# Mirror mode storms observed by Solar Orbiter

###### Abstract

Mirror modes are ubiquitous in space plasma and grow from pressure anisotropy.
Together with other instabilities, they play a fundamental role in
constraining the free energy contained in the plasma. This study focuses on
mirror modes observed in the solar wind by Solar Orbiter for heliocentric
distances between 0.5 and 1 AU. Typically, mirror modes have timescales from
several to tens of seconds and are considered quasi-MHD structures. In the
solar wind, they also generally appear as isolated structures. However, in
certain conditions, prolonged and bursty trains of higher frequency mirror
modes are measured, which have been labeled previously as mirror mode storms.
At present, only a handful of existing studies have focused on mirror mode
storms, meaning that many open questions remain. In this study, Solar Orbiter
has been used to investigate several key aspects of mirror mode storms: their
dependence on heliocentric distance, association with local plasma properties,
temporal/spatial scale, amplitude, and connections with larger-scale solar
wind transients. The main results are that mirror mode storms often approach
local ion scales and can no longer be treated as quasi-MHD, thus breaking the
commonly used long-wavelength assumption. They are typically observed close to
current sheets and downstream of interplanetary shocks. The events were
observed during slow solar wind speeds and there was a tendency for higher
occurrence closer to the Sun. The occurrence is low, so they do not play a
fundamental role in regulating ambient solar wind but may play a larger role
inside transients.

JGR: Space Physics

Swedish Institute of Space Physics, Uppsala, Sweden Departamento de Ciencias
Espaciales, Instituto de Geofísica, Universidad Nacional Autónoma de México,
Ciudad Universitaria, Ciudad de México, Mexico Division of Space and Plasma
Physics, School of Electrical Engineering and Computer Science, KTH Royal
Institute of Technology, Stockholm, Sweden IRAP UPS CNRS, Toulouse, France
Mullard Space Science Laboratory, University College London, UK

Andrew P<EMAIL_ADDRESS>
Mirror mode storms predominantly occurred during slow solar wind

Heliospheric plasma sheet crossings were effective at setting up MM unstable
conditions

Spatial scales of mirror mode structures approached and were smaller than ion-
scales

## Plain Language Summary

Plasma strives to be in equilibrium with little to no free energy. However,
this is often not the case, especially in close proximity to complex
structures such as shock waves and interplanetary coronal mass ejections. The
latter is an eruption of plasma from the Sun that propagates outward into the
solar system. In the presence of some free energy, instabilities will arise to
remove it, one example is the mirror mode instability. Instabilities such as
these are of extremely high importance to plasma physics as they act as a
feedback mechanism to the plasma. Nevertheless, there are many open questions
regarding the mirror mode instability, especially when their properties are
different from the most common scenarios. Typically, mirror modes in the solar
wind appear as dips that are isolated structures. However, this paper
investigates mirror modes when they appear as sudden bursts of magnetic peaks
and dips and typically have smaller temporal scales. These kinds of mirror
modes have been called mirror mode storms. This study aims to address at what
distances from the Sun they arise, what types of solar wind structures they
are associated with, quantify their physical properties, and understand what
local plasma conditions are important.

##  1 Introduction

Mirror modes (MMs) are fundamental plasma phenomena that are universal across
a diverse set of space plasma environments [Tsurutani . (19821), Neubauer .
(1993), Joy . (20061), Génot (2008), Génot, Budnik, Jacquey . (2009), Soucek .
(2008), Balikhin . (2009), Soucek . (2015)]. Analogous to other plasma
instabilities, MMs are essential to understanding both the global and local
kinetic behavior of plasma as they are a natural feedback mechanism that
drives the plasma towards marginal stability. Through theory, MMs were first
predicted [Chandrasekhar . (1958), Hasegawa (1969)] until the observational
evidence arrived soon after [Kaufmann Horng (1971)]. What ensued was a
multitude of MM observations [Tsurutani . (19822), Neubauer . (1993), Sahraoui
. (2004), Joy . (20062), Volwerk . (2008), Génot, Budnik, Hellinger . (2009),
Soucek . (2008), Balikhin . (2009), Soucek . (2015), Osmane . (2015), Dimmock
. (2015), Volwerk . (2016), Ala-Lahti . (2018), Karlsson . (2021)] in regions
such as the solar wind, planetary magnetosheaths, Interplanetary Coronal Mass
Ejections (ICMEs), and around comets. Furthermore, MMs have also been studied
in the context of local and global numerical simulations [Hoilijoki . (2016),
Ahmadi . (2017)].

Although they are commonly treated from a quasi magnetohydrodynamic (MHD)
perspective, they are kinetic structures by nature. They have zero phase
velocity in the plasma rest frame, appear as sharp peaks or dips in the
magnetic field that are anti-correlated with density, and are linearly
polarized. MMs grow when there is sufficient free energy from the ion pressure
anisotropy ($Pi_{A}=P_{\perp i}/P_{\parallel i}>1$) and the plasma $\beta_{i}$
is sufficiently high. The perpendicular pressure constructs local magnetic
mirror configurations analogous to a magnetic bottle. Particles undergo mirror
motion between the so-called bottlenecks, which results in the anti-
correlation between the magnetic field and particle density when traversed by
a spacecraft. hasegawa1969 derived a convenient threshold to describe mirror
unstable plasma ($T_{\perp i}/T_{\parallel i}>1+1/\beta_{\perp i}$) based on a
bi-Maxwellian cold electron fluid approximation, and thus is valid when
$T_{\perp e}\sim T_{\parallel e}\ll T_{\parallel i}$. This threshold is based
on a kinetic theory at the long-wavelength limit (see eqs 2-4 in
hasegawa1969). Thus, although a kinetic approach is used, its use is
applicable when spatial wavelengths are much greater than the ion gyroradius
(i.e. $L_{mm}\gg\rho_{p}$), where $L_{mm}$ is the spatial scale of one MM
structure and $\rho_{p}$ is the proton gyroradius. Thus, MMs are often
referred to as quasi-MHD. The MM threshold establishes that the local
$\beta_{i}=2\mu_{0}nk_{B}T_{i}/B^{2}$ and $T_{\perp i}/T_{\parallel i}$ are
necessary to quantify the degree of stability of plasma to MMs. Moreover, for
$T_{\perp i}>T_{\parallel i}$ conditions, the MM instability competes with the
Alfvén ion cyclotron (AIC) instability that dominates at lower values of
plasma $\beta_{i}$ [Gary (1992)]. For completeness, it is also worth
mentioning the firehose instability, which grows when $T_{\parallel
i}>T_{\perp i}$, implying it is mutually exclusive with the MM and AIC
instabilities. Nevertheless, the content of this paper will focus explicitly
on MMs.

MMs are frequently observed in planetary magnetosheaths as the shocked solar
wind plasma provides favourable conditions ($\beta_{i}>1$, $T_{\perp
i}>T_{\parallel i}$) for MM growth [Volwerk . (2008), Soucek . (2008), Génot,
Budnik, Hellinger . (2009), Dimmock . (2015)]. The readily available high-
cadence measurements from missions such as Cluster, THEMIS, and MMS have been
used to characterize and study MMs in the Earth’s magnetosheath [Génot,
Budnik, Hellinger . (2009), Soucek . (2015), Dimmock . (2015)]. In general,
MMs in the Earth’s magnetosheath appear in the form of continuous trains of
peaks or dips [Soucek . (2008), Génot, Budnik, Hellinger . (2009), Dimmock .
(2015)] with average temporal periods $\sim 13$ s [Soucek . (2008)].
Considering the average flow speeds in the magnetosheath [Dimmock Nykyri
(2013)], then the spatial extent of these structures approaches fluid scales.
The MM “peakness” is typically identified based on the skewness of the
probability distribution of the magnetic field; where negative values suggest
the existence of dips and vice versa in the case of peaks. The occurrence of
peaks or dips is understood to be related to the degree of instability of the
plasma [Soucek . (2008), Génot, Budnik, Hellinger . (2009), Dimmock . (2015)].
Peaks are associated with MM unstable plasma whereas dips appear around or
below marginal stability. Together, the MM and AIC instabilities, both a
function of $\beta_{i}$, put an upper bound on the ion temperature anisotropy
that produces a clear anti-correlation between $T_{\perp i}/T_{\parallel i}$
and $\beta_{i}$ [Gary Lee (1994), Fuselier . (1994)]. Regardless, MMs can also
be excited by electron anisotropies. yao2019 presented a case study of the
electron MM (scales below proton gyroradius) in the Earth’s magnetosheath
corresponding to the condition $T_{\perp e}/T_{\parallel e}>1+1/\beta_{\perp
e}$. In this event, there was no ion temperature anisotropy but a clear
electron temperature anisotropy was present that was in anti-correlation with
the electron pressure. These structures appeared as trains of dips. Kinetic
scale magnetic dips have also been reported in the magnetosheath as more
isolated structures [Yao . (2019)]

MMs are also observed inside the sheath regions of interplanetary coronal mass
ejections (ICMEs), occurring in around 70% of the cases at 1 AU behind the
leading IP shock [Ala-Lahti . (2018)]. Despite this high occurrence rate,
studies that have focused explicitly on MMs inside ICME sheaths are uncommon
(e.g. liu2006,ala-lahti2018). The recent statistical study by ala-lahti2018
estimated the occurrence and physical properties of MMs measured inside 96
ICME sheaths at 1 AU using the Wind spacecraft. The MMs displayed an average
temporal period between 11.6 s-13.7 s depending on if they were part of a MM
train or isolated structures; the general temporal width varied from around 6
s to over 40 s. Hence, the spatial scales should be on the order of thousands
of km, much larger than the hundreds of km expected from the ion gyroradius.
Thus, the long-wavelength approximation should be valid. There was also large
variability in the wave amplitudes (1 nT-14 nT). According to the statistical
distribution from the events considered, the structures had amplitudes of
approximately 3 nT and 96% of the time were dips. Although MMs inside ICME
sheaths can appear as trains, they are not as tightly packed and successive as
those seen in the Earth’s magnetosheath.

Structures in the solar wind called magnetic holes have been reported for
decades [Turner . (1977), Winterhalter . (1994), Xiao . (2010), Volwerk .
(2020), Karlsson . (2021)], and resemble MM structures. These generally differ
from the MM train-like structures seen in magnetosheaths since they are
especially more isolated and manifest at larger temporal and spatial scales.
Their scale sizes range from several seconds to minutes (see karlsson2021 and
references therein). At 1 AU, the occurrence rates are between 2.4 - 3.4 holes
per day, for those that are linear with no field rotations before and after
the hole [Pokhotelov . (2002)]. There are striking resemblances between linear
magnetic holes and MMs, such as the pressure balance, linear polarization, and
tendency to occur in regions unstable to the MM instability criteria
[Tsurutani . (2011)]. However, it has also been shown that magnetic holes can
occur in mirror stable plasma [Stevens Kasper (2007)], so open questions still
remain. Innately, it has been proposed that magnetic holes could be remnants
of the MM instability in localized regions [Winterhalter . (1994)].
Nevertheless, magnetic holes are frequently observed in the solar wind across
varied heliocentric distances. Yet, in some cases, MMs materialize in the
solar wind with properties that are significantly different from magnetic
holes.

MM structures also occur in the solar wind in the form of prolonged trains,
which are remarkably similar to those reported in planetary magnetosheath
regions [Russell . (2009), Enriquéz-Rivera . (2013)]. They maintain low
amplitudes ($\sim 1$ nT) and manifest as peaks or dips. These events have been
designated mirror mode storms (MM storms) [Russell . (2009)] but the
literature is scarce; to our knowledge, just a few studies have been published
(e.g. russell2009,enriquez_rivera2013) to date. Using STEREO measurements,
enriquez_rivera2013 reported on MM storms by characterizing 15 events and then
conducting a kinetic dispersion analysis. Most of their events were observed
for stream interaction regions (SIRs) and only one was associated with the
ambient solar wind. Interestingly, the authors note that alpha particle
density also increased for most of their MM storm events. Nevertheless, in
regions of high $\beta_{i}$, the ion temperature anisotropy needed for the
plasma to become mirror mode unstable diminishes, and SIRs can offer the ideal
conditions. Interestingly, the kinetic analysis suggested that ion cyclotron
waves should also be generated for similar conditions but were not observed.
They suggested that the differing phase velocities may be responsible for the
absence of concurrent observations. It has been understood for some time that
particularly in planetary magnetosheaths the MM and ion cyclotron
instabilities compete depending on the local $\beta_{i}$ [Soucek . (2015)].

The current study utilizes data from Solar Orbiter (SolO) to study MM storms
at heliospheric distances between 0.5-1 AU. The goal and motivation for this
study were to contribute to filling this gap and shed light on some unresolved
questions. This was achieved by employing the novel SolO observations to
investigate characteristics such as physical properties (e.g. amplitude,
frequency, peaks/dips, spatial scale), dependence on local plasma conditions,
and connection with solar wind structures (e.g. SIRs and shocks), and their
occurrence across heliocentric distances. The study firstly analyzed several
case studies in detail before conducting an automated search for events. This
produced 25 events that were used to investigate the occurrence rate,
dependence on solar wind conditions, and location in the inner-heliosphere.

##  2 Data & Instrumentation

SolO [Müller, D. . (2020)] measurements collected between 2020-04-15 and
2021-08-31 are used to conduct this investigation. The fluxgate magnetometer
instrument (MAG) [Horbury . (2020)] provides full 3D magnetic field vectors
and is used to characterize the magnetic field properties of the large-scale
structure and MM waves. The magnetic field data are also used to automatically
detect MMs later in the paper. The radio and plasma wave experiment (RPW)
[Maksimovic, M. . (2020)] measures the probe-to-spacecraft potential
($ScPot$), which can be calibrated to estimate the local electron density
($N_{e}$) [Khotyaintsev . (2021)]. Hereafter, $N_{e}$ refers to electron
density from $ScPot$ and is not calculated from moments of velocity
distributions. This high temporal resolution is needed since MM storms are
typically around 0.5-1 Hz in contrast to solar wind magnetic holes and typical
solar wind mirror modes that are above several seconds. The solar wind
analyzer (SWA) instrument [Owen . (2020)], particularly the proton alpha
sensor (SWA-PAS), is then employed to provide ion velocity distribution
functions (VDFs) and ion moments. Note that ground-based moments are
determined from the proton peak in the SWA-PAS VDFs. However, the proton peak
cannot always be easily distinguished and thus alphas may sometimes affect the
moment calculations. However, there is no dedicated flag to indicate when the
proton and alpha peaks are well resolved, so this study does not consider such
effects. The electron analyzer system (SWA-EAS) was also used to obtain
electron pitch angle distributions. Also note that the rtn (radial,
tangential, normal) coordinate system is used unless stated otherwise. The CDF
file versions used to analyze individual MM events (and compute solar wind
statistics) were V1-V3 (V1-V5) for MAG and V3 (V3) for SWA-PAS. Measurements
from the Magnetospheric Multiscale Mission (MMS) were also employed from the
fluxgate magnetometer [Russell . (2016)], Fast Plasma Investigation-Dual Ion
Spectrometer (FPI-DIS) [Pollock . (2016)]. OMNI data was used to infer ion
temperature for the MMS event since FPI-DIS was not intended to measure the
solar wind.

##  3 Case studies

###  3.1 Event 1: 2021-07-19

Plotted in Figure 1 are SolO measurements collected between 10:00 on
2021-07-18 and 23:10 on 2021-07-19 when the spacecraft was 0.84 AU from the
Sun. Panels (a & b) show the magnetic field while the remaining panels (c-g)
correspond to $\beta_{i}$, $N_{i,e}$, $|\mathbf{V_{i}}|$, $T_{i}$ and
omnidirectional differential energy flux (DEF).

Figure 1: Overview of the event for the interval on days 2021-07-18 and
2021-07-19. Panels (a-f) correspond to $|\mathbf{B}|$, $B_{rtn}$, $\beta_{i}$,
$N_{i}$, $|\mathbf{V_{i}}|$, and $T_{i}$, respectively. The bottom panel (g)
shows the omnidirectional DEF. The area highlighted in green shows the region
of interest, which contained abundant MM structures.

Near 18:00 on 2021-07-18, SolO measured a fast forward shock according to the
concurrent increase of $|\mathbf{B}|$, $N_{i}$, and $|\mathbf{V_{i}}|$ in
panels (a, d, & e). Later, about 08:30 on 2021-07-19, another fast forward
shock was measured according to comparable signatures in $|\mathbf{B}|$,
$N_{i}$, and $|\mathbf{V_{i}}|$. Before this event, the solar wind speed was
slow, below 300 kms-1 but then increased at the shock crossings to eventually
450 kms-1 at the end of the interval. $N_{i}$ was highly varying over the
entire event, rising to around 60 cm-3 at the first shock but afterward
increasing further to over 80 cm-3. The ions are also heated at both shock
crossings as shown by the sudden step increases and broadening of the
omnidirectional spectra in panel (g). Also meaningful are instances of
unusually small $|\mathbf{B}|$ to almost zero that creates large values in
$\beta_{i}$, which is discussed later. There are also substantial rotations of
the magnetic field in panel (b) signifying complex structures such as embedded
flux ropes and/or current sheets. The large-scale features and double shock
crossings are consistent with the passage of an SIR. The structured SIR region
results from the interaction between a slow wind stream and a fast wind
stream, where shock waves separate the unperturbed solar wind from the shocked
slow wind compressed by the incoming fast stream. Then another shock separates
the slowed down and compressed fast wind and the trailing undisturbed fast
stream. This picture corresponds to a two-stage increase in the density,
magnetic field magnitude and plasma temperature, along with the typical
transition from low to high plasma flow speed [Richardson (2018)].

In addition to the large-scale variations of the magnetic field and plasma
parameters seen in Figure 1, smaller-scale waves and structures were also
observed in concert. The large-scale variations appear inherently connected to
these smaller scales as they are responsible for significantly modifying the
local conditions that favor the growth of waves and instabilities. Of distinct
interest to this study is the area highlighted in green around 10:00-12:00 UT
on 2021-07-19. This interval contains a significant magnetic depression and a
polarity reversal of the radial and tangential magnetic field components. The
density also increases by a factor of two from 40 cm-3 to over 80 cm-3, and
although no substantial changes in velocity occurred, there were variations in
the temperature moments and intricate features in the DEF spectra. The
interpretation is that perhaps the spacecraft was crossing the heliospheric
plasma sheet (HPS). In Simunac2012 such crossings were identified by magnetic
field polarity changes, increased plasma density, a local decrease in the
alpha particle-to-proton number density, and a local increase in the ion
density. In our case, electron pitch-angle distributions support a crossing of
the heliospheric current sheet HCS at $\sim$ 10:18 UTC, where the electron
distributions change from anti-parallel to parallel (see Figure 3 later panel
b). Interestingly, inside this potential HPS encounter, numerous bursts of
linearly polarized structures were recorded. A more thorough analysis of these
structures is presented in Figure 2.

The HPS marked by the green highlighted region in Figure 1 is shown in Figure
2. A wavelet spectrogram of $\mathbf{B}$ is added in panel (c) and the
ellipticity of the magnetic field [Santolík . (2003)] is plotted in panel (d).
The ellipticity of $\pm$1 corresponds to right-/left-handed circular
polarization, and 0 to linear polarization. Ion temperature in magnetic field-
aligned coordinates is located in panel (g). $\beta_{i}$ is plotted in panel
(j) and the MM instability criterion $RMM$ is plotted in the bottom panel.
Note here (and in other figures), $\beta_{i}$ is plotted but $\beta_{\perp i}$
is used in the calculation for $RMM$. The quantity $RMM$ [Soucek . (2008)]
provides a measure of the variation from stability and is calculated as
follows:

$RMM=\beta_{\perp i}\left(\frac{T_{\perp i}}{T_{\parallel i}}-1\right).$ (1)

Here, and throughout the paper, the background magnetic field is determined
using a low-pass filter with a cut-off frequency of 0.01 Hz. Instability to
MMs corresponds to $RMM>1$ but mirror modes are also shown to appear when
$RMM<1$. In general, when $RMM>1$ (unstable plasma) MMs are peaks but appear
as dips when $RMM<1$. Equation 1 implies that MMs will grow when a temperature
anisotropy is present. In reality, the situation is more complex since MMs
compete with the ion cyclotron instability depending on the ion plasma
$\beta$. In general, the ion cyclotron instability will dominate for lower
$\beta$ [Soucek . (2015)].

Figure 2: MMs observed on 2021-07-19. Plotted in panels (a & b) are
$|\mathbf{B}|$ and $B_{rtn}$, a wavelet spectrogram of $\mathbf{B}$ is shown
in panel (c), and the ellipticity of the magnetic field is shown in panel (d).
Panels (e-k) depict $N_{i}$, $|\mathbf{V_{i}}|$, $T_{i}$, DEF, $\beta_{i}$,
and $RMM$, respectively. Regions that are highlighted in yellow correspond to
localized reductions in ellipticity and the manifestation of MM structures
since they should have zero ellipticity.

During the HPS encounter, there was a pronounced increase in the spectral
power of $\mathbf{B}$ as seen in Figure 2c, suggesting the presence of waves
and/or enhanced turbulence. More information is provided in panel (d) by
calculating the magnetic field ellipticity when the degree of polarization was
above 0.8. This unveiled multiple bursts of linearly polarized structures,
which were highlighted in yellow (1-4). As said previously, between 10:20 and
11:00, the ion density increased significantly ($\sim$ 40-80 cm-3) but what
was evident over this timescale is the complex behavior of the ion temperature
anisotropy. In general, $T_{\perp i}>T_{\parallel i}$ over this interval, and
the DEF intensity increased close to 1 keV while there appeared to be an
increase in alpha particle density. Evidence of an alpha particle density
increase is seen from the enhancement in DEF above the main population (i.e.
$>$1 keV). Upon closer inspection, the yellow highlighted intervals were
consistent with the characteristics of MM structures. According to panels (c &
d), the time period of these structures was around and slightly below 1
second, corresponding to wavelengths roughly 400 km assuming zero phase speed
in the plasma rest frame. The ion gyro-radius ($\rho_{p}$) within this
interval was around 32 km implying that $L_{mm}\sim 13\rho_{p}$, where
$L_{mm}$ is the spatial scale of an individual MM structure. Intervals 1 and 2
display enhancements in $RMM$ in panel (k), as expected from the concurrent
increase in $T_{\perp i}/T_{\parallel i}$ and $\beta_{i}$ from panels (g & j).
Interval 3 was MM stable ($RMM<1$) due to $T_{\parallel i}>T_{\perp i1}$,and
therefore surprising that MMs were so prevalent. On the other hand, since MMs
are convected with the plasma flow, such in situ conditions may not match the
plasma parameters at the moment when the MM structures were generated; this is
a plausible scenario considering the variations of $RMM$ over this brief
interval. The final interval was intriguing as $RMM\sim 0$, which will be
discussed later.

In Figure 3, VDFs were presented at five instances that were marked by the
vertical red lines in panel (a) by roman numerals I-V. These were chosen to
provide an overview of the changes in the VDFs across the event. The VDFs are
shown in two planes according to $\parallel-\perp 1$ and $\perp 1-\perp 2$,
which is derived with respect to the background magnetic field. The DEF was
shown for reference in panel (b) whereas the VDFs were placed below (d-m). The
plotted VDFs are averages of five distributions, which correspond to 16
seconds and the width of the red lines.

Figure 3: Evolution of ion VDFs and electron pitch angle. Panels (a c) show
$|\mathbf{B}|$, ion DEF, and electron pitch angle phase space density,
respectively. Panels (d-m) are VDFs at the time instances marked by I-V in
panel (a). The top row is a 2D reduced distribution in the
$V_{\parallel}$-$V_{\perp 1}$ plane whereas the bottom row is the $V_{\perp
1}$-$V_{\perp 2}$ plane.

According to Figure 3, there were noticeable variations of the ion VDFs
throughout this interval. For the majority of the event, the VDFs appeared
moderately gyrotropic. There is a field rotation (reversal in $B_{t}$) around
10:18 and this effect is seen in the VDFs from column II as elongated shapes
oblique to the local background magnetic field, which is present thereafter.
This did result in temperature anisotropy and favorable MM growth between
10:20 and 10:40 also due to the high $\beta_{i}$. However, column IV did
appear slightly different compared to II, III, and V, which coincided with a
sharp change in $B_{t}$ and larger localized parallel temperature according to
the moments, resulting in unfavorable conditions for MMs even though they are
seen in $\mathbf{B}$. VDF IV seemed to evolve to more gyrotropic but the
differences in the $V_{\perp 1}$-$V_{\perp 2}$ plane (k) constitute a few
pixels and thus it is not possible to draw strong conclusions from this. Thus,
for much of this interval, the MM growth condition seemed satisfied but still,
MMs were present in locations where the VDFs and moments did not provide a
clear explanation.

###  3.2 Event 2: 31 May 2021

Presented in Figure 4 is another period of intensive MM activity on 31 May
2021, when SolO was at a heliocentric distance of 0.95 AU. The layout of the
panels is equivalent to those shown in Figure 2. Occasionally the SWA-PAS
instrument would measure higher-cadence burst mode data for 5 minutes every 15
minutes, which is visible in the time series plot. The highlighted interval
denotes the period of MM activity, which started after an increase in
$|\mathbf{B}|$ at 08:08 due to abrupt changes in $B_{n}$ and $B_{t}$ (red and
blue traces). Interestingly, it is worth remarking that isolated MM structures
were also observed before this, such as the individual peak near 08:05, which
has been marked in panel (a). Thus, the plasma was likely to be marginally MM
unstable before the onset of the wave trains. Nevertheless, the advent of the
MM trains coincided with a small decrease in $\beta_{i}$, a small increase in
$N_{i}$, but no change in $V_{i}$ or $T_{i}$. Thus, this is not interpreted as
a shock crossing. The MM instability threshold was also below zero for the
majority of this interval.

Figure 4: MMs observed on 31 May 2021. Plotted in panels (a & b) are
$|\mathbf{B}|$ and $B_{rtn}$, a wavelet spectrogram of $\mathbf{B}$ is shown
in panel (c), and the ellipticity of the magnetic field is shown in panel (d).
Panels (e-k) depict $N_{i}$, $|\mathbf{V_{i}}|$, $T_{i}$, DEF, $\beta_{i}$,
and $RMM$, respectively.

Contrary to the event on 18 July 2021, these MMs appeared as extended trains
of structures for over 40 minutes (highlighted in yellow) rather than shorter
distinctive bursts of several minutes. The structures were linearly polarized,
appeared as peaks, had periods of around 1 second, and amplitudes of roughly
0.5 nT. Based on the local plasma conditions, the spatial scale of these MMs
was $L_{mm}\sim 4.5\rho_{p}$. Later in the interval, there was a polarity
reversal of $B_{n}$ and $B_{t}$, and the MMs appeared suppressed. Yet, they
began again soon afterward but were more bursty by nature, which could be
reflective of the variable $\beta_{i}$. Confusingly, there was no significant
temperature anisotropy over this period. As expected, $RMM$ remains
predominantly below zero meaning the plasma was stable or marginally stable
over this interval. These MMs also arose during a low energy slow solar wind
stream, which exhibited a low ion temperature and moderate density. Evidently,
the circumstances that led to this long train of MMs were different from the
previous example.

Figure 5 reveals the MMs in more detail and the sharp peak structures are
unmistakable from $|\mathbf{B}|$ in panel (a). Plotted in panel (b) is $N_{e}$
whereas panel (c) is a wavelet coherency spectra between $|\mathbf{B}|$ and
$N_{e}$, which represents the coherency and phase between these quantities for
the shown frequency range.

Figure 5: Wavelet coherency between $|\mathbf{B}|$ and $N_{e}$. Panels (a & b)
show $|\mathbf{B}|$ and $N_{e}$ during MM activity on 31 May 2021 and the
corresponding wavelet coherency spectra (c). The color in panel (c) depicts
the coherency (0-1) whereas the arrows pointing left suggest anti-phase.

A fundamental attribute of MMs is the anti-correlation between $\mathbf{B}$
and density. The bulging local magnetic field induced by pressure anisotropy
sets up bottle-like structures that create local magnetic mirror points. As a
spacecraft transits through these structures then it will measure a time
series of $|\mathbf{B}|$ and density that are anti-correlated [Soucek .
(2008), Dimmock . (2015)]. The wavelet coherency confirmed this, demonstrating
that the frequencies matching the MM time scales ($\sim 1Hz$) displayed
coherency values close to one and phase shifts around 180∘. The direction of
the arrows denotes the phase such that pointing to the right (left) is in-
phase (out of phase). Here, arrows are only plotted when the coherency
exceeded 0.85 and the prevailing trend of these arrows is that they are
pointing to the left and thus clearly demonstrate the anti-correlation over
this interval. Note that the anti-correlation is only visible in $N_{e}$ since
the cadence was sufficiently high compared to the ion moments. This anti-phase
behavior was also observed for different events, but in some circumstances, it
was not measurable due to the generally small amplitudes of MM storms.

Figure 6 shows VDFs for the 31 May 2021 event at the times marked in panel (a)
by the vertical red lines (I-V). Each VDF is an average of 5 VDFs, which is
equivalent to the thickness of the vertical red lines. For reference,
$|\mathbf{B}|$ and the DEF have been plotted in panels (a & b).

Figure 6: Evolution of ion VDFs, Panels (a & b) show $|\mathbf{B}|$ and and
the DEF, respectively. Panels (c-l) are VDFs at the time instances marked by
I-V in panel (a). The top row is a 2D reduced distribution in the
$V_{\parallel}$-$V_{\perp 1}$ plane whereas the bottom row is the $V_{\perp
1}$-$V_{\perp 2}$ plane.

The alpha particles are clear in panel (b) by the population around 1 keV
above the solar wind at 500 eV. The VDFs measured by SWA-PAS measure both ions
and alphas and in cases like this, the proton peak is well-defined. Thus, the
moments should not be affected by alpha contributions. This feature has been
labeled in panel (g) and is visible in the other VDFs (c-l). In contrast, to
the solar wind in Figure 3, the energy is lower, as expected due to the low
speed and temperature. As expected from Figure 4, there was no strong
anisotropy and the VDFs did not experience significant evolution across this
interval to account for the strong MM activity. Surprisingly, the VDFs I and
II were similar, which could suggest the change in $|\mathbf{B}|$ and
$\beta_{i}$ was not responsible for sufficiently altering the MM stability.
This is reasonable considering that the plasma appeared MM unstable or
marginally stable (according to the presence of isolated MM structures) before
the sharp onset of these waves. Thus, open questions were raised about this
event and there was no immediate local driving mechanism. It could be that
this MM criterion does not include key factors that were important to the
growth rate of these waves [Pokhotelov . (2002)]. On the other hand, it could
be that these waves were convected from a different location. These will be
addressed later in the discussion.

###  3.3 Event 3: 2021-08-14

On 2021-08-14, SolO observed another interval of prolonged MM activity lasting
approximately 50 minutes at 0.69 AU from the Sun. These measurements are shown
in Figure 7 and the panels are organized in the same manner as Figures 2 and
4.

Figure 7: Mirror modes observed on 2021-08-14. Plotted in panels (a & b) are
$|\mathbf{B}|$ and $B_{rtn}$, a wavelet spectrogram of $|\mathbf{B}|$ is shown
in panel (c), and the ellipticity of the magnetic field is shown in panel (d).
Panels (e-k) depict $N_{i}$, $|\mathbf{V_{i}}|$, $T_{i}$, DEF, $\beta_{i}$,
and $RMM$, respectively.

Remarkably comparable to the other events, the solar wind speed was still
unusually low ($<300$ kms-1). This event offered striking similarities to
Figure 2 where the MMs appeared in a magnetic depression, high density ($\sim
43$ cm-3), and enhanced $\beta_{i}$. The magnetic field spectral power up to 1
Hz was also visibly intensified. Within the magnetic dip, there were
negligible variations of the plasma parameters, but the magnetic field was
varying significantly, causing the $\beta_{i}$ to fluctuate and consequently
result in large changes in $RMM$. The ion VDF was also moderately gyrotropic.
What was also meaningful regarding this particular event was that the
timescales appeared larger than the previous event ($\sim 13$ sec) and
therefore $L_{mm}\sim 56\rho_{p}$. In addition, the distance between the dips
had grown and one can recognize individual MM structures even on this larger
time scale. In addition, the amplitudes exceed 1 nT, which is larger than the
aforementioned cases.

After scrutinizing multiple cases of MM storms, there seemed to be two
distinguishable types of events. Type one corresponds to intensive bursts with
timescales around 1 second and amplitude up to 1 nT, which manifested as peaks
or dips and had spatial scales of around one or several ion gyroradii. Type
two was consistent with extended trains of magnetic holes and had timescales
of several seconds, amplitudes larger than 1 nT, and larger spatial scales
that were several 10s of the local gyroradii. The features of this event could
also suggest another HCS encounter similar to Figure 2. Thus it seemed HCS
crossings were effective at setting up MM growth conditions.

###  3.4 Short-term temporal evolution of mirror mode structures

So far, the case studies that were presented revealed MM intervals in which
the individual structures were invariant in many properties, such as peakness,
frequency, and the spacing between peak/dip structures. Here, the peakness
refers to if the MMs were peaks or dips and was determined from the skewness
of the probability distribution of magnetic field calculated from:

$S=\frac{M_{3}}{\sigma^{3}},$ (2)

where

$M_{3}=\frac{1}{N}\sum^{N}_{i=1}(B_{i}-\bar{B})^{3}.$ (3)

For $S<0$, the MMs were dips and when $S>0$ the MMs were peaks. In this
section, examples are shown that demonstrate the evolution of peakness and
other MM properties.

Throughout this investigation, two types of MM intervals were discussed. These
types were defined based on the frequency and amplitude of the structures.
Yet, it is necessary to point out that these different types were not mutually
exclusive nor did they have to occur within completely separate events.
Depicted in Figure 8 is a period of intense MM activity on 2020-09-06. Panels
(a, b, and c) depict the magnetic field time series and a wavelet transform of
its magnitude. The ellipticity is plotted in panel (d) where the prolonged
linear polarization is easily identified by the nearly zero ellipticity. The
remaining panels (e-g) show $N_{i}$, $V_{i}$, and the DEF. The temperature was
reliable enough to draw conclusions from, so it was not included. Note that
there were some data gaps in the plasma measurements resulting in absent data
in the bottom three panels, which does not interfere with the investigation.

Figure 8: MMs on 2020-09-06. Panels (a & b) show the magnetic field, plotted
in panels (c & d) are a wavelet transform of $|\mathbf{B}|$ and ellipticity,
respectively. The remaining panels (e-g) are $N_{i}$, $|\mathbf{V_{i}}|$, and
DEF.

Comparable to the previous cases, the event took place during a slow solar
wind stream with speeds lower than 300 kms-1 and high densities ($N>20$ cm-3).
Between 10:40 and 11:40, the MMs were around 1 Hz with amplitudes around 0.5
nT. At 11:40 there was a polarity reversal of $B_{n}$ and a significant change
in $B_{r}$. Following this, there was a prominent change in the MM structures
resulting in larger amplitudes ($>1$ nT), larger periods, and increased
proximity between individual structures. Thus, from 11:30 - 11:40, the
physical nature of the MMs had dramatically changed, which seems triggered by
the field rotation and a slight increase in $|\mathbf{V_{i}}|$.

In addition to the evolution of amplitude and frequency as shown in Figure 8,
the peakness could similarly deviate. In the next example, this occurred over
approximately 10 minutes. Figure 9 shows a case where the MMs evolved from
peaks to dips. It is interesting to note that there was an interval with
circularly polarized waves, however, the analysis of such waves was not within
the scope of the present study. Panels (a, b, and & c) correspond to the
magnetic field time series and a wavelet transform of the magnetic field.
Plotted in panel (d) is the ellipticity whereas panel (e) is the skewness
calculated over a sliding window of 20 seconds that was advanced by one second
until the end of the interval was reached.

Figure 9: The evolution of MMs observed on 2021-03-04. The top two panels (a &
b) show the mirror mode structures in the magnetic field whereas a wavelet of
$|\mathbf{B}|$ is shown in panel (c). The ellipticity of the magnetic field is
shown in panel (d) and the skewness below in panel (e). The skewness ($S$)
demonstrates whether the mirror modes are peaks (skewness $S>0$) or dips
($S<0$) and panel (e) indicates a change in $S$ over this interval.

The skewness slowly transitions from positive to negative, indicating a shift
from peaks to dips, respectively. Yet, the frequency seemed to remain constant
throughout. Thus, there was no sharp change in conditions responsible,
contrary to the previous example; the change occurred more gradually. But,
there was a small rotation in $\mathbf{B}$ around 04:16, however, the peakness
evolution seems to be underway prior to this. Unfortunately, particle
measurements were not available during this time, so it is not possible to
interpret the plasma conditions. Based on earlier studies [Soucek . (2008),
Génot, Budnik, Hellinger . (2009), Soucek . (2015), Dimmock . (2015)], MM
peaks are associated with more MM unstable (larger value of $RMM$) conditions.
Hence, this evolution may signify a transition from MM unstable to marginally
MM stable conditions.

###  3.5 Mirror mode storm downstream of an IP shock by MMS

Since SolO is a single spacecraft, the assumption of zero propagation in the
plasma rest frame has been used to make conclusions regarding the spatial
scale of the MMs. However, MMS consists of 4 spacecraft with inter-spacecraft
separations that are sometimes similar to the spatial scale of the MMs that
were studied with SolO. Thus, MMS can be used to directly infer the spatial
scales. For this reason, this section describes MMS observations of MMs that
were observed directly downstream of an IP shock on 2017-10-24. This is shown
in Figure 10, and although IP shocks have been reported [Cohen . (2019),
Hanson . (2020)], this event is likely to be the earliest known IP shock
measured by MMS. Panels (a-d) show $|\mathbf{B}|$, $n_{e}$, $\mathbf{V_{i}}$,
and ellipticity. The bottom panel is a zoomed-in plot of $|B|$ but shows all
four MMS spacecraft. In this specific example, $n_{e}$ is a plasma moment as
opposed to derived from the spacecraft potential, which was the case with
SolO. The shock crossing was oblique ($\theta_{bn}\sim 52^{\circ}$) and low
Alfvén Mach number ($M_{A}\sim 1.7$).

Figure 10: A MM storm observed by MMS downstream of an IP shock on 2017-10-24.
Panels (a-d) show $|\mathbf{B}|$, $n_{e}$, $|\mathbf{V_{i}}|$ and the
polarization of $\mathbf{B}$, respectively. The bottom panel displays a
shorter interval where all four MMS spacecraft are plotted together.

Almost immediately downstream from the shock ramp, there is a sudden onset of
a MM train. Only burst mode is shown here but the MM structures can be
observed for around 3.5 minutes after the shock ramp. The structures are
linearly polarized and appear as sharp dips, similar to some of the other
events studied with SolO. What is valuable in this example is that the multi-
point measurement can be used to directly infer the spatial scales of the
individual structures. What is interesting here is that in panel (e), some MM
structures are observed by some MMS spacecraft, but not by others. This
implies that these structures are on the same scale, or smaller than the
spacecraft separation. Here, the average spacecraft separation is 27 km and
$\rho_{p}\sim 43$ km, confirming that these MMs are smaller than the local ion
scales. This is true in at least one direction, however, the 3D geometry of
MMs was not assessed here. In addition, the magnetic pressure is balanced by
the electron thermal pressure inside the dips. Thus, it is likely that these
were electron or kinetic MMs, which will be discussed in more detail below.

##  4 Statistical results

Using SolO, it is now feasible to investigate events ranging over heliocentric
distances ($|\mathbf{R}|$) without the reliance upon separate spacecraft
conjunctions. In addition, the onboard suite of instruments allowed the
investigation into the solar wind conditions that are key to the growth of
MMs. Regardless, a manual search is laborious and impractical. Therefore, an
automated search was employed. As established by these case studies and prior
literature, clear characteristics of these events where $\mathbf{B}$ had a
high degree of polarization ($>0.8$), linearly polarized (ellipticity = 0),
anti-correlated with density, and manifested as trains of structures
continuing for several minutes. It is also important to reiterate that the
purpose was to identify train-like MM events and not isolated magnetic holes.
Having stated that, the amplitudes of these structures could be low ($<$ 1 nT)
and although they were visible in the magnetic field, this was not always the
case for the plasma density. For this rationale, the lengthy period of a high
degree of polarisation concurrent with low ellipticity was used. It is also
necessary to mention that utilizing the local plasma conditions such as
temperature anisotropy may have been helpful in this search. However, these
measurements were not available for lengthy periods, whereas the magnetic
field is consistently available. In addition, it appears that the existence of
MMs does not always correspond with the anticipated in situ plasma conditions
such as $T_{\perp i}>T_{\parallel i}$. For these reasons, the automated search
was performed using solely magnetic field measurements.

###  4.1 Automated search

The search was conducted on measurements between 2020-04-15 and 2021-08-31.
The step-by-step procedure was as follows:

  1. 1.

Compute the magnetic field ellipticity ($\epsilon$) and degree of polarization
within a 5-minute window between frequencies 0.1-2 Hz (0.5s-10s).

  2. 2.

Apply a mask to points where the degree of polarization falls below 0.7.

  3. 3.

Require that 75% of $|\epsilon|<0.2$.

  4. 4.

Save the times of windows that satisfy the criteria of predominantly being
linearly polarized.

  5. 5.

Advance the window by 2.5 minutes (50% overlap) and repeat.

The above process delivered a set of 5-minute windows that fulfilled these
criteria. These windows were then manually arranged into separate events and
visually inspected for signatures of MM structures. If events were separated
by more than 1 hour, these were documented as separate intervals. The outcome
was 25 separate intervals, which are listed in table 1 as well as some
essential parameters.

Table 1: Prolonged MM events observed with Solar Orbiter detected by the automated search. Listed are some fundamental properties as well as the type of structure that the MMs were associated with. The acronyms SW-CS and SBC refer to solar wind current sheet and sector boundary crossing, respectively.  # | Date [UTC] | Type |  $|\mathbf{V_{i}}|$ [kms-1] |  $N_{i}$ [cm-3] | Ti [eV] | T⟂/∥ | $\beta_{i}$ | Event Type | QF |  $|R|$ [AU] | CDF ver [MAG, PAS]
---|---|---|---|---|---|---|---|---|---|---|---
1 | 2020-04-16 12:00 | 2 |  |  |  |  |  |  |  | 0.82 | [1, n/a]
2 | 2020-04-16 22:56 | 2 |  |  |  |  |  |  |  | 0.82 | [1, n/a]
3 | 2020-06-04 08:30 | 2 |  |  |  |  |  |  |  | 0.54 | [3, n/a]
4 | 2020-07-08 11:38 | 2 | 311 $\pm 1$ | 18 $\pm 1$ | 5 $\pm 0.4$ | 0.90 $\pm 0.28$ | 5.2 $\pm 5.1$ | SW-CS | 0.26 $\pm 0.15$ | 0.60 | [3, 3]
5 | 2020-07-29 10:33 | 1 | 297 $\pm 1$ | 13 $\pm 1$ | 3 $\pm 0.2$ | 0.54 $\pm 0.09$ | 4.5 $\pm 1.3$ | SW-CS | 7.98 $\pm 1.49$ | 0.73 | [3, 3]
6 | 2020-08-09 11:02 | 1 | 326 $\pm 3$ | 23 $\pm 1$ | 3 $\pm 0.1$ | 0.97 $\pm 0.33$ | 3.5 $\pm 1.8$ | SW-CS | 0.04 $\pm 0.06$ | 0.80 | [2, 3]
7 | 2020-08-27 02:55 | 1 |  |  |  |  |  |  |  | 0.89 | [2, n/a]
8 | 2020-09-06 10:32 | 1, 2 | 297 $\pm 3$ | 26 $\pm 3$ | 2 $\pm 0.5$ | 0.90 $\pm 0.24$ | 3.2 $\pm 1.3$ | SIR | 0.59 $\pm 0.71$ | 0.93 | [2, 3]
9 | 2020-09-06 16:31 | 2 | 336 $\pm 3$ | 45 $\pm 2$ | 3 $\pm 0.3$ | 1.07 $\pm 0.09$ | 3.1 $\pm 1.1$ | SIR | 0.00 $\pm 0.0$ | 0.93 | [2, 3]
10 | 2020-12-02 12:20 | 2 |  |  |  |  |  |  |  | 0.87 | [4 ,n/a]
11 | 2020-12-31 21:40 | 1 |  |  |  |  |  |  |  | 0.70 | [4 ,n/a]
12 | 2021-01-27 18:58 | 2 |  |  |  |  |  |  |  | 0.53 | [4 ,n/a]
13 | 2021-01-30 04:40 | 2 |  |  |  |  |  |  |  | 0.52 | [3 ,n/a]
14 | 2021-02-20 12:44 | 1 |  |  |  |  |  |  |  | 0.51 | [2 ,n/a]
15 | 2021-03-04 04:00 | 1 |  |  |  |  |  |  |  | 0.57 | [2 ,n/a]
16 | 2021-04-07 16:14 | 1 |  |  |  |  |  |  |  | 0.79 | [2 ,n/a]
17 | 2021-04-19 13:55 | 1 |  |  |  |  |  |  |  | 0.85 | [3 ,n/a]
18 | 2021-05-31 08:05 | 1 | 321 $\pm 5$ | 16 $\pm 1$ | 3 $\pm 0.2$ | 0.92 $\pm 0.1$ | 1.5 $\pm 0.3$ | SW-CS | 0.00 $\pm 0.00$ | 0.95 | [2, 3]
19 | 2021-05-31 10:24 | 1 | 319 $\pm 3$ | 31 $\pm 3$ | 7 $\pm 2.8$ | 0.51 $\pm 0.17$ | 4.7 $\pm 1.8$ | SW-CS | 0.10 $\pm 0.10$ | 0.95 | [2, 3]
20 | 2021-06-27 08:46 | 1 |  |  |  |  |  |  |  | 0.92 | [1, n/a]
21 | 2021-06-28 03:42 | 1, 2 |  |  |  |  |  |  |  | 0.92 | [1 ,n/a]
22 | 2021-07-19 10:18 | 1 | 427 $\pm 9$ | 61 $\pm 14$ | 12 $\pm 2.4$ | 1.17 $\pm 0.24$ | 1.5 $\pm 2.1$ | SIR & SBC | 0.00 $\pm 0.00$ | 0.84 | [1, 3]
23 | 2021-08-14 04:10 | 2 | 278 $\pm 1$ | 43 $\pm 1$ | 2 $\pm 0.1$ | 1.00 $\pm 0.09$ | 10.9 $\pm 20.6$ | SBC | 0.31 $\pm 0.12$ | 0.69 | [2, 3]
24 | 2021-08-14 07:05 | 2 | 270 $\pm 1$ | 41 $\pm 1$ | 2 $\pm 0.1$ | 1.02 $\pm 0.08$ | 10.3 $\pm 15.8$ | SBC | 0.65 $\pm 0.19$ | 0.69 | [2, 3]
25 | 2021-08-14 18:57 | 2 | 286 $\pm 1$ | 33 $\pm 1$ | 4 $\pm 0.3$ | 1.54 $\pm 0.13$ | 2.8 $\pm 2.0$ | SBC | 0.23 $\pm 0.09$ | 0.69 | [2, 3]
$\pm$ calculated as one standard deviation |  |  |  |

The quantities documented in table 1 represent the mean values over the
interval that MM structures were visually perceptible. As demonstrated by the
case studies above, this can be a variable period between several minutes to
an hour. Thus, quantities can deviate significantly. For this reason, this
variability has been denoted by adding $\pm$ one standard deviation. The PAS
quality factor (QF) has also been included to provide readers with a proxy for
the trustworthiness of these values.

From table 1, there are several intriguing results to point out. Surprisingly,
all of the events (when PAS data was available) were identified during
moderate/slow solar wind streams. This could also be a statistical effect
caused by the average speed of the solar wind in the studied period, which
will be discussed later. However, an anomaly is the SIR on 2021-07-19
($|\mathbf{V}_{i}|\sim 430$ kms-1) compared to the remaining events where
$|\mathbf{V}_{i}|<350$ kms-1. As anticipated from the slow solar wind, $N_{i}$
was also high and as depicted in the case study shown above that surpassed 80
cm-3. For most events, $T_{i}$ was low, but the temperature anisotropy could
be highly complex and variable during each event. To some extent, this
explains why the anisotropy values in table 1 are generally moderate and why
in some cases $T_{\perp i}<T_{\parallel i}$. A caveat to interpreting these
values properly is that this cannot be considered the ambient solar wind, that
is, solar wind that is not clearly associated with some known transient such
as an SIR and ICME. Even so, the occurrence during slow solar wind is
striking.

###  4.2 Dependence on heliocentric distance and solar wind conditions

The criteria adopted in this automated search were intended to identify
prolonged intervals of linearly polarised structures that are indicative of
MMs. From a period of 16.5 months, only 25 intervals were detected. Although
further events would be desired to more accurately calculate the occurrence
rates of these events, one prominent result was that their presence is not
frequent. It was also possible to calculate the probability at which
$|\mathbf{R}|$ these events are identified. This is plotted in Figure 11.

Figure 11: Occurrence rate of prolonged mirror mode trains across heliocentric
distances between 0.5-1 AU.

The values in Figure 11 are calculated based on the availability of
$|\mathbf{B}|$ such that event counts were normalized by the availability of
MAG data at each $|\mathbf{R}|$ bin. Although the number of events is limited
to 25, Figure 11 implies that the likelihood of identifying these events
declines with raising $|\mathbf{R}|$. Having said that, this trend was not
strong, and additional events will be required for confirmation.

To properly interpret the values provided in table 1, they have to be put into
context with the typical values of the solar wind. However, these will vary
with $|\mathbf{R}|$, which was investigated in Figure 12.

Figure 12: Solar wind statistics measured by SolO between $0.5<R<1$. Panel (a)
shows the availability of data when MAG and MAG+PAS data are available. Panels
(b, c, e, & f) show $|\mathbf{V_{i}}|$, $N_{i}$, and $|\mathbf{B}|$ as a
function of $|R|$ in which the color shows the bin counts and the red crosses
are the values for specific events. Panel (d) is the cumulative distribution
function for the PAS quality factor.

Panel (a) shows the availability of MAG and MAG+PAS data for bins of
$|\mathbf{R}|$. Thus, the spacecraft occupied $|\mathbf{R}|\sim 1$ longer than
$|\mathbf{R}|\sim 0.5$. This demonstrated why this had to be taken into
account in the occurrence rates of these events. The red crosses show the
values of $|\mathbf{R}|$ for each of the 25 events (note the placement on the
y-axis is arbitrary). Plotted in panels (b, c, e, and f) are 3D histograms of
various quantities for bins of $|\mathbf{R}|$. The 3D histograms are used here
to demonstrate the availability of solar wind data at various heliospheric
distances. The red crosses again show the values for each event and the error
bars correspond to $\pm$ one standard deviation. The cumulative distribution
function (CDF) of the PAS quality factor is located in panel (d).

According to panel (b), and as expected, the solar wind speed naturally
increased with $|\mathbf{R}|$ [Khabarova . (2018)], however in general the MM
events stayed at the lower range of $|\mathbf{V}|$ regardless of
$|\mathbf{R}|$. Thus, based on the criteria that were adopted here, the events
were identified within the slow solar wind for each heliocentric distance.
However, there is a lack of faster solar wind speed observations at some
$|\mathbf{R}|$, particularly around 0.7 AU and 0.85 AU, and the bin density
corresponding to slower speed is higher at 1 AU. Thus, it cannot be ruled out
completely that there could be some statistical influence. Panel (c) also
demonstrated that events also occurred during cold ion temperatures, and
according to panel (e), higher than typical ion densities. This could be
related to the fundamental characteristics of the fast and slow solar wind,
i.e. the slow solar wind is usually denser and colder. However, it should be
noted that these events were not identified in the ambient solar wind. There
was a tendency for $|\mathbf{B}|$ to decrease with $|\mathbf{R}|$, but there
was no clear reliance on the magnetic field strength of the events depicted in
panel (f). The case studies presented in detail above were selected partly
based on low-quality factor values from PAS (i.e. high-quality data), however,
panel (d) suggested that several events suffer from higher quality factors
which are unavoidable due to the low solar wind speed for each event. It
should also be noted that many events appeared as outliers for ion density and
for a couple of cases in the magnetic field. This is expected since these MM
storms did not seem to appear in the ambient solar wind but during disturbed
intervals such as SIRs and current sheets.

##  5 Discussion

For the first time, missions such as SolO and Parker Solar Probe (PSP) have
enabled the study of the dependence of kinetic instabilities and other complex
structures such as MMs on heliocentric distance ($<$ 1 AU) and solar wind
properties. This paper has concentrated on continuous MMs, referred to in
prior studies as mirror mode storms [Russell . (2009), Enriquéz-Rivera .
(2013)] that differ from the more isolated magnetic hole structures that are
examined in numerous earlier studies [Turner . (1977), Winterhalter . (1994),
Xiao . (2010), Volwerk . (2020), Karlsson . (2021)]. The objective here was to
understand the connection with the solar wind, structures/transients,
heliocentric distance, and local plasma conditions while shedding light on
their physical properties. Throughout this investigation, several case studies
were analyzed followed by statistical results. Statistics were compiled
utilizing an automated search exploiting the linearly polarized nature of
these types of structures. There were multiple physical mechanisms/structures
over a wide variety of temporal scales that created the conditions favorable
for MM growth. Yet, although significant questions remain for some cases,
several clear and novel conclusions could be reached. Below, the physical
interpretations of these results are discussed, and explanations for difficult
events are offered, which are also put into context with the existing
literature.

According to the MM growth condition (equation 1), the larger the $\beta_{i}$,
the smaller the temperature anisotropy needed for the plasma to become mirror
unstable [Hasegawa (1969), Soucek . (2008)]. In two events studied, HCS
crossings resulted in simultaneous magnetic field decreases and density
increases [Simunac . (2012)], which created sudden and large enhancements of
$\beta_{i}$. Such conditions should then require only small temperature
anisotropies to set up MM unstable plasma conditions. It seems HCSs can
sometimes be embedded within SIRs and CMEs, and one result obtained from this
study demonstrated that they were highly efficient at setting up conditions
for MM growth. This also implied that the plasma parameters (e.g. temperature
anisotropy and $\beta_{i}$) associated with large-scale solar wind transients
such as SIRs and CMEs were also constrained to some extent by these
instabilities. Similar to planetary magnetosheaths [Soucek . (2015), Génot
(2008), Dimmock . (2015)], solar wind transients also offer a rich natural
laboratory for investigating these structures.

For the events when $\beta_{i}$ was low, the temperature anisotropy did not
appear to reach exceptionally large values and appeared constrained between 1
and 1.2, sometimes even below 1. These events, therefore, appear marginally
stable or near the stability threshold; some were noticeably below. At Earth,
and planetary magnetosheaths in general, MMs are mainly driven by the large
temperature anisotropy created by the quasi-perpendicular bow shock [Dimmock
Nykyri (2013), Dimmock . (2015), Soucek . (2015), Osmane . (2015)], which also
increases the $\beta_{i}$. Interplanetary shocks also produce temperature
anisotropies, which can result in mirror modes [Ala-Lahti . (2018)]. However,
in that study, they appeared as more isolated magnetic hole structures and not
as the MM storms that were examined here. Two interplanetary shocks occurred
for the event presented in Figures 1 and 2 and both shocks did appear to
generate moderate temperature anisotropy downstream. However, the increase in
$\beta_{i}$ is not significant since the density and magnetic field both
increase across the shocks, and the ion temperature change is inconsequential.
This was evident from panel (c) in Figure 1 as no sharp changes in $\beta_{i}$
occurred across the shock fronts. It seemed that the interplanetary shocks
studied here do not seem to be efficient in generating the conditions
resulting in MM storms. On the other hand, russell2009 confirmed that MM
storms can be generated downstream of weak interplanetary shocks, so this is
not always the case. No MM storms were observed directly downstream from SolO
shocks in this study but were driven by large-scale changes in field and
plasma properties associated with other structures. However, one cannot rule
out shorter MM intervals that fall outside of the search criteria adopted
here. Analyzing other SolO interplanetary shocks (not shown) also implied that
MM storms are not a common feature. enriquez_rivera2013, also proposed that
shocks were not essential to MM storm growth in their investigation, which
used STEREO data. Nevertheless, this could be highly specific to shock
parameters (e.g. Mach number, geometry) and it is worthy of more research as
SolO assembles a diverse shock catalog over the nominal mission phase and
beyond.

enriquez_rivera2013 also reported that the alpha particle density increased
for most of the MM storms that they studied. Although alpha particle moments
were not directly available for this study, there was some evidence to support
enhanced alpha particle density in some events (e.g. Figures 1 and 7) during
the enhanced $\beta_{i}$ intervals. It has been established in previous
studies [Price . (1986), Hellinger Trávníček (2005), Lee (2017)] that
different particle species can play a significant role in modifying the mirror
mode instability criteria while also having the effect of suppressing the
competing ion cyclotron instability. However, it was not possible to directly
investigate that in this study since the instrument does not separate ions and
alphas.

It is worth commenting that some of the events identified in this study showed
no evident local mechanisms for MM growth, particularly because the ion
temperature anisotropy was around one (e.g. Figure 4) while $\beta_{i}$ was
also small. There are some conceivable explanations for these events. Firstly,
the events identified in this study occurred during low solar wind speeds,
which can lead to instrumental problems. Explicitly, this can result in
nonphysical ion VDF features due to low solar wind energies. This issue is
quantified to some capacity by the PAS quality factor, which serves as a proxy
that is anti-correlated to the trustworthiness of the data. As a rule-of-
thumb, the PAS quality factor increases for low solar wind speeds, and the
data becomes less reliable. In addition, the temperature is a higher-order
moment and is especially susceptible to artifacts in the ion VDFs. As a
result, estimating the correct temperature anisotropy becomes challenging in
specific situations. Secondly, in cases when the data is reliable, the in-situ
plasma measurements may not reflect the MM growth conditions at the
moment/location that the structures were generated. The reason was that MMs
are convected with the plasma flow, therefore, it is conceivable that the
source region could be located elsewhere. Another interpretation is a temporal
variation of the source region plasma parameters, such as a relaxation of the
temperature anisotropy as a result of the MMs. The final reason stems from the
variety of these events in terms of the plasma conditions, spatial scales, and
their presence in different solar wind transients. Therefore, a growth
condition that incorporates additional factors (e.g electron temperatures,
smaller wavelengths, non-Maxwellian VDFs, and other particle species) may be
required.

The mirror instability threshold expressed in equation 1 [Hasegawa (1969)] is
a cold electron bi-Maxwellian fluid approximation, assuming the low frequency
and long wavelength limit such that $\omega\ll\omega_{ci}$, $\omega\ll
k_{\parallel}v_{A}$, and $k_{\parallel}/k_{\perp}\ll 1$. This set a quasi-MHD
constraint on the spatial scales, meaning MMs were required to be much larger
than the local ion scales. In the terrestrial magnetosheath, mirror mode
spatial scales are typically a few thousand km (15 sec duration with 150-200
kms-1 plasma flow) [Soucek . (2008)], which results in scales at least an
order of magnitude beyond the usual ion gyroradii. For most of the events
studied here, this condition seemed appropriate, however, some MM structures
approached this limit. For example, plotted in Figure 13 are several
individual MM structures over approximately 10 seconds. Panels (a-c) portrayed
$|\mathbf{B}|$, $N_{e}$ and a wavelet transform of $\mathbf{B}$. Note in panel
(b), for clarity, the red trace indicated a 2Hz low-pass filter of $N_{e}$. As
expected, $N_{e}$ was anti-correlated with $|\mathbf{B}|$.

Figure 13: Shortened interval from the event on 2021-08-14. Panels (a-c) show
$|B|$, $N_{e}$, and a wavelet of $\mathbf{B}$. The anti-correlation between
$|B|$ and $N_{e}$ is clear when viewed on this timescale.

By measuring the duration of each structure, the spatial scale could be
estimated from the plasma flow since MMs have zero phase speed in the plasma
rest frame. During this interval, $\rho_{p}\sim 75$ km and $L_{mm}=167-276$
km, hence these MM structures were approaching the ion kinetic scales.
Although the unusually slow solar wind raised the quality factor and reduced
the reliability of the data, it is plausible to consider the solar wind speed
was slow as it is expected from the other events. To confirm and strengthen
this result, a MM train was found in MMS data. Using multiple spacecraft,
directly confirms that these MM trains can be smaller than local ion scales.
Moreover, even in this case, the solar wind speed was $<400$ kms-1, which is
consistent with the SolO events. As a result, these cases may test the low-
frequency limit assumption and a fully kinetic MM threshold may be demanded.

For cases when $L_{mm}\gg\rho_{p}$, it has been indicated that finite electron
temperature effects in the long-wavelength limit also modify the instability
threshold [Pantellini Schwartz (1995), Pokhotelov . (2002)]. This occurs due
to the electron pressure gradient that in turn generates an $E_{\parallel}$
[Pantellini Schwartz (1995)], increasing the mirror mode instability threshold
and lowering the growth rate. Nevertheless, it is not anticipated that this
would shed meaningful light on the ambiguous events reported here since the
ion anisotropy was weak for these cases; which, would only contribute to
explaining a lack of MMs during large anisotropies. Another key consideration
was that during CMEs and SIRs, the particle distributions are expected to
deviate from the non-bi-Maxwellian shape due to the existence of
characteristics such as shocks, sheaths, and current sheets. Prior work
[Pokhotelov . (2002)] had sought to address this by understanding the
consequences of arbitrary distribution functions (within the long-wavelength
limit). The consensus from that study was that distributions such as loss
cones and tails from energetic particles can reduce the instability threshold
and increase the growth rate. On the other hand, although the VDFs examined
here did present slight deviations from non-Maxwellianity, there was no
evidence of significant features such as energetic particles and/or supra-
thermal tails. Although these effects cannot be ruled out entirely, it was not
expected to play a considerable role here; but they may become more
consequential in explaining MM growth in additional plasma regimes or solar
wind transients. Note that PAS VDFs are collected every 4 seconds, so the
purpose was to check the underlying distribution properties in relation to the
occurrence of the MM trains. For larger-scale MMs lasting several seconds, it
could be possible to investigate the ion dynamics and particle interactions
with MM structures (e.g. soucek2011).

When $L_{mm}\sim\rho_{p}$ or below, the electron-scale mirror mode threshold
$RMM_{e}=(T_{e\perp}/T_{e\parallel})/(1+1/\beta_{e\perp})>1$ [Pokhotelov .
(2013)] can explain the generation of MMs. This was shown experimentally by
yao2019 who studied such structures upstream of the Earth’s bow shock using
MMS. The authors showed that even though there was no ion temperature
anisotropy, the presence of an electron temperature anisotropy was understood
to provide the sufficient free energy required. But it should be pointed out
that the structures analyzed in that study were 0.1 $\rho_{p}$, which is
smaller than the approximately $2\rho_{p}$ that were calculated for the SolO
cases here. Although it should be noted that the MMS case presented in Figure
10 was significantly less than the $\rho_{p}$. Electron MMs do not apply to
all the events in this investigation, but only in cases when the spatial
scales approach or are below ion scales. One feature consistent with the cases
in this study was the clear anti-correlation with $N_{e}$. Nevertheless, an
investigation into the physics of kinetic MM structures is outside the scope
of this study, but it should be considered for future investigations of these
structures using electron data.

The statistical analysis has revealed several intriguing results. Firstly, all
but one of the 25 events were found when $|\mathbf{V}|<400$ kms-1. A
straightforward explanation is that the median solar wind speed for the data
set that was analyzed was 340 kms-1, thus the probability of finding events
for $|\mathbf{V}|<400$ kms-1 was not so unreasonable. Thus, one explanation
could be statistical. Yet, this does not justify the lack of event detection
when the solar wind speed was faster, since there were data available
according to Figure 12, especially at 1 AU. Some solutions could be discovered
from the various studies that have devoted efforts to understanding the radial
evolution of solar wind parameters (e.g. Khabarova2018,echer2020), and the
inter-dependency of properties during fast and slow solar wind streams. Yet,
this is not readily applied to the current study and will not be explored
further here. The reason is that MM storms did not tend to appear in the
ambient solar wind, but were associated with transients such as SIRs, HCS, and
other field and plasma structures. Hence, the ambient solar wind properties
could be misleading in this regard as they are more applicable to isolated MMs
such as magnetic holes, which are abundantly found in the ambient solar wind.
Thus, this remains an open question, but as SolO collects more data into the
following solar cycle, forthcoming studies will shed more light on this. These
results also imply that the probability of detecting MM storms is higher
closer to the Sun. This could be an indication of the tendency for events to
occur for lower solar wind speeds ($|\mathbf{V_{i}}|$ increases with radial
distance), but the same arguments above are valid and it is problematic to
apply undisturbed solar wind conditions. Thus, future studies could
concentrate on the evolution of solar wind transients and determine if
“younger” SIRs and/or CMEs are more prone to these instabilities. Opportune
radial alignments (e.g. SolO, PSP, BepiColombo, ACE/Wind) may also shed light
on this topic. The final point to make is that MM storms are not common. Just
25 events were identified between 15 April 2020 and 31 August 2021. The
broader implications of this propose that MM storms should not play a
meaningful role in regulating and/or constraining the ambient solar wind
properties. On the other hand, MM storms should be more crucial to solar wind
transients and complicated structures, primarily during high $\beta_{i}$
conditions.

This study also showed that MM trains can also undergo significant deviations
in terms of their amplitudes and frequency. This was especially pronounced in
two examples that were highlighted (see Figures 8 and 9). According to earlier
studies [Soucek . (2008), Génot (2008), Dimmock . (2015)], peaks are
associated with MM unstable plasma ($RMM>1$), whereas dips tend to occur for
marginally stable MM conditions ($0<RMM<1$); but are able to survive the
transition to MM stable plasma. Thus, the interpretation of these events is
that the change in peakness (peaks-dips) is owed to local changes in plasma
conditions that deviate to more marginally MM stable conditions. The change
from peaks to dips was also noted by enriquez_rivera2013. In Figure 8 the
temporal width of the MMs increased from 0.7 seconds to 3.2 seconds across the
event even though the plasma speed remained stable. The time between
individual structures also increased from $<$1 second to $>$1 minute. In the
immediate vicinity, there is a reversal in $B_{n,t}$, an increase of
$|\mathbf{B}|$, and a decrease in $N_{i}$. Thus, $\beta_{i}$ decreases, which
could push the plasma to more mirror stable conditions, explaining the change
from peaks-dips. The difference in frequency is also connected to the above
discussion, where the initial spatial scales are $\sim 2\rho_{p}$, implying
other factors may need to be assessed. Thus, the change of frequency, in this
case, could demonstrate an evolution of electron temperature and/or the move
toward satisfying the long-wavelength limit assumption. russell2009 postulated
that MM storms may evolve as they are carried outward by the solar wind, as
they similarly behave in the magnetosheath when moving towards the
magnetopause. Although deviations in properties seem to take place for
individual events as debated above, there was no clear evidence yet to point
towards a fundamental discrepancy between the properties of these waves at
smaller heliocentric distances compared to those at 1 AU. An important caveat
to consider in this work is the criteria for the automated search, which
analyzed 5 minutes windows. Thus, the search could have missed shorter
interval MM trains that were notably shorter than the window length.

The present study has achieved its goals by shedding significant light on the
properties of MM storms in the solar wind, their dependence/occurrence with
heliocentric distance, and their connection to large-scale transients. The
study has also highlighted the complex nature of MM storm and their occurrence
across a wide variety of plasma structures. Naturally, some open questions
remain, especially when MMs violate the long-wavelength assumption and the
mechanisms responsible for their growth are unclear. With increasing catalogs
of inner-heliospheric observations from SolO, PSP, and BepiColombo, these data
are, and will, be a rich source for advancing understanding of the coupling
between kinetic instabilities and large-scale structures. In addition, closer-
than-before perigees ($\sim$ 0.3 AU) will provide new insights into where and
when such instabilities develop and the importance of the “age” of solar wind
transients.

##  6 Summary & conclusions

The objective of this study was to shed important light on continuous mirror
mode activity in the solar wind, previously called mirror mode storms. The
main motivation was the scarcity of literature on the topic, which the Solar
Orbiter mission is ideally placed to fill. The study has utilized Solar
Orbiter data from 2020-04-15 - 2021-08-31 between heliocentric distances of
0.5-1 AU, resulting in 25 events. Several events were studied in detail
whereas some statistical analysis was presented later. From this work, the
main conclusions can be summarized as follows:

  1. 1.

A statistical search based on magnetic field data only detected MM storms
during moderate-slow solar wind speeds.

  2. 2.

Heliospheric current sheet, interplanetary current sheets, and extended
magnetic field minima appear to be efficient at setting up conditions for MM
growth due to sudden enhancements of $\beta_{i}$.

  3. 3.

MM storms manifest over a range of spatial scales, but in some situations
approach the local ion gyroradii, which challenges the long-wavelength limit
assumption.

  4. 4.

Based on the events considered here, interplanetary shocks were not the
dominant driver of MM storms. However, with increasing solar activity this
could change as more shocks are expected.

  5. 5.

MM storms demonstrate visible evolution in terms of peakness, spatial scale,
and amplitude.

  6. 6.

MM storms typically arise in two categories, the first has a higher frequency
(1-2 Hz) and smaller amplitudes ($<$1 nT) and can appear as peaks. The second
has amplitudes $>$1 nT and frequencies $<$ 1 Hz and seems to appear as dips.

  7. 7.

The typical temporal scales of individual MMs are between 0.5 - 1.5 seconds,
but this can be larger.

  8. 8.

MM storms are not common, and only 25 events were detected between 2020-04-15
and 2021-08-31.

  9. 9.

Due to the low occurrence, MM storms likely do not play a major role in
modifying the ambient solar wind properties. However, their importance in
terms of regulating the plasma should increase during large-scale disturbed
intervals such as SIRs and CMEs.

  10. 10.

There is evidence to suggest that MM storms are more likely to be observed at
smaller heliocentric distances between 0.5-1 AU. However, more events will be
required to provide a definitive confirmation.

  11. 11.

For some events, it was not clear what plasma conditions were responsible. One
interpretation was that finite electron temperatures, kinetic scales, and non-
Maxwellian distribution functions need to be accounted for. Or it could be
that the MMs were generated elsewhere. Another likely possibility was that the
alpha particle population may play a strong role. However, currently, there
are no readily available alpha particle moments to properly assess their role,
which could be addressed in a future study.

###### Acknowledgements.

APD received financial support from the Swedish National Space Agency (Grant
2020-00111) and the EU Horizon 2020 project SHARP: SHocks: structure,
AcceleRation, dissiPation 101004131. Solar Orbiter is a space mission of
international collaboration between ESA and NASA, operated by ESA. 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
Centre National d’Etudes Spatiales (CNES, France), the Centre National de la
Recherche Scientifique (CNRS, France), 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. We thank the entire
MMS team and instrument PIs for the access and use of MMS. Solar Orbiter data
is publicly available at the ESA Solar Orbiter archive
(https://soar.esac.esa.int/soar/). MMS data is freely available from the MMS
science data center (https://lasp.colorado.edu/mms/sdc/public/). The OMNI data
were obtained from the GSFC/SPDF OMNIWeb interface at
https://omniweb.gsfc.nasa.gov.

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