizawa 11 C. Foullon 77 V. Génot 11 R. F. Pinto 1188 N.
Fargette 11 P. Louarn 11 A. Rouillard 11 A. Fedorov 11 E. Penou 11 C.J. Owen
99 T. Horbury 66 H. O’Brien 66 V. Evans 66 and V. Angelini
66112233445566778899

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. The Kelvin-HeImholtz instability (KHI) is a nonlinear shear-driven
instability that develops at the interface between shear flows in plasmas. KHI
has been inferred in various astrophysical plasmas, and has been observed in
situ at the magnetospheric boundaries of solar-system planets and through
remote sensing at the boundaries of coronal mass ejections.

Aims. KHI is also expected to develop at flow shear interfaces in the solar
wind. While it was hypothesized to play an important role in the mixing of
plasmas and in triggering solar wind fluctuations, its direct and unambiguous
observation in the solar wind was still lacking.

Methods. We report in-situ observations of ongoing KHI in the solar wind using
Solar Orbiter during its cruise phase. The KHI is found in a shear layer in
the slow solar wind in the close vicinity of the Heliospheric Current Sheet,
with properties satisfying linear theory for its development. Analysis is
performed to derive the local configuration of the KHI. A 2-D MHD simulation
is also set up with the empirical values to test the stability of the shear
layer. In addition, magnetic spectra of the KHI event are analyzed.

Results. We find that the observed conditions satisfy the KHI onset criterion
from the linear theory analysis, and its development is further confirmed by
the simulation. The current sheet geometry analyses are found to be consistent
with KHI development. Additionally, we report observations of an ion jet
consistent with magnetic reconnection at a compressed current sheet within the
KHI interval. The KHI is found to excite magnetic and velocity fluctuations
with power law scalings that approximately follow $k^{-5/3}$ and $k^{-2.8}$ in
the inertial and dissipation ranges, respectively. Finally, we discuss reasons
for the lack of in-situ KHI detection in past data.

Conclusions. These observations provide robust evidence of KHI development in
the solar wind. This sheds new light on the process of shear-driven turbulence
as mediated by the KHI with implications for the driving of solar wind
fluctuations.

###### Key Words.:

Solar-wind shear flows – Kelvin-Helmholtz instability – Slow solar-wind –
Heliosphere – Space plasmas

##  1 Introduction

The magnetic Kelvin-Helmholtz instability (KHI) is a magnetohydrodynamic (MHD)
shear-driven instability frequently observed in solar system plasmas. KHI can
be induced at the surface between two media with different flow velocity and
plasma conditions. This shear instability is fundamental and can be found in
many flow shear systems throughout the Universe. The KHI that has been most
studied in situ is at the Earth’s magnetopause and low-latitude boundary
layers where periodic fluctuations of magnetic fields and plasma parameters
are observed (e.g., Hones et al., 1981; Fairfield et al., 2000; Hasegawa et
al., 2004; Foullon et al., 2008; Eriksson et al., 2016). This is because the
Earth’s flank magnetopause is prone to the KHI onset condition which prefers
weak magnetic field in the direction of the shear flow. Nevertheless, KHI has
been observed in strong magnetic field environments such as in the solar
corona (Ofman & Thompson, 2011) and at the flank of a coronal mass ejection
(CME) (e.g., Foullon et al., 2011, 2013; Möstl et al., 2013) via Extreme
Ultraviolet (EUV) imaging by the Solar Dynamics Observatory (SDO). Variations
of the KHI such as a sinusoidal mode have also been observed in a solar
prominence (Hillier & Polito, 2018) using Interface Region Imaging
Spectrograph (De Pontieu et al., 2014).

Recently, remote observations above the solar corona using Heliospheric Imager
instrument on board the Solar-Terrestrial Relations Observatory (STEREO/HI1)
have revealed a transition in texture of the solar wind from highly
anisotropic coronal plasma, or “striae”, to more isotropic, or “flocculated”,
solar-wind plasma (DeForest et al., 2016). The transition is found to be
consistent with the onset of hydrodynamic and MHD instabilities leading to
development of turbulence. Qualitatively, this transition is found to occur
near the first surface of plasma $\beta=1$, where the $\beta$ changes from
$\beta\ll 1$ near the Sun to $\beta\approx 1$, and the Alfvén critical surface
where the solar wind speed reaches the Alfvén speed ($V=V_{A}$) (Chhiber et
al., 2018). Ruffolo et al. (2020) propose that the transition from striae to
flocculation of the young solar wind is powered by shear-driven instabilities
such as the KHI, caused by the relative velocities of adjacent coronal
magnetic flux tubes. This is supported by compressible MHD numerical
simulations in which several features observed by Parker Solar Probe (PSP; Fox
et al., 2016) including the magnetic “switchback” signatures near perihelia
are reproduced. It is argued that the KHI can be triggered when the relative
velocity between flux tubes is larger than the local Alfvén velocity ($\Delta
V>V_{A}$) when the magnetic field is along the velocity shear, leading to
shear-driven turbulence just outside the Alfvén critical zone.

In theory, the KHI may develop at tangential discontinuities (TDs) because
there are relative changes in velocity field ($\mathbf{V}$) and magnetic field
($\mathbf{B}$) across them and no normal magnetic field threading through
them, which would otherwise stabilize the instability (as for rotational
discontinuities). The solar wind is full of TDs (e.g., Burlaga et al., 1977;
Knetter et al., 2004; Neugebauer & Giacalone, 2010), which separate different
plasma regions. TDs are thought of as surfaces that separate adjacent solar
wind flux-tubes (e.g., Hollweg, 1982) that originated from granules, or
meso/super granules in the Sun’s photosphere (e.g., Roudier & Muller, 1986;
Axford & McKenzie, 1992) with different plasma properties and composition and
spread out in the heliosphere (e.g., Bruno & Carbone, 2005; Borovsky, 2008).
It was demonstrated theoretically by many authors that TDs can support MHD
surface waves (e.g., Hollweg, 1982). In particular, TDs could support the KHI
(e.g., Burlaga, 1972; Neugebauer et al., 1986). As for the flux tube picture,
it was suggested that the KHI should be induced when adjacent flux tubes move
relative to each other with a speed greater than the Alfvén speed (Burlaga,
1972). Zaqarashvili et al. (2014) consider the topology of magnetic flux tubes
and found that, while the axial $\mathbf{B}$ of the flux tubes stabilizes the
KHI, a slight twist in the $\mathbf{B}$ (i.e., as for when the $\mathbf{B}$
does not align with the $\mathbf{V}$) may allow the surface between them with
$\Delta V<V_{A}$ to become unstable to the KHI.

Across TDs, an alignment between the change in velocity ($\Delta V$) and the
change in magnetic field ($\Delta B$) are commonly observed (Neugebauer, 1985;
Neugebauer et al., 1986; Neugebauer, 2006; Knetter et al., 2004). This
alignment is not expected from the MHD discontinuity theory, unlike at
rotational discontinuities where this alignment is expected (Hudson, 1970).
This alignment of $\Delta V$ and $\Delta B$ is observed independent of the
type of solar wind stream between 1 and 2.2 AU based on IMP 8 and Voyager 2
data (Neugebauer, 1985). Based on Helios data, Neugebauer et al. (1986) report
that this alignment is also observed as close as $0.3$ AU. They further
consider the possibility that the KHI may have developed across TDs and
destroyed the random alignment between $\Delta V$ and $\Delta B$. Since the
number of TDs per unit time decrease with distance from the Sun (e.g.,
Tsurutani & Smith, 1979; Lepping & Behannon, 1986), Neugebauer et al. (1986)
suggested that the KHI may have destroyed TDs as the KHI growth rate becomes
larger with decreasing Alfvén speed.

Despite all these postulations, the KHI has not been observed in situ in the
solar wind to our knowledge. Here we report unambiguous in-situ KHI detection
in Solar Orbiter observations (SolO; Müller et al., 2020) during its cruise
phase. SolO is an ESA mission, launched on February 10, 2020, aimed to study
the Sun and inner heliosphere from out-of-ecliptic vantage points. The cruise
phase started in June 2020 with the in-situ instruments operated nominally
after a successful commissioning. SolO was at $0.69$ AU during the
observations presented here when several periodic fluctuations in plasma
parameters are observed.

In this work, we report quasi-periodic magnetic and plasma variations within a
shear layer in the slow solar wind consistent with the development of the KHI,
supported by the linear theory analysis and numerical simulations. The paper
is organized as follows. First, we present the instrumentation and overview,
context, and KHI observations in Section 2. Additionally, we report
observations of magnetic reconnection signatures in the KHI. Then, we focus on
the linear theory, boundary layer analyses, numerical simulations, and
magnetic spectra of the KHI in Section 3. As this event shows clear evidence
of the KHI in the solar wind, we discuss why it was not observed in past data,
as well as its implications for solar wind dynamics in Section 4. We summarize
our findings and discussion in Section 5.

##  2 Observations

###  2.1 Instrumentation and overview

Figure 1: Overview of SolO and other spacecraft positions on 23 July 2020 at
20:01:08 UT in the RTN coordinates. Orange solid lines represent solar wind
Parker spirals (Parker, 1958) calculated using a constant velocity of $300$ km
s-1. This figure was obtained using the 3DView software (Génot et al., 2018).

We use magnetic field data from the fluxgate vector magnetometer (MAG; Horbury
et al., 2020). MAG continuously samples the magnetic field with the rate up to
16 vector/s in the normal mode and up to 128 vector/s in the burst mode with a
precision of about 5 pT. We also use particle data from the Proton and Alpha
Particle Sensor (PAS) that is part of the Solar Wind Analysis instrument suite
(SWA; Owen et al., 2020). PAS provides high-cadence measurements of 3D
velocity distribution function of solar wind particles (electrons, protons,
alpha particles, and heavy ions). We use the Radial Tangential Normal (RTN)
coordinate system throughout this paper unless stated otherwise. In this
system, the coordinates are centred at the spacecraft where $R$ is directed
radially outward from the Sun to spacecraft, $T$ is longitudinal along the
cross product of the Sun’s rotation vector with $R$, and $N$ completes the
right-handed orthogonal set, which points in the latitudinal direction.

Figure 2: Connectivity map of the observations on July 23, 2020, produced from
the Connectivity Tool (Rouillard et al., 2020) to provide a global context.
The background is a combined image from remote observation using SDO/AIA and
STEREO-A/EUVI 193 $\AA$ Carrington maps. The SolO position is projected to the
area of observation and is shown as a blue cross (see text). The Heliospheric
Current Sheet (HCS) is marked with a red dashed line.

On July 23 - 24, 2020, SolO was at the distance 0.69 AU from the Sun. From
July 23 at 12:00 UT to July 24 at 12:00 UT, SolO was moving from the distance
$r=1.03\times 10^{8}$ km ($0.68$ AU) to $r=1.04\times 10^{8}$ km ($0.70$ AU)
from the Sun. Fig. 1 shows the projected position of SolO onto the equatorial
plane with the solar wind Parker spiral (orange lines) (Parker, 1958)
calculated using a constant solar wind speed of $300$ km s-1. This figure is
obtained using the 3DView software (Génot et al., 2018) available at
http://3dview.cdpp.eu/. The radial speed of SolO was $V_{R}=11.3$ km s-1
throughout this interval.

Fig. 2 shows the global context on July 23, 2020 from 18 to 24 UT of the solar
observations with the SolO position (blue cross) projected back to the Sun’s
surface. This figure is obtained from the Magnetic Connectivity Tool
(Rouillard et al., 2020), accessible at http://connect-tool.irap.omp.eu/, that
can be used to connect between remote observations and in-situ observations.
The background image is produced by combining images taken by SDO and STEREO-A
in EUV at $193$ $\AA$, using the Heliospheric Imagers (HI) instruments on
board. The green and red dots indicate the points at the surface that are most
likely to be magnetically connected to SolO, assuming uniform slow and fast
wind flows. The SolO position is estimated to be at the blue cross position.
The connected positions are projected on magnetograms from the ADAPT (Air
Force Data Assimilative Photospheric Flux Transport) flux transport model
(Arge et al., 2010; Hickmann et al., 2015). On this image, the SolO position
appears at low-latitude ($\sim 0^{o}$) and near the Heliospheric Current Sheet
marked by a red dashed line. In this work, we report SolO observations of the
KHI near an edge of the HCS.

Figure 3: Solar Orbiter observations between 16:00 UT on 23 July 2020 and 6:00
UT on 24 July 2020 showing the context of the Kelvin-Helmholtz instability
observation (shaded area). (a) Magnetic fields in the RTN coordinates. (b) Ion
bulk velocity $V_{R}$ component. (c) Ion bulk velocity $V_{T}$ and $V_{N}$
components. (d) Ion number density. (e) Alfvén Mach number. (f) Normalized
cross-helicity. (g) The angle between $\delta\mathbf{v}$ and
$\delta\mathbf{b}$ (see text). The Heliospheric Current Sheet (HCS) is
observed between 22:00 UT on 23 July 2020 and 2:00 UT on 24 July 2020, marked
by purple dotted lines. The Magnetic Increase with Central Current Sheet
(MICCS) is observed between 17:35 and 18:08 UT on 23 July 2020, marked by
orange dotted lines.

###  2.2 12-hour context

Fig. 3 shows SolO observations between 16:00 UT on July 23, 2020 and 6:00 UT
on July 24, 2020, covering a 12-hour interval. In Fig. 3a, we show the
magnetic field components and magnitude. Fig. 3b displays the radial component
($V_{R}$) and Fig. 3c the tangential ($V_{T}$) and out-of-ecliptic ($V_{N}$)
components of the ion bulk flow velocity. Two main HCS crossings are observed
between 22:00 UT on July 23 and 2:00 UT on July 24 in between the purple
vertical dotted lines. The HCS is characterized by a large-scale
($\sim$several hours) change in the polarity of the radial magnetic field
($B_{R}$). Around 18:00 UT on July 23, we mark a meso-scale structure ($\sim$a
few hours) between vertical orange dotted lines, characterizing bipolar
magnetic fields with a sharp magnetic transition (i.e., current sheet) at the
center. At this central current sheet, there is an ion jet clearly seen in
$V_{R}$ and $V_{N}$, probably indicating ongoing magnetic reconnection. This
structure is known as Magnetic Increase with Central Current Sheet (MICCS) and
it has been observed with Parker Solar Probe (Fargette et al., 2021); this
signature corresponds to two magnetic flux tubes that become interlinked and
with magnetic reconnection at the interface (e.g., Louarn, 2004; Kacem et al.,
2018; Øieroset et al., 2019; Kieokaew et al., 2020). Although mentioned here
for context, this feature is not discussed any more in our study.

Figure 4: Solar Orbiter observations on 23 July 2020 between 20:10 UT and
21:50 UT. (a) Magnetic field in the RTN coordinates. (b) Ion bulk velocity
$V_{R}$ component. (c) Ion bulk velocity $V_{T}$ and $V_{N}$ components. (d)
Alfvén speed $|\mathbf{V}_{A}|$ and Alfvén velocity components
($V_{A,r},V_{A,t},V_{A,n}$). (e) Ion number density. (f) Plasma pressure
$P_{p}$, magnetic pressure $P_{m}$, and total pressure $P_{tot}$. (g) Plasma
beta $\beta$. The KH waves can be noticed between 20:45 and 21:30 UT. The
vertical grey dashed lines (1) - (7) mark the boundary layer crossings
corresponding to KH trailing edges. The magenta dashed-dotted lines mark times
when the plasma parameters reach asymptotic values on Side 1 (20:38 UT) and
Side 2 (21:29 UT) of the shear layer.

From 20:45 to 21:30 UT on July 23 in Fig. 3c, we observe a velocity shear of
about $30$ km s-1 in the $V_{T}$ (green) and about $20$ km s-1 in the $V_{N}$
(blue) components. This velocity shear interval features periodic fluctuations
in the magnetic fields (Fig. 3a) and the radial velocity ($V_{R}$; Fig. 3b).
The shear layer is also accompanied by changes in the ion number density shown
in Fig. 3d. This interval as shaded in grey exhibits KH-like signatures; we
show a zoom-in of this interval in the next section.

Figs. 3e, 3f, and 3g show the Alfvén Mach number ($M_{A}$), the normalized
cross-helicity ($\sigma_{c}$), and the angle between $\mathbf{V}$ and
$\mathbf{B}$ changes ($\theta_{VB}$), respectively. During the shaded
interval, the $M_{A}$ is $12$, indicating that the solar wind is super-
Alfvénic. The $\sigma_{c}$ (Matthaeus & Goldstein, 1982; Roberts et al., 1992)
is calculated from
$\sigma_{c}=2\langle\delta\mathbf{v}\cdot\delta\mathbf{b}\rangle/\langle|\delta\mathbf{v}|^{2}+|\delta\mathbf{b}|^{2}\rangle$,
where $\delta\mathbf{v}=\mathbf{V}-\langle\mathbf{V}\rangle_{20min}$ and
$\delta\mathbf{b}-\langle\mathbf{b}\rangle_{20min}$ are the velocity field and
magnetic field fluctuations from the $20-$min running averages of $\mathbf{V}$
and $\mathbf{b}=\mathbf{B}/\sqrt{\mu_{0}\rho}$, where $\mu_{0}$ is the vacuum
permeability and $\rho$ is the average proton mass density, respectively. The
magnetic field $\mathbf{b}$ is measured in Alfvén speed units in km s-1. The
sum ($\langle\rangle$) brackets are also taken over $20$-min running averages.
$\sigma_{c}$ relates to the cross helicity
$H_{c}=\frac{1}{2}\int(\mathbf{V}\cdot\mathbf{b})d^{2}x$ and the energy per
unit mass $E=\frac{1}{2}\int(\mathbf{V}^{2}+\mathbf{b}^{2})d^{2}x$ through the
relation $\sigma_{c}=2H_{c}/E$. This quantity measures the Alfvénicity with
$\sigma_{c}=\pm 1$ indicating Alfvénic fluctuations, where the signs $+,-$
correspond to Alfvénic propagation anti-parallel or parallel to the mean
field, respectively. During the shaded interval, $\sigma_{c}$ is fluctuating
around $0.5$, indicating that the solar wind is not strongly Alfvénic. The
angle $\theta_{VB}$ is calculated from
$\arccos(\delta\mathbf{v}\cdot\delta\mathbf{b}/(|\delta\mathbf{v}||\delta\mathbf{b}|))$.
For the shaded time period, Fig. 3g shows that the angle between
$\delta\mathbf{v}$ and $\delta\mathbf{b}$ is spreading from $90^{o}$ to
$120^{o}$. The non-alignment of $\delta\mathbf{v}$ and $\delta\mathbf{b}$
during this interval indicates the possibility for this shear layer to be
unstable to the KHI. The KHI would be suppressed if the magnetic field change
aligns with the velocity field change due to the magnetic tension exerted in
the direction of the shear flow (e.g., Chandrasekhar, 1961), i.e., when
$\theta_{VB}$ is near $0^{o}$ or $180^{o}$.

###  2.3 KHI observations

Fig. 4 shows a zoom-in around the shaded time period in Fig. 3 between 20:45
and 21:30 UT on July 23, 2020. The quasi-periodic fluctuations in several
parameters resemble in-situ KH waves at the interface between shear flows
(e.g., Hasegawa et al., 2004). The magnetic field in Fig. 4a clearly shows
repeated, homologous fluctuations in all components. We mark sharp magnetic
rotations (1) - (7) with vertical dashed lines. Fig. 4b shows $V_{R}$ and Fig.
4c shows $V_{T}$ and $V_{N}$ components of the ion bulk velocity. The velocity
shear is clearly seen in the $V_{T}$ component. To mark the shear layer, we
define Side 1 at 20:38 UT as the beginning of the shear interval and Side 2 as
the end of the shear interval at 21:29 UT, marked by magenta dotted-dashed
lines. Sides 1 and 2 are characterized mainly by the tangential velocity
$V_{T}$ which reached local minimum and maximum, respectively. The periodic
features are also seen in the velocity fields especially for the tangential
($V_{T}$) component in Fig. 4c in which sudden, sharp transitions colocate
with the changes in the magnetic field marked by the vertical dashed lines. To
facilitate discussion, we define the magnetic rotations (1) - (7) as “wave
edges” that mark sudden changes in both magnetic and velocity fields. Table 1
notes times of these wave edges and time differences between these edges.
These time differences correspond to the period of the waves. The average time
difference between wave edges is $7$ minutes $17$ seconds (s) and the standard
deviation is $55$ s. We analyze the wave edges in detail in Section 3.3.

Table 1: Timing of the KH wave edges marked in Fig. 4 and time difference between them. Numbers | Times (UT) | Time difference (mm:ss)
---|---|---
(1) | 20:45:59 | -
(2) | 20:53:43 | 07:44
(3) | 21:02:11 | 08:28
(4) | 21:09:56 | 07:45
(5) | 21:17:32 | 07:36
(6) | 21:23:39 | 06:07
(7) | 21:29:38 | 05:59

Fig. 4d shows Alfvén speed and Alfvén velocity components
($\mathbf{V}_{A}=\mathbf{B}/\sqrt{\mu_{0}n_{i}m_{p}}$, where $n_{i}$ is the
ion number density and $m_{p}$ is the proton mass). The average Alfvén speed
in this interval is $26$ km s-1. Fig. 4e shows the ion number density
($n_{ion}$). Despite some fluctuations, $n_{ion}$ gradually changes from $30$
cm-1 at Side 1 to $22$ cm-1 at Side 2. Fig 4f shows the magnetic ($P_{m}$),
thermal ($P_{p}$), and total pressures ($P_{tot}=P_{m}+P_{p}$). The total
pressure is approximately constant, indicating approximate pressure balance
across the shear layer. Fig. 4g shows the ion beta ($\beta$). The $\beta$ is
changing from $\beta\approx 2$ at Side 1 to $\beta\approx 1$ at Side 2. A
strong peak in $\beta$ of about $\beta\approx 7$ is observed adjacent to wave
edge (3). At wave edge (3), there is an ion jet in $V_{N}$ (blue in Fig. 4c)
with $\Delta V_{N}\sim 11$ km s-1 that colocates with the magnetic rotation
observed in $B_{N}$ (blue in Fig. 4a). We explain that this ion jet may be
produced by magnetic reconnection, in Section 2.4.

Table 2 summarizes the values of $\mathbf{V},\mathbf{B}$ and $n_{ion}$ at
Sides 1 and 2. Considering the velocity change across the shear layer, we
define $\Delta\mathbf{V}=\mathbf{V}_{2}-\mathbf{V}_{1}$, where the subscripts
$1$ and $2$ label Sides 1 and 2, respectively. The velocity change is
$\Delta\mathbf{V}=(\Delta V_{R},\Delta V_{T},\Delta V_{N})=(30,30,-11)$ km
s-1, with $|\Delta\mathbf{V}|=44$ km s-1. The ratio of the velocity change
across the shear layer to the average Alfvén speed is therefore $\Delta
V/V_{A}=1.7$. An important criterion of the KHI from the linear theory of
stability of a shear layer in plasmas considered by Chandrasekhar (1961), Sen
(1963), and Landau & Lifshitz (1960) is that the KHI would be suppressed for
$\Delta V<{V_{A}}$, i.e., when the velocity shear is less than the local
Alfvénic speed. Note that this condition is obtained in a simplistic
configuration of an infinitely thin shear layer embedded in the uniform
density and magnetic field and with the magnetic field being exactly along the
velocity shear. The observed shear velocity exceeds the local Alfvén speed for
this approximate consideration and thus the KHI is not suppressed. In Section
3.1, to test whether the observed local shear conditions support the KHI
growth, we consider the KHI onset condition by taking into account the
nonuniform conditions on either side of the shear layer derived from the
linear theory. We also set up a local MHD simulation using these empirical
conditions in Section 3.2.

Table 2: Parameter values on Side 1 and Side 2 of the shear layer marked in Fig. 4. Parameters | Side 1 | Side 2 | Units
---|---|---|---
$V_{R}$ | $285$ | $305$ | km s-1
$V_{T}$ | $-20$ | $10$ | km s-1
$V_{N}$ | $-5$ | $-16$ | km s-1
$B_{R}$ | $4$ | $4$ | nT
$B_{T}$ | $2$ | $-2$ | nT
$B_{N}$ | $0$ | $3$ | nT
$n_{ion}$ | $30$ | $22$ | cm-3

###  2.4 Magnetic reconnection

Figure 5: Magnetic reconnection signatures at wave edge (3) in Fig. 4. (a)
Magnetic field strength. (b) Magnetic field reconnecting $L$-component. (c)
Magnetic field $M,N$ components. (d) Ion bulk velocity $V_{L}$ component with
the predicted velocity jet ($V_{A,L}$; cyan). (e) Ion number density. (f) Ion
temperature. Dashed orange lines mark the extent of the ion jet. A dashed grey
line marks the maximum ion velocity during the jet.

Fig. 5 shows a zoom-in at wave edge (3) in Fig. 4 that contains an ion jet
colocated with the change in magnetic field seen in the $N$ component (blue).
The current sheet interval at the wave edge is delineated by orange dashed
lines between 21:02:00 and 21:02:40 UT. The magnetic field magnitude in Fig.
5a shows a drop at the centre of the current sheet of about $2.5$ nT. To
clearly see the ion jet, we transform the magnetic field into local current
sheet $LMN$ coordinates using the hybrid Maximum Variance Analysis (MVA)
technique (Gosling & Phan, 2013). In this coordinate system, $L$ points in the
magnetic shear direction (i.e., the reconnecting component), $N$ points in the
direction normal to the shear plane, and $M=N\times L$ points in the out-of-
plane direction (i.e., the guide-field direction). We transform the magnetic
field from the $RTN$ coordinates to $LMN$ coordinates as follows. First, the
current sheet normal $N$ is obtained from the cross-product of the asymptotic
$10$-s averaged magnetic fields just outside the current sheet interval.
Second, the $M$ component is obtained from $N\times L_{1}$, where $L_{1}$ is
the maximum variance direction obtained from the MVA technique (Sonnerup &
Cahill, 1967) applied in the current sheet interval. Finally, $L=M\times N$
completes the right-handed orthonormal system. We obtain
$L=[-0.285,0.245,-0.924],M=[0.882,0.44,-0.156]$, and $N=[0.37,-0.863,-0.343]$.

The magnetic field rotation is clearly seen in the $B_{L}$ component in Fig.
5b. $B_{L}$ rotates from positive to negative during the current sheet
interval in colocation with the ion jet seen in the $V_{L}$ component in Fig.
5e. Note that the velocity data have lower cadence ($4$ s) than the magnetic
field data ($1$ s), thus the velocity is sampled only at a few points in the
vicinity of the current sheet interval. The velocity peak is marked by a grey
dashed line. This ion jet has a magnitude of $\Delta V=13$ km s-1. Although
there is only one velocity measurement associated with the peak inside the
jet, one should note that the operation cycle of the PAS instrument is such
that each sample is made over only $1$ s every $4$ s. The measurement is thus
made over a limited time exactly in the center of the current sheet, rather
than over $4$ s which would have led to significant time aliasing across the
current sheet.

The change in $B_{L}$ correlates with the change in $V_{L}$ negatively on the
inbound side and positively on the outbound side. This sequence of
correlations is consistent with a jet that is produced by magnetic
reconnection. Although we acknowledge that there is only one velocity
measurement in the jet, there are additional signatures that are consistent
with the interpretation of magnetic reconnection, such as the enhanced ion
number density shown in Fig. 5e, consistent with mixing of ions from either
side of the current sheet in the reconnection exhaust (Gosling et al., 2005)
and the enhanced ion temperature consistent with plasma heating within the
reconnection exhaust (Phan et al., 2014) in Fig. 5f.

To more quantitatively assess whether this jet is consistent with
reconnection, we consider the Walén relation:
$\Delta\mathbf{V}_{A}\sim\pm\Delta\mathbf{B}/(\mu_{0}m_{p}n_{ion})^{1/2}$,
where $+$ or $-$ is applied for a positive or negative correlation between
$\mathbf{B}$ and $\mathbf{V}$, respectively, (Hudson, 1970; Paschmann et al.,
1986) within the exhaust as bounded by dashed orange lines in Fig. 5. The
predicted jet velocity is shown as cyan in Fig. 5d. The predicted velocity
produces a trend that resembles the observed jet. However, the predicted
velocity jet from the Walén relation is estimated to be $\Delta V_{A}=33$ km
s-1. Thus, the observed jet has a velocity that is $40$% of the predicted jet
velocity. A sub-Alfvénic reconnection jet is not unusual in observations. In
the literature, sub-Alfvénic jet speeds are found when the spacecraft crosses
the reconnection exhaust near the X-line. The development of secondary
instabilities due to high plasma-$\beta$ was also found to lower the
reconnection jet speed (Haggerty et al., 2018). The presence of reconnection
within KH waves may imply that it is produced as a consequence of vortex-
induced-reconnection (e.g., Nykyri & Otto, 2001; Nakamura et al., 2006;
Karimabadi et al., 2013; Eriksson et al., 2016), which can be triggered when
KH vortices develop and create thin current sheets between them. We further
discuss this possibility in the discussion section.

##  3 Results

###  3.1 Linear theory analysis

To test whether the observed local conditions in Table 2 satisfy the KHI onset
criterion, we consider the stability of a shear layer derived using the linear
theory of Chandrasekhar (1961). The KHI onset criterion derived for an
infinitely thin boundary layer in an incompressible plasma can be written
(Hasegawa, 1975) as

$\left[\mathbf{k}\cdot(\mathbf{V}_{1}-\mathbf{V}_{2})\right]^{2}>\frac{n_{1}+n_{2}}{\mu_{0}m_{p}n_{1}n_{2}}\left[(\mathbf{k}\cdot\mathbf{B}_{1})^{2}+(\mathbf{k}\cdot\mathbf{B}_{2})^{2}\right]$
(1)

where $\mathbf{k}$ is the wave vector, $\mathbf{V}$ is the velocity field,
$\mathbf{B}$ is the magnetic field, and $n$ is the ion number density, with
the subscripts $1,2$ representing either side of the shear layer. The phase
velocity of the KH mode, associated with the real part of the KH dispersion
relation $V_{ph}=\omega/k$, where $\omega$ is the wave frequency and $k$ is
the wave number, is given as

$V_{ph}=\dfrac{n_{1}\mathbf{k}\cdot\mathbf{V}_{1}+n_{2}\mathbf{k}\cdot\mathbf{V}_{2}}{k(n_{1}+n_{2})}$
(2)

The growth rate of the KHI, associated with the imaginary part of the
dispersion relation, can be written as

$\gamma=[\alpha_{1}\alpha_{2}[(\mathbf{V}_{1}-\mathbf{V}_{2})\cdot\mathbf{k}]^{2}-\alpha_{1}(\mathbf{V_{A,1}}\cdot\mathbf{k})^{2}-\alpha_{2}(\mathbf{V_{A,2}}\cdot\mathbf{k})^{2}]^{1/2}$
(3)

where $\alpha_{1}=n_{1}/(n_{1}+n_{2})$ and $\alpha_{2}=n_{2}/(n_{1}+n_{2})$,
and $\mathbf{V_{A,1}},\mathbf{V_{A,2}}$ label the Alfvén speeds on either side
of the boundary.

Figure 6: Magnetic and velocity fields in the local shear frame obtained from
the application of the MVA to the ion bulk velocity. (a) Magnetic field in the
maximum ($X$), intermediate ($Y$), and minimum ($Z$) variance directions and
its magnitude. (b, c, d) Velocity field in the $X$, $Y$, and $Z$ directions,
respectively. (e) Simplified shear boundary layer configuration obtained from
the transformation to the maximum shear frame.

To simplify the configuration of the observed shear layer, we transform the
velocity field using the application of the MVA technique to the ion bulk
velocity from 20:14 to 21:50 UT. The maximum, intermediate, and minimum
variance directions are found to be $[0.53,0.79,-0.32],[0.84,-0.53,0.07]$, and
$[0.12,0.31,0.95]$, respectively. The ratios of the maximum to the
intermediate eigenvalues ($\lambda_{1}/\lambda_{2}$) and the intermediate to
the minimum eigenvalues ($\lambda_{3}/\lambda_{2}$) are $7.7$ and $2.0$,
respectively, indicating reliable estimations (Siscoe & Suey, 1972). The
maximum variance direction is the direction where we find the velocity jump
(i.e., the wave amplitude direction); it is assigned as $Y$. The minimum
variance direction is the invariant direction; it is assigned as $Z$. The
intermediate variance direction is the wave propagation direction; it is
assigned as $X$. The transformed $\mathbf{B}$ and $\mathbf{V}$ are shown in
Figs. 6a - 6d. The velocity jump is clearly seen in Fig. 6b with $\Delta V=40$
km s-1. Fig. 6e shows a simplified configuration of this shear layer in the
$X-Y$ plane. Note that the $V_{y}$ and $V_{z}$ are nearly constant and thus
not shown in this figure.

We may calculate a KH growth rate from the simplified configuration in Fig.
6e. In this frame, the shear velocity is
$\Delta\mathbf{V}=\mathbf{V}_{2}-\mathbf{V}_{1}\approx(40,0,0)$ km s-1. Since
the magnetic field perpendicular to the shear direction, i.e., the $B_{z}$
component (the invariant direction), does not impact the KHI (Chandrasekhar,
1961), we can ignore this component. Assuming that the wave vector
$\mathbf{k}$ is in the $X-Y$ plane and makes an angle $\phi$ from the
$Y$-direction such that $\mathbf{k}=(k\cos\phi,k\sin\phi,0)$, Eq. 3 can be
written as

$\displaystyle\left(\frac{\gamma}{k}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{\rho_{1}\rho_{2}}{(\rho_{1}+\rho_{2})^{2}}\left[\Delta
V_{x}\cos\phi+\Delta V_{y}\sin\phi\right]^{2}$ (4)
$\displaystyle-\frac{1}{\mu_{0}(\rho_{1}+\rho_{2})}\left[B_{1,x}\cos\phi+B_{1,y}\sin\phi\right]^{2}$
$\displaystyle-\frac{1}{\mu_{0}(\rho_{1}+\rho_{2})}\left[B_{2,x}\cos\phi+B_{2,y}\sin\phi\right]^{2}$

where $\rho_{i}=m_{p}n_{i}$, $i=1,2$. We find positive growth rates for an
arbitrary angle $\phi$ with a maximum growth rate ($\gamma/k$) of $16$ km s-1.
This means that the observed conditions of this event are unstable to the KHI.
In brief, the linear theory analysis supports the KHI interpretation.

Assuming that the wave vector $\mathbf{k}$ is in the same direction as the
flow, we obtain the KH phase velocity from Eq. (2) to be $V_{ph}=152$ km s-1.
Since the average wave period from Table 1 is found to be $437\pm 55$ s
($7.3\pm 0.9$ minutes), the KH wavelength is estimated to be
$\lambda_{KH}=66,400\pm 8,400$ km or $0.10\pm 0.01$ solar radii. The linear
theory analysis of a finite-thickness shear layer by Miura & Pritchett (1982a)
predicts that the fastest growing mode occurs for $kL\sim 0.5-1.0$, where $L$
is the initial shear layer thickness. Using $k=2\pi/\lambda$, the fastest
growing mode should have the wavelength of $2\pi L-4\pi L$. Using our
estimated $\lambda_{KH}$, we estimate the initial shear layer thickness to be
$L\approx 5,300$ \- $10,500$ km. Note that this estimate ignores any vortex
merging or potential influence of pre-existing turbulence that could influence
the size of the vortices that are observed.

###  3.2 KH simulation

Figure 7: A snapshot of the numerical simulation of the KHI using empirical
values of the SolO event from Side 1 and Side 2. The color-scale represents
values of the out-of-plane flow vorticity ($\omega$). The KHI quickly reaches
the non-linear stage where rolled-up KH vortices form and coalesce. The
stripes in vorticity in the lower part of the simulation are shocks produced
by the supersonic flow as the Mach number $\sim 3$ (see text).

To further test whether the observed conditions would support the KHI, we
performed a numerical simulation using SolO observations for Side 1 and Side 2
as boundary conditions. We exploit a numerical simulation that solves
compressible MHD equations via a hybrid compact-weighted essentially non-
oscillatory (WENO) scheme (Yang et al., 2016b). This hybrid scheme couples a
sixth-order compact finite difference scheme for smooth regions and a fifth-
order WENO scheme in shock regions, suitable for capturing strong
discontinuities in MHD systems. The time stepping is performed by the third-
order Runge-Kutta scheme. This code has been used to study compressible MHD
turbulence (Yang et al., 2016a, 2017) and shear-driven turbulence by the KHI
near the Sun (Ruffolo et al., 2020).

To simulate our event, we consider the local shear frame in Fig. 6e, derived
in Section 3.1. Moreover, we consider the local KHI frame that travels with
the KH phase speed at $V_{ph}\approx 150$ km s-1. In this frame, the speeds on
Sides 1 and 2 are $U_{X,1}=-15$ km s-1 and $U_{X,2}=25$ km s-1, respectively.
Since the magnetic field perpendicular to the shear flow does not impact the
KH growth (Chandrasekhar, 1961), we only include magnetic field in the shear
flow direction. The ion number density values on Sides 1 and 2 are $n_{1}=30$
cm-3 and $n_{2}=22$ cm-3, respectively. The ion temperature values on Sides 1
and 2 are set to $T_{1}=1.32$ eV and $T_{2}=3.63$ eV, respectively. The ion
$\beta$ is set to $1$. The magnetosonic Mach number across the shear layer is
$\Delta U/c_{s,1}=2.75$, where $\Delta U=40$ km s-1 and $c_{s,1}$ is the sound
speed on Side 1. The Alfvén Mach number is $\Delta U/V_{A,1}=2.52$, where
$V_{A,1}$ is the Alfvén speed on Side 1.

The numerical simulation is performed using a $L_{x}\times L_{y}=8\pi\times
4\pi$ domain with $n_{x}\times n_{y}=1024\times 512$ resolution with periodic
boundary conditions in the $X$-direction. For simplicity, equal viscosity and
resistivity $\mu=\eta$ are used, i.e., the magnetic Prandtl number is set to
unity. We solve the dimensionless form of the MHD equations by introducing
several reference scales. The normalizations are $U_{0}=100$ km s-1,
$n_{0}=30$ cm-3, $B_{0}=25$ nT, and $T_{0}=66$ eV. The simulation is 2-D as we
ignore the invariant direction and only impose the magnetic field in the
direction of the shear flow.

We set up double shear layers in the simulation domain similar to those of
Ruffolo et al. (2020). The velocity and magnetic profiles are only set in the
$X$-direction and both are colocated. The velocity profile is given by

$\displaystyle
u_{x}=U_{\alpha}\left[1-\tanh\left(\frac{y-L_{y}/4}{d}\right)+\tanh\left(\frac{y-3L_{y}/4}{d}\right)\right]+U_{\beta},$
(5)

where $U_{\alpha}=\left(\frac{U_{1}-U_{2}}{2}\right)$,
$U_{\beta}=\left(\frac{U_{1}+U_{2}}{2}\right)$, $U_{1}=-0.15U_{0}$,
$U_{2}=0.25U_{0}$, and $d=0.003L_{y}$ is the half thickness of the shear
layer. The magnetic profile is given in a similar way as

$\displaystyle
B_{x}=B_{\alpha}\left[1-\tanh\left(\frac{y-L_{y}/4}{d}\right)+\tanh\left(\frac{y-3L_{y}/4}{d}\right)\right]+B_{\beta},$
(6)

where $B_{\alpha}=\left(\frac{B_{1}-B_{2}}{2}\right)$,
$B_{\beta}=\left(\frac{B_{1}+B_{2}}{2}\right)$, $B_{1}=0.16B_{0}$, and
$B_{2}=0.0$. The density is set with $\rho_{1}=\rho_{0}$ and
$\rho_{2}=0.73\rho_{0}$ in normalized units. The initial temperature profile
is set such that the total (magnetic plus thermal) pressure is balanced across
the shear layer, where $T_{1}=0.02T_{0}$ and $T_{2}=0.055T_{0}$. Finally, the
background shear is initially perturbed by adding a small compressive velocity
field in the $Y$-direction in the form

$\displaystyle du_{y}=\delta
u_{0}\left[e^{\left(\frac{y-L_{y}/4}{4d}\right)^{2}}-e^{-\left(\frac{y-3L_{y}/4}{4d}\right)^{2}}\right]\text{ran}(x)$

where $\delta u_{0}=0.008(U_{2}-U_{1})$ (i.e., less than $10\%$ of the shear
flow magnitude) and $\text{ran}(x)$ represents a random number generator in
the range [-0.5, 0.5] at each grid value.

Fig. 7 shows a snapshot of the simulation at $\sim 13$ large-eddy turnover
time for one of the shear layers (the two shear layer develop similar KHI
structures). The color represents the flow vorticity ($\omega$) in the out-of-
plane direction. The KH waves are seen to develop in the simulation. They
quickly reach the non-linear stage where rolled-up KH vortices clearly form
(from $\sim 6$ large-eddy turnover times onwards) with visible vortex merging.
This confirms that the solar wind observations by SolO are consistent with the
KHI growth. In the lower part of the simulation (Side 1), there are features
seen as stripes in vorticity. These features are shocks that are produced by
the supersonic flow on Side 1 (Mach number $\sim 3$) as the speed difference
between the recirculating vortex and the nearby passing flow exceeds the sound
speed (e.g., Landau & Lifshitz, 1987). Future work ought to determine whether
such features are sometimes observed in spacecraft data. For the present work,
the key point would remain that the KHI does develop.

###  3.3 Boundary layer analysis

Figure 8: SolO observations on July 23, 2020 with marked outbound and inbound
crossings (i.e., wave edges) of the in-situ KH waves and a schematic sketch of
the event. (a) Magnetic fields. (b, c) Magnetic field fluctuations from the
averages in the $R-T$ and $R-N$ planes, respectively. (d) Ion bulk velocity
$V_{R}$ component. (e) Ion bulk velocity $V_{T}$ and $V_{N}$ components. (f,
g) Velocity field perturbations in the $R-T$ and $R-N$ planes, respectively.
Cyan dashed lines mark times of the outbound crossings, which correspond to KH
trailing edges. Magenta dashed lines mark times of the inbound crossings,
which correspond to KH leading edges. Normal directions of the inbound and
outbound crossings are listed in Table 3. (h) Schematic sketch of the KH waves
based on the normal angle (red arrows) at the wave edges.

To understand local configurations of the KH waves, we characterize
orientations of the observed wave edges in the RTN coordinates. In Fig. 4, the
magnetic rotations are clearly defined for wave edges (2), (3), and (4). Fig.
8 shows $\mathbf{B}$ and $\mathbf{V}$ and their fluctuations from the average
values between 20:35 and 21:20 UT. Wave edges (2) - (3), defined by clear
rotations of $\mathbf{B}$ in Fig. 8a and $\mathbf{V}$ in Figs. 8d and 8e, are
marked by cyan vertical dashed lines. We define these sharp rotations as the
“outbound crossings”, which correspond to crossings of KH wave trailing edges
that separate adjacent vortices. A KH trailing edge is typically thin, seen as
a sharp transition, as a consequence of the compression between two waveforms
or vortices. Smaller magnetic field rotations are also visible between these
outbound crossings. We mark these smaller rotations as (2’), (3’) , and (4’)
with magenta vertical dashed lines in Fig. 8. These edges show reverse
transitions compared to the outbound crossings in $B_{R}$ (red) and $B_{N}$
(blue) components. We define these smaller transitions as “inbound crossings”,
which correspond to crossings of KH wave leading edges. A KH leading edge is
typically less sharply defined in in-situ data as it is in the vicinity of KH
vortices that are regions of plasma mixing, so the transitions between two
regions of the shear layer are less clear (e.g., Chen & Kivelson, 1993;
Fairfield et al., 2000).

To analyze the orientations of the wave edges, we calculate the boundary
normals of the inbound and outbound pairs marked in Fig. 8. The normal of a
discontinuity (i.e., current sheet) can be obtained from the cross-product of
magnetic fields on either side of the discontinuity, i.e.,
$\mathbf{n}=\pm(\langle\mathbf{B}_{1}\rangle\times\langle\mathbf{B}_{2}\rangle)/|\langle\mathbf{B}_{1}\rangle\times\langle\mathbf{B}_{2}\rangle|$,
where $\langle\mathbf{B}_{1}\rangle$ and $\langle\mathbf{B}_{2}\rangle$ are
time averages of asymptotic magnetic fields before and after the current sheet
interval, respectively. The obtained normal direction has a sign ambiguity
($\pm$); we assign a direction outward from the Sun, i.e., by forcing the
radial component of $\mathbf{n}$ to be positive. The time-averaged
$\langle\mathbf{B}_{i}\rangle$, where $i=1,2$, are defined as $10$-s averages
of the magnetic fields. We obtain the normal orientations
($\mathbf{n}=[n_{R},n_{T},n_{N}]$) of the marked inbound and outbound
crossings in Fig. 8. In addition, we define an angle
$\phi=\arctan(-n_{T}/n_{R})$ to be the angle of the current sheet normal from
the $R$ direction in the $RTN$ coordinate system. The angle $\phi$ is in range
$[-90^{o},90^{o}]$ where $\phi=0^{o}$ is parallel to the Sun-SolO line and
$\phi=-90^{o}$ and $90^{0}$ are perpendicular to the Sun-SolO line in the $T$
and $-T$ directions, respectively.

Table 3: Normal directions and orientations of the inbound (leading edge) and outbound (trailing edge) crossings with their normal directions ($\mathbf{n}$) and associated angle $\phi=\arctan(-n_{T}/n_{R})$. Wave edges | Times (UT) | Normal direction | Angle $\phi$
---|---|---|---
Inbound (2’) | 20:49:20 | [0.64, -0.57, -0.51] | $41.7^{o}$
Outbound (2) | 20:53:43 | [0.78, -0.56, -0.28] | $35.7^{o}$
Inbound (3’) | 20:58:08 | [0.71, -0.63, -0.31] | $41.5^{o}$
Outbound (3) | 21:02:11 | [0.91, -0.36, -0.21] | $21.6^{o}$
Inbound (4’) | 21:06:50 | [0.70, -0.68, 0.2] | $44.2^{o}$
Outbound (4) | 21:09:56 | [0.58, -0.67, 0.45] | $49.1^{o}$

Table 3 notes times of the inbound and outbound pairs, their normal directions
($\mathbf{n}$), and the normal angle $\phi$. The normal directions for all
inbound and outbound crossings are in between the $R$ and $-T$ directions;
these translate to the normal angles between $0^{o}$ and $90^{o}$. The normal
angles ($\phi$) of an inbound-outbound pair were used to characterize in-situ
waves at the Earth’s magnetopause by (Plaschke et al., 2016). By considering
the angles of an inbound and outbound pair, we can determine whether the in-
situ waveform is consistent with a sinusoidal wave, a KH wave, or another type
of wave. Of particular relevance is that, when the waveform is a KH vortex
(i.e., a rolled-up KH wave), the angles of the inbound and outbound pair
should be in the same range (cf. Fig. 1 of Plaschke et al. (2016)). The
observed angles of all the inbound-outbound pairs in Table 3 are well
consistent with the steepening of KH vortices. This result supports the
argument that the KHI is developed at the shear layer. We sketch the wave
trains based on the normal angles with a hypothetical relative SolO trajectory
in Fig. 8h. Note that, in reality, the spacecraft is relatively static as the
structures flow past with the solar wind.

To further see the vortex-type variations in $\mathbf{B}$ and $\mathbf{V}$, we
define their fluctuation vectors from the averages ($\delta\mathbf{B}$ and
$\delta\mathbf{V}$) between 20:35 and 21:20 UT. Figs. 8b and 8c show
$\delta\mathbf{B}$ in the $R-T$ and $R-N$ planes, respectively. Around the
inbound crossing (3’), $\delta\mathbf{B}$ in the $R-N$ plane rotates smoothly
in a clockwise sense. This pattern is also observed at other inbound
crossings, although it is less pronounced. Figs. 8f and 8g show
$\delta\mathbf{V}$ in the $R-T$ and $R-N$ planes, respectively. A clockwise
pattern is also visible between (2) and (3’) in Fig. 8f. This pattern of
rotation seen in both $\delta\mathbf{B}$ and $\delta\mathbf{V}$ is consistent
with a vortical structure (i.e., a rolled-up KH vortex). The pattern is not as
obvious at other inbound crossings; this may indicate that SolO does not pass
centrally through the KH vortices in these cases.

###  3.4 Magnetic spectra

We now examine turbulence properties of the KH event. Fig. 9 shows the KHI
interval with KH sub-regions V1 to V6 highlighted with colors (middle)
together with time periods before and after the KH interval (see top). Fig. 9h
shows a spectrogram of magnetic spectrum. The magnetic spectrum shows
enhancement within the KH region compared to before or after the interval. The
enhancement is visually strongest in V2 compared with other vortices. This V2
is the same interval as between (2) and (3’) in Fig. 8 where we see a clear
clockwise rotation in the magnetic field and velocity field perturbations
consistent with a rolled-up KH vortex. Thus, the strong enhancement in
magnetic wave power provides evidence of enhanced activity in this vortex as
plausibly facilitated by the development of a nonlinear KH vortex. The
enhancement of the magnetic spectrum in V4 is also strong but less than for
V2.

Figure 9: Overview of the intervals for magnetic spectrum analysis. The KH
region is marked in the middle (see top) together with the intervals before
(top left) and after (top right). The KH vortices V1 to V6, marked between the
compressed current sheets, are shaded in colors. (a) Total magnetic field. (b)
Magnetic field in the RTN system. (c, d, e) Velocity field $R$, $T$, and $N$
components, respectively. (f) Ion number density. (g) Ion temperature. (h)
Magnetic spectrum.  Figure 10: Magnetic spectra of all regions marked in Fig.
9. The Kolmogorov power law $k^{-5/3}$ and the dissipation range scalings
$k^{-2.8}$ are plotted for reference as straight black dashed lines. The
vortex size is noted by an arrow (top left). The ion gyroradius scale
($\rho_{p}$) and the ion inertial length ($d_{p}$) are marked by vertical
black solid and dashed lines, respectively.

To quantitatively assess the magnetic field fluctuations in the full KHI
region and in each of the six sub-regions V1-V6 marked in Fig. 9, we computed
magnetic field spectra, which are shown in Fig. 10. Additionally, the KH-
stable regions immediately before and after the KHI event are shown for
reference. The vortex size is marked by an arrow (top left). The spectrum for
the full KH region is shown in black while the spectra of the intervals before
and after are shown in grey. As references, power law scalings of $k^{-5/3}$
(Kolmogorov, 1941) at MHD scales and $k^{-2.8}$ at ion scales (i.e.,
Alexandrova et al., 2009) are plotted as black dashed lines. The scale of a
thermal ion gyroradius ($\rho_{p}$) and an ion inertial length ($d_{p}$) based
on the average properties in the KH region are marked with vertical solid and
dashed black lines, respectively. The spectrum of the entire KH interval has
more power than the intervals before and after, which both have similar low-
intensity spectra. This suggests that the KHI is exciting additional
fluctuations. The magnetic spectrum of the KH region approximately follows
both power laws with a spectral breakpoint at $f\sim 0.2$ Hz. The spectrum
essentially follows the power law $k^{-2.8}$ for scales smaller than the ion
gyroradius. This indicates that the magnetic spectrum of the KHI interval is
consistent with a classic turbulence cascade down to the kinetic scales (e.g.,
Bruno et al., 2014, 2017). This result provides evidence of shear-driven
turbulence as driven by the local KHI.

Fig. 10 also shows magnetic spectra for individual vortices V1 - V6. Note that
the compressed current sheets are excluded for the analyses of these vortices
(this is why the colored regions are not exactly contiguous on Fig. 9). The
powers of the magnetic spectra of all vortices are weaker than that of the
entire KH region, indicating that the current sheets are key regions for
enhancing the power spectrum. V2 (blue) and V4 (green) appear to have higher
powers compared to other vortices and almost reach the power of the entire KH
region (black). The enhanced power in these vortices may be related to the
excitation of turbulent fluctuations through secondary instabilities and the
nonlinear evolution of the KHI. The difference in the power spectrum between
the different vortices may indicate that SolO was crossing different parts of
KH vortices while crossing through the shear layer from Side 1 to Side 2. It
is likely that SolO was passing through the center of a rolled-up vortex in
V2, for reasons noted earlier. The lower powers of V1, V5, and V6 may indicate
that SolO was skimming through the trough or crest parts of KH vortices. These
results are in agreement with the assumed spacecraft trajectory through the
KHI structure using the boundary layer analysis in Section 3.3.

##  4 Discussion

We have reported observations of the KHI within a shear layer embedded in the
slow solar wind close to an HCS using Solar Orbiter. The event is observed in
the inner heliosphere at a distance of $\sim 0.69$ AU. Despite several
theoretical postulations (e.g., Parker, 1963; Sturrock & Hartle, 1966; Miura &
Pritchett, 1982a; Korzhov et al., 1984; Neugebauer et al., 1986; Hollweg,
1987) and spacecraft missions in the inner heliosphere, direct evidences for
the KHI were not reported in past in-situ observations of the solar wind. We
discuss why this event may be favorable for a KHI detection as well as
implications of the KHI in the solar wind as follows.

###  4.1 KHI criterion in the solar wind

First, we consider solar wind conditions that are favorable for the KHI. Based
on the KHI onset condition (Eq. 1), the shear layer can more easily become
unstable to the KHI when $B$ is low and $n$ is high because it requires a
velocity jump across the shear layer greater than the local Alfvén speed
($\Delta V>V_{A}$). Near the Sun, $B$ and $V_{A}$ are typically large. This
inhibitory effect on the KHI criterion should be stronger near the Sun.
Nevertheless, there may be several situations where the local conditions allow
the KHI. For example, remote sensing by DeForest et al. (2018) show a
particularly strong shear values of a $\Delta V_{R}\sim 200$ km s-1 across
streamer structures in the young solar wind. Ruffolo et al. (2020) also
propose the KHI development near the Alfvén critical zone, at $R<0.17$ AU. The
event we analyze here was observed near the HCS with many coherent structures
and shears. Besides, this event is found in the slow wind, which is generally
dense, making the conditions to meet the KHI criterion easier.

Second, we consider the magnetic and velocity field configurations across
shear layers. The KHI is suppressed when $\mathbf{B}$ is strong in the
direction of $\Delta\mathbf{V}$ due to the stabilization by magnetic tension
in the direction of the shear flow. In this interplanetary medium, we
typically expect a shear interface along the Parker spiral direction (see Fig.
11) so $\mathbf{B}$ may usually be aligned with $\Delta\mathbf{V}$. However,
near the Sun, there may be velocity shear due to the solar-wind corotation
with the Sun. Several studies have shown that the KHI may occur in various
situations, i.e., at the edge of a CME (e.g., Foullon et al., 2013; Möstl et
al., 2013) and at the interfaces between CME and sheath and between sheath and
solar wind (e.g., Páez et al., 2017).

Third, Eq. 1 is derived by assuming an ideal MHD plasma with an infinitely
thin shear layer. In reality, non-ideal MHD effects such as the
compressibility can stabilize the KHI (e.g., Sen, 1964); for example, the KHI
only grows for a limited range of the velocity jump across a shear flow for a
1-D TD in homogeneous plasmas and magnetic fields (e.g., Talwar, 1964; Pu &
Kivelson, 1983). The solar wind is indeed compressible and thus we expect some
stabilizing effects. In addition, shear layers have finite thicknesses. A
finite thickness of the shear layer can also stabilize the KH mode for small
wavelength perturbations (i.e., for large wave number $k$). A combination of
the compressibility and the finite thickness can stabilize the KHI such that
only certain modes of $k\Delta L$, where $\Delta L$ is the shear layer
thickness, are KHI unstable (Miura & Pritchett, 1982b). Although these two
factors can impact the shear-layer stability, we do not expect their effects
to be large, nor to be specifically dependent on distance from the Sun.

To summarize, there are factors that can impact the KHI development in the
solar wind. The magnitude of $B$ and $V_{A}$ depend on distance from the Sun.
As $V_{A}$ is higher closer to the Sun, the KHI criterion should be more
difficult to satisfy, except where the shear $\Delta V$ is particularly
strong. Often $\mathbf{B}$ may be parallel to the velocity shear, tending to
inhibit the KHI, except in some circumstances, e.g., when there is a CME that
changes the local conditions. Compressibility of the solar wind and a finite
thickness of the shear region can help stabilize the KHI.

Since the observed conditions during our event are not particularly unusual
for dense solar wind near the HCS, KHI development should not be rare. We now
consider arguments related to KHI timescale and configuration as follows.

###  4.2 KHI timescale

A first fact to consider now is that when the KHI develops at a shear layer,
it quickly reaches the non-linear stage (i.e., the rolled-up stage). The
periodic features and vortical structures then get rapidly destroyed as the
plasmas from either side of the shear layer mix and vortices coalesce. At such
late stage, they would be indistinguishable from other solar wind types,
albeit likely associated with higher levels of fluctuations as we have
actually found in Section 3.4 for this event. The timescale for the decay of a
KH vortex is on the order of one to a few eddy turnover time. The eddy
turnover time of a KH vortex scales as a fraction of the KH wavelength
($\lambda_{KH}$) divided by the KH wave phase speed ($V_{ph}$). Assuming that
the KH vortex size is about $\lambda_{KH}/3$ to $\lambda_{KH}/2$ ($22,000$ to
$33,000$ km) for the fastest growing mode (i.e., Walker, 1981; Miura &
Pritchett, 1982a), with the $V_{ph}=152$ km s-1 (see Section 3.1), we obtain a
KH vortex turnover time of $2.4$ to $3.6$ minutes. Since the conditions of our
event are not unusual for dense solar wind near the HCS, this short timescale
for turnover time (before the vortex decay) may contribute to the rarity of
KHI detection.

Figure 11: (top) Schematic illustrations of scenarios for which the KHI
develops (a) near SolO position and (b) at $1$ AU, shown in the ecliptic plane
with the Parker spirals shown with orange solid lines (similar to Fig. 1). At
(a), we project the derived $X-Y$ plane onto the $R-T$ plan as described in
the text. The main flow is radially outward from the Sun as marked with
vectors $\@vec{r_{a}}$ and $\@vec{r_{b}}$. As the Parker spiral becomes more
bent at (b), the spacecraft relative spacecraft trajectory is more
perpendicular to the shear interface, making the likelihood of crossing
several vortices much lower than in the inner heliosphere at (a). (bottom, c)
A simplistic 3-D view of the KHI in (a) with the out-of-plane ($Z$) magnetic
field shown with arrows on Sides 1 and 2, with corresponding values in Fig.
6e.

It is also possible that KH waves were detected by past missions but their
signatures were not resolved. Several periodic oscillations in magnetic field
strength were observed by Burlaga (1968) using the Pioneer-6 spacecraft,
launched in 1965, at $\sim 0.8$ AU. One of the cases considered was found to
have sinusoidal $|\mathbf{B}|$ oscillations with a period of $\sim 5$ minutes,
embedded in a velocity shear layer. Although no other fluctuations were seen
in the data, it was suggested that these waves were generated by a KHI.

###  4.3 KHI configuration

When TDs form in the solar wind, they are typically at the boundary between
solar wind flux tubes that move with relative velocity. Since the solar wind
flux tubes bend following Parker spiral arcs to first order, we expect the
normal of shear interfaces to be generally perpendicular to the Parker’s
spiral. Fig. 11 shows a schematic illustration of expected KHI configurations
with respect to the solar wind Parker spiral in the ecliptic plane, produced
using a constant velocity of $300$ km s-1 close to the observed speed. We
illustrate two scenarios of the KHI configuration (a) in the interplanetary
medium between $0.3$ and $0.7$ AU and (b) near $1$ AU. We describe a scenario
for which the KHI develops relative to the Parker spiral based on our data as
follows.

For our event at $\sim 0.7$ AU depicted at (a), we found that the velocity
jumps for a displacement is in the direction $Y=[0.53,0.79,-0.32]_{RTN}$ (the
maximum variance direction of $\mathbf{V}$). This direction is indeed
perpendicular to the Parker spiral, which is rather oriented along what we
call the $X$ direction, as shown at (a) (see frame definition in upper right
part of the Fig. 11). The KH wave amplitude spreads in the Y direction as it
is perpendicular to the shear interface. The wave propagation direction is
$X=[0.84,-0.53,0.07]_{RTN}$ (the intermediate variance direction of
$\mathbf{V}$). We expect that the KHI configuration in the solar wind will
often be similar to this geometry. As explained next, such a KHI configuration
relative to a more slowly moving spacecraft in the inner heliosphere impacts
the likelihood of in-situ observations of the KHI.

We now consider two scenarios of KHI development (a) near SolO position at
$\sim 0.7$ AU and (b) near 1 AU in Fig. 11. As the main solar wind flow is in
the radial direction, a static spacecraft at (a) would sample several vortices
within the shear layer only if it has an appropriate relative trajectory
through the layer. Note that SolO was relatively static (with a speed of $11$
km s-1) compared to the KH wave phase speed ($152$ km s-1). At a further
distance from the Sun, i.e., near $1$ AU as shown in (b), the Parker’s spiral
becomes more bent. The shear interface thus becomes more tilted relative to
the outward solar wind flow as compared to the geometry observed at locations
in the inner heliosphere (as show in Fig. 11a). Consequently, a spacecraft
located in the inner heliosphere has a relative trajectory that makes it more
likely to sample several vortices than a spacecraft that is farther out in the
heliosphere, and which will make a more perpendicular cut across the shear
layer, as depicted in Fig. 11b. This may be a prime explanation as to why
despite the many spacecraft near $1$ AU, there were no reports of in-situ KHI
detection.

Note that the above considerations are based on a simplistic 2-D view. In
reality, 3-D effects such as out-of-ecliptic magnetic and velocity fields can
significantly impact the KHI configuration and development with respect to the
Parker spiral. For example, we show magnetic fields perpendicular to the shear
plane in Fig. 11c and these are in fact required to avoid the alignment of the
velocity and magnetic shear, which otherwise has a stabilizing effect on the
KHI. We acknowledge that our considerations are to first order only; future
work should address this in more detail.

###  4.4 Implications of the KHI in the solar wind

The KHI is expected to play important roles, such as allowing for plasma
mixing, generating turbulence, or producing Alfvénic fluctuations in the solar
wind as mediated by KH vortex dynamics. During the present event, SolO
observed an ion jet consistent with magnetic reconnection (see Section 2.4).
An interesting question is whether this reconnection is produced due to
dynamics of a KH vortex. For vortex-induced reconnection (VIR), we expect
reconnection to be produced at a thin current sheet in between two vortices.
At the Earth’s magnetopause, VIR jets were found to orient themselves along KH
trailing edges (e.g., Eriksson et al., 2016) as the vortex evolves and further
enhances the magnetic shear. In our case, we found that the jet is in the out-
of-ecliptic direction while the KH trailing edges (see Table. 3) and the shear
layer are in the $R,T$ directions. Thus, it is unclear whether the observed
jet is a VIR. We think that the $N-$ directed jet is rather a consequence of
the inclination of the local current sheet. Nevertheless, the KH vortex may
further increase the magnetic shear at the KH edge and make the current sheet
thin enough to trigger magnetic reconnection.

We now discuss the magnetic and velocity field fluctuations. We found that the
KHI enhances fluctuations compared to outside the interval and the magnetic
spectrum of the KHI region approximately follows the power law scalings of
$k^{-5/3}$ and $k^{-2.8}$ at inertial and kinetic scales, respectively
(Section 3.4). These enhanced fluctuations are consistent with turbulence
generation by the KHI at the magnetopause (e.g., Stawarz et al., 2016;
Nakamura et al., 2017). Therefore, the magnetic spectrum is consistent with a
classical turbulence cascade down to the kinetic scales. These observations
are consistent with an enhancement of turbulence in the solar wind as driven
by the local KHI (e.g., Goldstein et al., 1989). We also note that current
sheets are key structures that contribute to the power spectrum, as power
spectra of only vortex regions are lower than the overall spectrum. In
addition, several of the vortices have enhanced power within them, which may
be due to secondary instabilities, perhaps supporting the idea that the KHI
helps to drive some fraction of the turbulent fluctuations in the solar wind.

One important implication of KHI in the solar wind is that it can contribute
to the evolution of the magnetic and velocity fluctuations. Near the Sun, the
KHI is believed to be a mechanism that leads to shear-driven turbulence at the
Alfvén critical zone where $V=V_{A}$ and in the vicinity of the $\beta=1$
surface (DeForest et al., 2016; Chhiber et al., 2018), leading to more
isotropic solar-wind streams. Furthermore, the dynamical evolution invoked by
shear-driven instabilities such as the KHI is found to be able to account for
features observed by PSP including magnetic “switchbacks” near perihelia
(Ruffolo et al., 2020). This topic needs to be investigated further but it is
beyond the scope of the present study.

##  5 Conclusions

We report observations of the KHI with SolO on July 23, 2020 at $0.69$ AU,
during the cruise phase. The KH waves are observed within the velocity shear
layer with periodic fluctuations in several parameters in the slow solar wind
near an HCS. Several KH waveforms are observed with a period of $\sim 7$
minutes but only a few vortices are clearly noticed. One of them is found to
have magnetic and velocity field perturbations consistent with a rolled-up KH
vortex in the non-linear stage. We test the observed conditions on either side
of the shear layer with linear theory and find that the shear layer is indeed
KHI-unstable. The maximum variance direction of the shear flow, associated to
the KH wave amplitude direction, is found in the direction approximately
perpendicular to the Parker spiral. The intermediate variance direction of the
shear flow, associated with the net flow and thus the direction of the KH wave
propagation, is along the Parker spiral. Using linear theory, the wave phase
speed is estimated to be $152$ km s-1. The KH wavelength is approximately
$66,400$ km or $0.1$ solar radii. We also comfirm the local KHI development by
exploiting a 2-D MHD simulation with the empirical values.

Additionally, we report the observation of an ion jet consistent with magnetic
reconnection at one of the outbound (trailing) edges, likely as a result of
current sheet compression in between two KH vortices. The ion jet has $\Delta
V=11$ km s-1 along the maximum variance direction, consistent with magnetic
reconnection with a sub-Alfvénic jet. Nevertheless, we found other signatures
consistent with magnetic reconnection, namely a drop in magnetic field
strength, an ion number density enhancement (Gosling et al., 2005), and plasma
heating (Phan et al., 2014). It is unclear whether this jet is produced due to
KH vortex-induced reconnection (e.g., Nakamura et al., 2006) or the local
inclination of the magnetic field, as the jet direction does not correspond to
the KH trailing edge orientation (Eriksson et al., 2016).

We also report the enhancement of the magnetic and velocity field fluctuations
within the KHI interval compared to intervals before and after. The power of
the magnetic spectrum of the entire KHI interval approximately follows the
power law scalings of $k^{-5.3}$ and $k^{-2.8}$ in the inertial and kinetic
ranges, respectively, consistent with the turbulent cascade in the solar wind.
This provides evidence for the local enhancement of turbulence as driven by
the KHI. Moreover, we find that current sheets within the KHI interval are key
structures that enhance the power, as the magnetic spectra of individual KH
vortices (excluding compressed current sheet intervals) generally have less
power.

As our reported event here is an unambiguous in-situ observation of the KHI in
the solar wind, we discuss possible reasons why the KHI was not reported in
past in-situ observations. First, the KHI onset criterion requires a velocity
jump across the shear layer that is larger than the local Alfvén speed
($\Delta V>V_{A}$) and weak magnetic field in the direction of the shear flow
(i.e., low $\mathbf{B}\cdot\mathbf{V}$). Second, the KHI is estimated to
quickly reaches the nonlinear stage where KH vortices roll up and merge. This
timescale should be on the order of a few eddy turnover. The observed
conditions are typical in the solar wind, and we estimate the timescale of the
KHI to be on the order of minutes. In other words, when KHI develops in the
solar wind, it evolves rapidly and is thus rather ephemeral. Third, the
configuration in which the KHI develops may not be optimal for detection by a
relatively slow-moving spacecraft near $1$ AU owing to the geometry of the
Parker spiral. The KHI develops at discontinuities formed in between solar
wind flux tubes that move with a relative velocity. In this picture, the KH
wave amplitude should spread in a direction perpendicular to the Parker
spiral, albeit often with significant north-south components as in our case.
Owing to its more skimming trajectory, a spacecraft in the inner heliosphere
such as SolO may more easily pick up several waveforms in the shear layer.
Further from the Sun, in contrast, the Parker spiral becomes more bent. A
spacecraft near $1$ AU is less likely to sample several vortices along its
trajectory which has a larger angle to the discontinuity, making it difficult
to recognize KH waves. We think these are main factors that contribute to the
rarity of in-situ KHI detection in the solar wind.

This event provides evidence for the existence of the KHI in the solar wind.
It sheds new light on solar wind shear processes in the interplanetary medium
with direct applications to shear-driven turbulence mediated by the KHI,
likely contributing to the solar wind fluctuations observed at $1$ AU (e.g.,
Ruffolo et al., 2020). As the Alfvén speed decreases away from the Sun, the KH
growth rate becomes higher (e.g., Neugebauer et al., 1986) and thus the KHI
may be more common. Due to the short timescale of the linear KHI, there may be
more chances to detect nonlinear KHI or its remnants. Recently, techniques for
detecting kinetic features of the KHI during the nonlinear and turbulent stage
of the KHI were proposed (Settino et al., 2021). Further studies would be
needed to study secondary processes induced by the KHI such as vortex-induced
reconnection and other kinetic mechanisms, as the KHI is rich with magnetic
and plasma structures as is well known for the case of the magnetopause.

###### Acknowledgements.

Work at IRAP was supported by the Centre National de la Recherche Scientifique
(CNRS, France), the Centre National d’Etudes Spatiales (CNES, France), and the
Université Paul Sabatier (UPS). Solar Orbiter data are publicly available at
http://soar.esac.esa.int/soar/. We acknowledge science teams of the
Connectivity Tool (http://connect-tool.irap.omp.eu/) and 3DView
(http://3dview.irap.omp.eu/) at IRAP and CNES. Y. Y. is supported by grant No.
11902138 from the National Natural Science Foundation of China. The computing
resources were provided by the Center for Computational Science and
Engineering of Southern University of Science and Technology. D.R. is
supported by grant RTA6280002 from Thailand Science Research and Innovation.
J.E.S is supported by the Royal Society University Research Fellowship
URF$\backslash$R1$\backslash$201286\. C.J.O. is funded under STFC grant number
ST/5000240/1. Solar Orbiter Solar Wind Analyser (SWA) data are derived from
scientific sensors which have been designed and created, and are operated
under funding provided in numerous contracts from the UK Space Agency (UKSA),
the UK Science and Technology Facilities Council (STFC), the Agenzia Spaziale
Italiana (ASI), the CNES, the CNRS, the Czech contribution to the ESA PRODEX
programme and NASA. Solar Orbiter SWA work at UCL/MSSL is currently funded
under STFC grants ST/T001356/1 and ST/S000240/1.

## References

  * Alexandrova et al. (2009) Alexandrova, O., Saur, J., Lacombe, C., et al. 2009, Phys. Rev. Lett., 103, 165003
  * Arge et al. (2010) Arge, C. N., Henney, C. J., Koller, J., et al. 2010, in American Institute of Physics Conference Series, Vol. 1216, Twelfth International Solar Wind Conference, ed. M. Maksimovic, K. Issautier, N. Meyer-Vernet, M. Moncuquet, & F. Pantellini, 343–346
  * Axford & McKenzie (1992) Axford, W. I. & McKenzie, J. F. 1992, in Solar Wind Seven Colloquium, ed. E. Marsch & R. Schwenn, 1–5
  * Borovsky (2008) Borovsky, J. E. 2008, Journal of Geophysical Research (Space Physics), 113, A08110
  * Bruno & Carbone (2005) Bruno, R. & Carbone, V. 2005, Living Reviews in Solar Physics, 2, 4
  * Bruno et al. (2017) Bruno, R., Telloni, D., DeIure, D., & Pietropaolo, E. 2017, MNRAS, 472, 1052
  * Bruno et al. (2014) Bruno, R., Trenchi, L., & Telloni, D. 2014, ApJ, 793, L15
  * Burlaga (1968) Burlaga, L. F. 1968, Sol. Phys., 4, 67
  * Burlaga (1972) Burlaga, L. F. 1972, Microstructure of the Interplanetary Medium, ed. C. P. Sonett, P. J. Coleman, & J. M. Wilcox, Vol. 308, 309
  * Burlaga et al. (1977) Burlaga, L. F., Lemaire, J. F., & Turner, J. M. 1977, J. Geophys. Res., 82, 3191
  * Chandrasekhar (1961) Chandrasekhar, S. 1961, Hydrodynamic and hydromagnetic stability (Clarendon Press, Oxford)
  * Chen & Kivelson (1993) Chen, S. H. & Kivelson, M. G. 1993, Geochim. Res. Lett., 20, 2699
  * Chhiber et al. (2018) Chhiber, R., Usmanov, A. V., DeForest, C. E., et al. 2018, The Astrophysical Journal, 856, L39
  * De Pontieu et al. (2014) De Pontieu, B., Title, A. M., Lemen, J. R., et al. 2014, Sol. Phys., 289, 2733
  * DeForest et al. (2018) DeForest, C. E., Howard, R. A., Velli, M., Viall, N., & Vourlidas, A. 2018, ApJ, 862, 18
  * DeForest et al. (2016) DeForest, C. E., Matthaeus, W. H., Viall, N. M., & Cranmer, S. R. 2016, The Astrophysical Journal, 828, 66
  * Eriksson et al. (2016) Eriksson, S., Lavraud, B., Wilder, F. D., et al. 2016, Geophysical Research Letters, 43, 5606
  * Fairfield et al. (2000) Fairfield, D. H., Otto, A., Mukai, T., et al. 2000, J. Geophys. Res., 105, 21,159
  * Fairfield et al. (2000) Fairfield, D. H., Otto, A., Mukai, T., et al. 2000, Journal of Geophysical Research: Space Physics, 105, 21159
  * Fargette et al. (2021) Fargette, N., Lavraud, B., Rouillard, A. P., et al. 2021, forthcoming
  * Foullon et al. (2008) Foullon, C., Farrugia, C., Fazakerley, A., et al. 2008, Journal of Geophysical Research (Space Physics), 113, A11203
  * Foullon et al. (2011) Foullon, C., Verwichte, E., Nakariakov, V. M., Nykyri, K., & Farrugia, C. J. 2011, Astrophysical Journal Letters, 729, 2
  * Foullon et al. (2013) Foullon, C., Verwichte, E., Nykyri, K., Aschwanden, M. J., & Hannah, I. G. 2013, ApJ, 767, 170
  * Fox et al. (2016) Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, Space Sci. Rev., 204, 7
  * Génot et al. (2018) Génot, V., Beigbeder, L., Popescu, D., et al. 2018, Planetary and Space Science, 150, 111, enabling Open and Interoperable Access to Planetary Science and Heliophysics Databases and Tools
  * Goldstein et al. (1989) Goldstein, M. L., Roberts, D. A., & Matthaeus, W. H. 1989, Washington DC American Geophysical Union Geophysical Monograph Series, 54, 113
  * Gosling & Phan (2013) Gosling, J. T. & Phan, T. D. 2013, Astrophysical Journal Letters, 763, 1987
  * Gosling et al. (2005) Gosling, J. T., Skoug, R. M., McComas, D. J., & Smith, C. W. 2005, Journal of Geophysical Research: Space Physics, 110
  * Haggerty et al. (2018) Haggerty, C. C., Shay, M. A., Chasapis, A., et al. 2018, Physics of Plasmas, 25, 102120
  * Hasegawa (1975) Hasegawa, A. 1975, Plasma instabilities and nonlinear effects
  * Hasegawa et al. (2004) Hasegawa, H., Fujimoto, M., Phan, T.-D., et al. 2004, Nature, 430, 755
  * Hickmann et al. (2015) Hickmann, K. S., Godinez, H. C., Henney, C. J., & Arge, C. N. 2015, Sol. Phys., 290, 1105
  * Hillier & Polito (2018) Hillier, A. & Polito, V. 2018, ApJ, 864, L10
  * Hollweg (1982) Hollweg, J. V. 1982, Journal of Geophysical Research, 87, 8065
  * Hollweg (1987) Hollweg, J. V. 1987, ApJ, 317, 918
  * Hones et al. (1981) Hones, E. W., Birn, J., Bame, S. J., et al. 1981, J. Geophys. Res., 86, 814
  * Horbury et al. (2020) Horbury, T. S., O’Brien, H., Carrasco Blazquez, I., et al. 2020, Astronomy and Astrophysics, 642, 1
  * Hudson (1970) Hudson, P. D. 1970, Planet. Space Sci., 18, 1611
  * Kacem et al. (2018) Kacem, I., Jacquey, C., Génot, V., et al. 2018, Journal of Geophysical Research: Space Physics, 1779
  * Karimabadi et al. (2013) Karimabadi, H., Roytershteyn, V., Wan, M., et al. 2013, Physics of Plasmas, 20, 012303
  * Kieokaew et al. (2020) Kieokaew, R., Lavraud, B., Foullon, C., et al. 2020, Journal of Geophysical Research: Space Physics, 125, 1
  * Knetter et al. (2004) Knetter, T., Neubauer, F. M., Horbury, T., & Balogh, A. 2004, Journal of Geophysical Research (Space Physics), 109, A06102
  * Kolmogorov (1941) Kolmogorov, A. 1941, Akademiia Nauk SSSR Doklady, 30, 301
  * Korzhov et al. (1984) Korzhov, N. P., Mishin, V. V., & Tomozov, V. M. 1984, Planet. Space Sci., 32, 1169
  * Landau & Lifshitz (1960) Landau, L. D. & Lifshitz, E. M. 1960, Electrodynamics of continuous media
  * Landau & Lifshitz (1987) Landau, L. D. & Lifshitz, E. M. 1987, Fluid Mechanics
  * Lepping & Behannon (1986) Lepping, R. P. & Behannon, K. W. 1986, J. Geophys. Res., 91, 8725
  * Louarn (2004) Louarn, P. 2004, Geophysical Research Letters, 31, L19805
  * Matthaeus & Goldstein (1982) Matthaeus, W. H. & Goldstein, M. L. 1982, J. Geophys. Res., 87, 6011
  * Miura & Pritchett (1982a) Miura, A. & Pritchett, P. L. 1982a, Journal of Geophysical Research, 87, 7431
  * Miura & Pritchett (1982b) Miura, A. & Pritchett, P. L. 1982b, Journal of Geophysical Research, 87, 7431
  * Möstl et al. (2013) Möstl, U. V., Temmer, M., & Veronig, A. M. 2013, The Astrophysical Journal, 766, L12
  * Müller et al. (2020) Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, Astronomy and Astrophysics, 642, 1
  * Nakamura et al. (2006) Nakamura, T. K. M., Fujimoto, M., & Otto, A. 2006, Geochim. Res. Lett., 33, L14106
  * Nakamura et al. (2017) Nakamura, T. K. M., Hasegawa, H., Daughton, W., et al. 2017, Nature Communications, 8, 1582
  * Neugebauer (1985) Neugebauer, M. 1985, Journal of Geophysical Research, 90, 6627
  * Neugebauer (2006) Neugebauer, M. 2006, Journal of Geophysical Research (Space Physics), 111, A04103
  * Neugebauer et al. (1986) Neugebauer, M., Alexander, C. J., Schwenn, R., & Richter, A. K. 1986, J. Geophys. Res., 91, 13694
  * Neugebauer & Giacalone (2010) Neugebauer, M. & Giacalone, J. 2010, AIP Conference Proceedings, 1216, 194
  * Nykyri & Otto (2001) Nykyri, K. & Otto, A. 2001, Geochim. Res. Lett., 28, 3565
  * Ofman & Thompson (2011) Ofman, L. & Thompson, B. J. 2011, ApJ, 734, L11
  * Øieroset et al. (2019) Øieroset, M., Phan, T. D., Drake, J. F., et al. 2019, Geochim. Res. Lett., 46, 1937
  * Owen et al. (2020) Owen, C. J., Bruno, R., Livi, S., et al. 2020, Astronomy and Astrophysics, 642
  * Páez et al. (2017) Páez, A., Jatenco-Pereira, V., Falceta-Gonçalves, D., & Opher, M. 2017, ApJ, 851, 112
  * Parker (1958) Parker, E. N. 1958, ApJ, 128, 664
  * Parker (1963) Parker, E. N. 1963, Interplanetary dynamical processes.
  * Paschmann et al. (1986) Paschmann, G., Papamastorakis, I., Baumjohann, W., et al. 1986, J. Geophys. Res., 91, 11099
  * Phan et al. (2014) Phan, T. D., Drake, J. F., Shay, M. A., et al. 2014, Geochim. Res. Lett., 41, 7002
  * Plaschke et al. (2016) Plaschke, F., Kahr, N., Fischer, D., et al. 2016, Geophysical Research Letters, 43, 7373
  * Pu & Kivelson (1983) Pu, Z.-Y. & Kivelson, M. G. 1983, Journal of Geophysical Research, 88, 841
  * Roberts et al. (1992) Roberts, D. A., Goldstein, M. L., Matthaeus, W. H., & Ghosh, S. 1992, J. Geophys. Res., 97, 17115
  * Roudier & Muller (1986) Roudier, T. & Muller, R. 1986, Sol. Phys., 107, 11
  * Rouillard et al. (2020) Rouillard, A. P., Pinto, R. F., Vourlidas, A., et al. 2020, Astronomy & Astrophysics, 642, A2
  * Ruffolo et al. (2020) Ruffolo, D., Matthaeus, W. H., Chhiber, R., et al. 2020, The Astrophysical Journal, 902, 94
  * Sen (1963) Sen, A. K. 1963, Physics of Fluids (New York), 6
  * Sen (1964) Sen, A. K. 1964, Physics of Fluids, 7, 1293
  * Settino et al. (2021) Settino, A., Perrone, D., Khotyaintsev, Y. V., Graham, D. B., & Valentini, F. 2021, arXiv e-prints, arXiv:2102.04117
  * Siscoe & Suey (1972) Siscoe, G. L. & Suey, R. W. 1972, J. Geophys. Res., 77, 1321
  * Sonnerup & Cahill (1967) Sonnerup, B. U. Ö. & Cahill, L. J. 1967, Journal of Geophysical Research, 72, 171
  * Stawarz et al. (2016) Stawarz, J. E., Eriksson, S., Wilder, F. D., et al. 2016, Journal of Geophysical Research (Space Physics), 121, 11,021
  * Sturrock & Hartle (1966) Sturrock, P. A. & Hartle, R. E. 1966, Phys. Rev. Lett., 16, 628
  * Talwar (1964) Talwar, S. P. 1964, Journal of Geophysical Research, 69, 2707
  * Tsurutani & Smith (1979) Tsurutani, B. T. & Smith, E. J. 1979, J. Geophys. Res., 84, 2773
  * Walker (1981) Walker, A. D. M. 1981, Planetary and Space Science, 29, 1119
  * Yang et al. (2017) Yang, Y., Matthaeus, W. H., Shi, Y., Wan, M., & Chen, S. 2017, Physics of Fluids, 29, 035105
  * Yang et al. (2016a) Yang, Y., Shi, Y., Wan, M., Matthaeus, W. H., & Chen, S. 2016a, Phys. Rev. E, 93, 061102
  * Yang et al. (2016b) Yang, Y., Wan, M., Shi, Y., Yang, K., & Chen, S. 2016b, Journal of Computational Physics, 306, 73
  * Zaqarashvili et al. (2014) Zaqarashvili, T. V., Vörös, Z., & Zhelyazkov, I. 2014, Astronomy & Astrophysics, 561, A62