# On the Role of Alfvénic Fluctuations as Mediators of Coherence within
Interplanetary Coronal Mass Ejections: Investigation of Multi-Spacecraft
Measurements at 1 au

Camilla Scolini Institute for the Study of Earth, Oceans, and Space,
University of New Hampshire, Durham, NH, USA  Noé Lugaz Institute for the
Study of Earth, Oceans, and Space, University of New Hampshire, Durham, NH,
USA  Réka M. Winslow Institute for the Study of Earth, Oceans, and Space,
University of New Hampshire, Durham, NH, USA  Charles J. Farrugia Institute
for the Study of Earth, Oceans, and Space, University of New Hampshire,
Durham, NH, USA  Norbert Magyar Centre for mathematical Plasma Astrophysics,
Department of Mathematics, KU Leuven, Leuven, Belgium  Fabio Bacchini Centre
for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven,
Leuven, Belgium  Royal Belgian Institute for Space Aeronomy, Solar-Terrestrial
Centre of Excellence, Uccle, Belgium

(Received July 24, 2023; Revised November 15, 2023; Accepted November 18,
2023)

###### Abstract

Interplanetary coronal mass ejections (ICMEs) are defined as “coherent” if
they are capable of responding to external perturbations in a collective
manner. This implies that information must be able to propagate across ICME
structures, and if this is not the case, single-point in-situ measurements
cannot be considered as indicative of global ICME properties. Here, we
investigate the role of Alfvénic fluctuations (AFs) as mediators of ICME
coherence. We consider multi-point magnetic field and plasma measurements of
10 ICMEs observed by the ACE and Wind spacecraft at 1 au at longitudinal
separations of $0.5^{\circ}-0.7^{\circ}$. For each event, we analyze the
Alfvénicity in terms of the residual energy and cross helicity of
fluctuations, and the coherence in terms of the magnetic correlation between
Wind and ACE. We find that $\sim 65$% and 90% of ICME sheaths and magnetic
ejecta (MEs), respectively, present extended AFs covering at least 20% of the
structure. Cross helicity suggests AFs of solar and interplanetary origin may
co-exist in the ICME population at 1 au. AFs are mainly concentrated
downstream of shocks and in the back of MEs. The magnetic field is poorly
correlated within sheaths, while the correlation decreases from the front to
the back of the MEs for most magnetic field components. AFs are also
associated with lower magnetic field correlations. This suggests either that
ICME coherence is not mediated by Alfvén waves, implying that the coherence
scale may be smaller than previously predicted, or that the magnetic field
correlation is not a measure of coherence.

Solar coronal mass ejections (310) — Solar wind (1534) — Interplanetary
magnetic fields (824) – Alfvén waves (23)

††journal: ApJ

##  1 Introduction

Interplanetary coronal mass ejections (ICMEs; Bothmer & Schwenn, 1998; Cane &
Richardson, 2003; Kilpua et al., 2017) are said to be “coherent” if they are
capable of responding to external perturbations in a collective manner
(Burlaga et al., 1981; Owens et al., 2017). Whether ICMEs behave coherently at
global or only at local scales is a topic of debate (see e.g. Lugaz et al.,
2018; Owens, 2020; Al-Haddad et al., 2022; Scolini et al., 2023) due to its
implications for the global evolution of ICME magnetic structures and the
interpretation of single-point in-situ measurements. Throughout this work, we
use the term “magnetic ejecta” (ME; Winslow et al., 2015) to refer to the
magnetically dominated portion of an ICME, which is identified by an enhanced
magnetic field and low levels of magnetic fluctuations compared to the
preceding and following interplanetary magnetic field. This definition of ME
includes as subset the structures called “magnetic clouds” (MCs; Burlaga et
al., 1981), which require the presence of smoothly rotating magnetic fields,
low density and low temperature in addition to the ME signatures. Further
clarifications on how this choice of terminology applies to the specific
events considered in this study are provided below.

To date, a number of complementary approaches have been pursued to shed light
on the scale of coherence within ICMEs, yielding partial, and sometimes
inconsistent, results. On the one hand, early observations by Burlaga et al.
(1981), for example, presented evidence that ICMEs can exhibit comparable
plasma and magnetic field properties across a longitudinal separation of $\sim
30^{\circ}$ at least. Other multi-spacecraft measurements indicate that some
of the ICME properties may be global, and that a comparable magnetic
configuration can be measured within magnetic ejecta (MEs) by spacecraft at
separations $\geq 30^{\circ}$ (e.g., Cane et al., 1997; Kilpua et al., 2011;
Good & Forsyth, 2016; Lugaz et al., 2022). Evidence in favor of a global
coherence of ICME structures is well represented by the famous cartoon by
Zurbuchen & Richardson (2006) (see Figure 2 therein), of which several
variations have since been proposed (as summarized in Figure 1 by Owens,
2016). On the other hand, based on considerations on the shape, expansion
speeds, and Alfvén speed of ICMEs, Owens et al. (2017) estimated that ICMEs
(and particularly their MEs) may cease to behave as globally coherent
structures at heliocentric distances of $0.2-0.5$ au, and maintain a coherent
scale of around $26^{\circ}$ by the time they reach 1 au. Such an estimate is
intimately linked to the assumption that ICMEs maintain their angular width
during propagation, in such a way that non-radial flows should not be present,
neither away nor towards the ICME axis. Al-Haddad et al. (2022) challenged
such a scenario revealing the presence, at least in two case studies observed
at 1 au, of small non-radial expansion rates consistent with an ICME cross
section more elliptical than implied by the kinematic model assumed by Owens
et al. (2017). If applicable to the general ICME population, such results may
indicate that at 1 au, ICME coherence could only be sustained at even smaller
scales than previously estimated. However, no steady, large non-radial flows
inside MEs were detected over a set of 48 ICMEs considered by Al-Haddad et al.
(2022), which is more consistent either with an ICME cross section that
becomes highly elliptical due to kinematic effects, or with a scenario where
the entire paradigm of MEs described in terms of flux ropes (FR) with an axial
invariance might be too limited. Importantly, Al-Haddad et al. (2022) also
raised the issue of distinguishing between ICME lateral expansion and
deflection from single-point measurements, stressing the need for multi-point
observations at two opposite sides of an ICME to advance our understanding of
ICME coherence from an observational standpoint.

In addition to the scarcity of multi-point ICME observations, estimating the
scale of ICME coherence is further complicated by the fact that coherence is
not a physical observable, and as such, assumptions have to be made when
interpreting observational data with respect to the coherence of ICME
structures. Observationally, the correlation of magnetic field components as
measured by different spacecraft crossing the same ICME through different
trajectories has been commonly assumed as a proxy for ICME coherent behavior
(Matsui et al., 2002; Farrugia et al., 2005; Lugaz et al., 2018; Ala-Lahti et
al., 2020). The investigation by Lugaz et al. (2018) suggested the existence
of two characteristic scales of correlation within ICMEs near 1 au: one
related to the magnetic field components (around $4^{\circ}-7^{\circ}$) and
one for the total magnetic field (around $14^{\circ}-20^{\circ}$). However, so
far limited multi-point ICME observations prevented a direct and systematic
exploration of the parameter space, particularly at scales between $1^{\circ}$
and $10^{\circ}$. Additionally, such observational studies have been so far
only possible near 1 au due to the lack of spacecraft reaching close angular
separations at inner heliocentric distances prior to the current solar cycle.
In an attempt to overcome these observational limitations, in Scolini et al.
(2023) we performed 3-D numerical simulations of ICMEs in the inner
heliosphere which highlighted the role of interactions with other large-scale
structures, such as high speed streams (HSSs) and stream interaction regions
(SIRs), as a primary mechanism acting to decrease the correlation scale of the
magnetic field components within ICMEs. This study revealed how, in such
cases, the correlation is progressively lost by ICMEs during propagation
between 0.1 and 2 au.

Overall, while previous works agree in considering ICMEs, and particularly
MEs, as coherent structures only at scales around or smaller than $\sim
25^{\circ}$ at 1 au, how this depends on the ICME heliocentric distance and
interaction history, remains unassessed from an observational standpoint. At
the same time, the fundamental nature of the relationship between ICME
coherence and the correlation of magnetic field time profiles measured across
different locations within the same ICME remains unexplored, and to be
answered, requires first of all a physical understanding of how information
about the acting external forces is transmitted across ICME structures.

One fundamental question with respect to the coherence of ICMEs relates to the
nature of the information carrier required to propagate information about an
external perturbation across an ICME structure. In fact, the very definition
of magnetic coherence implies that a coherent behavior can only be exhibited
if information about the acting perturbation is able to propagate across an
ICME structure, but the nature of such an information carrier is still an open
question in its own right. In this paper, we explore the possibility that
Alfvén waves (Alfvén, 1942) are the main mediators of coherent behavior across
ICME structures. In the theory of magnetohydrodynamics (MHD), Alfvén waves
consist of low-frequency (i.e. much lower than the ion cyclotron frequency,
which is typically around $10^{-2}-10^{0}$ Hz in the solar wind at 1 au and
inner heliocentric distances; e.g. Bale et al., 2016; Zhao et al., 2021), non-
compressive (shear), in-phase oscillations of plasma and magnetic field lines
perpendicular to the local magnetic field direction, propagating along
magnetic field lines. In the solar wind, Alfvén waves are generally called
Alfvénic fluctuations (AFs). AFs are extensively present in both the solar
atmosphere and the solar wind (see, for example, Belcher et al., 1969; De
Pontieu et al., 2007; Tomczyk et al., 2007; Wang et al., 2012; D’Amicis et
al., 2021) and can be generated through a variety of mechanisms including
magnetic reconnection (e.g. Kigure et al., 2010; Cranmer, 2018) and various
changes in the force-balanced state of a magnetic flux-rope structure (e.g.
Hollweg et al., 1982; Velli & Liewer, 1999; Longcope & Welsch, 2000).
Particularly in the solar wind, AFs are observed to propagate predominantly
along the anti-sunward direction, suggesting AFs of solar origin are weakly
damped in the interplanetary medium and can efficiently propagate up to 1 au
and beyond (Belcher & Davis, 1971; Chen, 2016). Thus, our intuition is that
due to their ubiquitous presence in the solar wind and the broad range of
physical mechanisms able to generate them, AFs are the most prominent
candidates for the propagation of information throughout ICME structures. Such
a hypothesis was also implicitly suggested by Owens et al. (2017); Owens
(2020), but was not followed up by any quantitative verification based on
theoretical nor observational evidence. General discussions on the role of AFs
as propagators of information in other contexts in space plasma can be also
found in classic textbooks, see, e.g., Kivelson & Russell (1995). Only in
recent years have large-amplitude (comparable to the average background
magnetic field) AFs within MEs been first detected in Helios data between 0.3
au and 0.7 au (Marsch et al., 2009; Yao et al., 2010). A handful of studies
(e.g. Yao et al., 2010; Liang et al., 2012) related AFs within MEs to solar
formation mechanisms, particularly in relation to solar prominence eruptions.
Solar prominences are common progenitors of ICMEs (e.g., St. Cyr & Webb, 1991;
Scolini et al., 2018) and routinely exhibit the presence of MHD waves and
oscillations (e.g., Okamoto et al., 2007; Arregui et al., 2018), which
suggests many CMEs may be filled with AFs already in the solar corona. On the
other hand, several studies also highlighted a general scarcity of AFs within
prominence-erupted ICMEs compared to the solar corona, suggesting AFs tend to
quickly dissipate as ICMEs propagate away from the Sun (Marsch et al., 2009;
Yao et al., 2010; Li et al., 2016a, 2017). Another possibility, yet less
explored, is given by interplanetary formation of AFs within MEs via
ubiquitous plasma processes such as magnetic reconnection (e.g. Gosling et
al., 2005) and/or the interaction of plasma structures with velocity shears
(e.g. Bavassano et al., 1978; Roberts et al., 1992).

Based on the above discussion, the goals of this study are twofold: first, we
want to characterize and quantify the Alfvénicity of ICMEs at 1 au. Second, we
want to answer the question of whether AFs are significant mediators of
coherent behavior within ICME structures, in a scenario where a coherent
behavior is measured by the correlation of magnetic field profiles across
different ICME locations. In other words, we want to understand if and how AFs
alter the internal structure of ICMEs at large to “meso” (intermediate between
the ICME size and kinetic) scales. In this first study on the topic, we aim to
answer these questions by uncovering general trends that exist within the ICME
population. Such an approach requires the consideration of a homogeneous set
of ICMEs observed by multiple spacecraft at comparable heliocentric distances
and angular separations, and for which multi-point in-situ plasma and magnetic
field data are available. At the time of writing, such a data set is only
available at 1 au. Therefore, in this study we specifically consider a set of
10 ICMEs observed near Earth by the ACE and Wind spacecraft at longitudinal
separations between 0.5∘ and $0.7^{\circ}$. We investigate the Alfvénicity of
their plasma and magnetic field fluctuations in two regimes of the
fluctuations’ power spectrum: (i) the injection range, covering wavenumbers of
the largest-scale fluctuations, excited by macroscopic dynamics, such as
prominence oscillations in the corona (typically covering frequencies
$<10^{-3}$ Hz for MEs at 1 au; Good et al., 2022), and (ii) the inertial
range, which corresponds to the range of wavenumbers where self-similar
(fluid-like, i.e. MHD) cascades transport the energy injected at the injection
scales towards progressively smaller scales (typically covering frequencies at
$10^{-3}-10^{-2}$ Hz for MEs at 1 au; Good et al., 2022). In this study, we
focus specifically on frequencies between $2.3\times 10^{-5}-3.3\times
10^{-3}$ Hz (corresponding to temporal scales between 12 hours and 5 minutes),
as they cover the injection range and low-frequency end of the inertial range,
which correspond to the large-to-meso ICME scales. Additionally, we
investigate the correlation of magnetic field profiles at 0.5∘ to
$0.7^{\circ}$ of angular separation in order to draw general conclusions
regarding the role of AFs as mediators of ICME coherence and magnetic field
correlation.

The paper is structured as follows. Section 2 provides an overview of the data
sets and methods used to quantify the Alfvéncity and correlation between in-
situ ICME properties at different spacecraft locations. Section 3 presents the
results of the analysis of a set of 10 ICMEs detected at 1 au. In Section 4 we
summarize our results and discuss them in the context of understanding the
relationship between coherence and AFs within ICMEs near 1 au.

##  2 Data and methods

###  2.1 Data

We start our event selection from the list of 35 ICMEs observed by ACE and
Wind at 1 au analyzed by Lugaz et al. (2018). All these ICMEs present MC or
MC-like signatures (i.e. listed as “2” or “1” in Richardson & Cane, 2010,
respectively). Thus, we refer to the ejecta part of these ICMEs as MEs,
whether they are MCs or MC-like. We filter out events observed when the two
spacecraft had longitudinal separation larger than $0.5^{\circ}$
(corresponding to $\sim 200$ Earth radii at 1 au) in order to ensure the
observing spacecraft sampled the ICME structures across sufficiently different
directions (these separations correspond to an Alfvén propagation time of
about 4 hours between the spacecraft). Previous studies established that the
typical ME duration at 1 au ranges between 18 and 26 hours (Gopalswamy et al.,
2015; Nieves-Chinchilla et al., 2018), while small flux ropes (SFRs) can have
duration up to 12 hours (Yu et al., 2014). To ensure sample homogeneity and
prevent contamination, we thus discard all ICMEs that have an ME duration of
less than 12 hours at 1 au, and that lack a clear solar counterpart. Such a
choice was based on the fact that weak and slow interplanetary magnetic flux
ropes such as SFRs may be ascribed to other formation mechanisms than large-
scale solar eruptions (e.g., through streamer blow-outs or interplanetary
reconnection; Moldwin et al., 1995; Cartwright & Moldwin, 2010; Sanchez-Diaz
et al., 2017; Lavraud et al., 2020). Due to their different formation and
propagation histories, they can therefore be expected to yield a different
Alfvénic content at 1 au compared to ICMEs, particularly if AFs are formed
during solar eruptive events such as in relation to prominence eruptions.

This selection criteria results in 10 ICMEs observed at longitudinal
separations between $0.5^{\circ}$ and $0.7^{\circ}$ (corresponding to $\sim
200-300$ Earth radii, or $\sim 0.009-0.013$ au, at 1 au), so that the typical
Alfvén travel time within the ICME between the two spacecraft locations was
typically around or larger than 4 hours. As a result of these selection
criteria, 9 out of the 10 ICMEs selected were propagating fast enough to have
driven a shock and sheath by the time they reached 1 au. For each event, we
analyze the Alfvénicity within the ICMEs using the wavelet analysis described
in Section 2.2.1 below, and the correlation between the magnetic field time
series at Wind and ACE using the methodology described in Section 2.2.2 below.

For each ICME driving a shock, we cross-check our identifications of the ICME
start time with the Heliospheric Shock Database, generated and maintained at
the University of Helsinki (Kilpua et al., 2015, http://ipshocks.fi). For each
event in the list, we determine the ME boundaries as follows. The ME start and
end times are initially selected based on the boundaries listed in the
HELIO4CAST ICMECAT (Möstl et al., 2017, https://helioforecast.space/icmecat)
and Wind ICME (Nieves-Chinchilla et al., 2018,
https://wind.nasa.gov/ICME_catalog/ICME_catalog_viewer.php) catalogs, but are
then adapted visually as follows. First, Wind and ACE magnetic field and
plasma measurements are investigated separately, and ME boundaries are chosen
independently at the two spacecraft. In case of data gaps in the magnetic
field and/or plasma data right where one of the boundaries is expected to
occur (as in the case of ACE plasma data around the ME end boundary for Event
6), features at the other spacecraft are used to guide the selection of the ME
boundaries at the spacecraft affected by the data gaps. As a result, the ME
boundaries are chosen as consistently as possible between ACE and Wind. These
ICME boundaries are used for the rest of the analysis, and are provided in
Table 1.

Throughout our analysis, we use ACE magnetic field data at 16-s cadence taken
by the Magnetic Field Experiment (MAG; Smith et al., 1998), and measurements
of the solar wind plasma and suprathermal electron properties at 64-s cadence
from the Solar Wind Electron Proton Alpha Monitor (SWEPAM; McComas et al.,
1998). At Wind, we use measurements of the magnetic field at 1-min cadence
taken by the Magnetic Field Investigation (MFI; Lepping et al., 1995),
complemented by measurements of the solar wind plasma properties at 3-s
cadence, and of suprathermal electron pitch angle distribution data at 24-s
cadence from the Three-Dimensional Plasma and Energetic Particle Investigation
(3DP; Lin et al., 1995). When appropriate, we compare results obtained from
the Wind/3DP data set against results obtained by considering solar wind
plasma properties measured at 92-s cadence from the Solar Wind Experiment
(SWE; Ogilvie et al., 1995). The magnetic field and velocity data used in this
paper are given in RTN coordinates.

|  | Wind |  |  | ACE |
---|---|---|---|---|---|---
Event no. | ICME start time | ME start time | ME end time | ICME start time | ME start time | ME end time
1 | 2000-10-12 22:33 | 2000-10-13 17:24 | 2000-10-14 19:12 | 2000-10-12 21:45 | 2000-10-13 16:10 | 2000-10-14 17:20
2 | 2001-03-27 18:07 | 2001-03-27 20:00 | 2001-03-28 14:23 | 2001-03-27 17:15 | 2001-03-27 19:00 | 2001-03-28 13:40
3 | 2001-04-11 14:09 | 2001-04-11 22:48 | 2001-04-12 17:58 | 2001-04-11 13:14 | 2001-04-11 22:20 | 2001-04-12 17:25
4 | 2001-04-21 15:29 | 2001-04-22 00:10 | 2001-04-23 01:11 | 2001-04-21 15:06 | 2001-04-21 23:30 | 2001-04-23 00:15
5 | 2001-04-28 05:00 | 2001-04-28 14:50 | 2001-04-29 16:00 | 2001-04-28 04:31 | 2001-04-28 14:00 | 2001-04-29 15:00
6 | 2001-07-10 10:30 | 2001-07-10 10:30 | 2001-07-11 11:00 | 2001-07-10 09:18 | 2001-07-10 09:18 | 2001-07-11 08:35
7 | 2001-12-29 05:17 | 2001-12-30 01:00 | 2001-12-30 19:09 | 2001-12-29 04:47 | 2001-12-30 00:10 | 2001-12-30 18:30
8 | 2002-02-28 05:06 | 2002-02-28 19:11 | 2002-03-01 23:15 | 2002-02-28 04:00 | 2002-02-28 16:55 | 2002-03-01 23:15
9 | 2002-05-18 19:45 | 2002-05-19 03:30 | 2002-05-20 03:34 | 2002-05-18 19:19 | 2002-05-19 03:00 | 2002-05-20 02:50
10 | 2002-05-20 03:35 | 2002-05-20 11:15 | 2002-05-21 21:00 | 2002-05-20 02:57 | 2002-05-20 11:00 | 2002-05-21 20:30
Table 1: Summary of the ICME times at ACE and Wind.

###  2.2 Methods

####  2.2.1 Characterization of Alfvénicity

To identify periods of AFs in and around ICMEs (including MEs, preceding
sheaths, and surrounding solar wind), velocity and magnetic field fluctuations
are explored through continuous wavelet spectrograms of the normalized
residual energy

$\sigma_{r}(k,t)=\frac{E_{v}(k,t)–E_{B}(k,t)}{E_{v}(k,t)+E_{B}(k,t)},$ (1)

where $E_{v}(k,t)$ and $E_{B}(k,t)$ are the sum of the power of the wavelet
transforms (Torrence & Compo, 1998) of the components of the velocity
$\vec{v}(t)$ and magnetic field $\vec{B}(t)$ vectors, respectively (Telloni et
al., 2012, 2013, 2021; Good et al., 2020, 2022), and are functions of time $t$
and of the wavenumber $k$. $\sigma_{r}(k,t)$ measures the imbalance between
the kinetic and magnetic energies and is expected to be close to zero in a
reference frame co-moving with the solar wind due to the equipartition of
magnetic and kinetic energy of AFs. This method enables us to investigate
large periods of data through visual inspection across wide frequency ranges.
Information on the predominant direction of propagation of candidate AF
periods with respect to the local magnetic field direction is derived from the
normalized cross helicity:

$\sigma_{c}(k,t)=\frac{W_{+}(k,t)–W_{–}(k,t)}{W_{+}(k,t)+W_{–}(k,t)},$ (2)

where $W_{\pm}(k,t)$ are the sum of the power of the wavelet transforms of the
components of the Elsässer variables
$\vec{z}_{\pm}(t)=\vec{v}(t)\pm\vec{v}_{A}(t)$, with $\vec{v}_{A}(t)$ being
the Alfvén velocity of the plasma. The Elsässer variables represent a useful
formalism to identify the dominant direction of propagation of AFs along a
background magnetic field. $\sigma_{c}(k,t)$ is expected to be $<0$ ($>0$) for
dominant propagation parallel (anti-parallel) to the local magnetic field
direction, and $\sim 0$ for a balanced propagation along both directions.

For each ICME, we perform the wavelet analysis using the Paul wavelet (due to
its better time localization capability than the Morlet wavelet; Telloni et
al., 2012) and considering a period of $2$ days before the ICME start and $2$
days after the ME end in order to avoid effects related to the cone of
influence (the region of the wavelet spectrum where edge effects become
important) at the edges of the time period of interest. We perform the
analysis on both ACE and Wind data, and for Wind we consider two different
plasma data sets, i.e. from the SWE and 3DP instruments, to check for
instrumental/processing and temporal resolution effects that might affect the
identification of AFs. Before applying the wavelet transforms to magnetic
field and plasma time series, we resample them to a common cadence at both ACE
and Wind. The resampling cadence is dictated by the largest cadence available
across all data sets at both Wind and ACE, which in our case is 92-s based on
Wind/SWE plasma data.

Based on these results, at each time step we integrate $\sigma_{r}(k,t)$ and
$|\sigma_{c}(k,t)|$ across different scales by computing their median values
across scales $k_{i}$ corresponding to periods between 5 min and 12 hours
(corresponding to $2.3\times 10^{-5}-3.3\times 10^{-3}$ Hz, and falling within
the injection range and the low-end of the inertial range of the power
spectrum; see Good et al., 2020). The sample points used to determine these
averages are equally spaced across the linear frequency range. In this way we
obtain time-dependent medians for $\sigma_{r}(k,t)$ and $|\sigma_{c}(k,t)|$
which are purely functions of time $t$. Additionally, we also define an
“Alfvénicity parameter” as

$p_{A}(t)=\mathrm{median}_{k}(|\sigma_{c}(k,t)|)(1-\mathrm{median}_{k}(|\sigma_{r}(k,t)|)),$
(3)

which runs between 0 and 1 and measures the Alfvénicity of the structure at
time $t$. The closer $p_{A}(t)$ is to 1, the stronger the Alfvénic content of
the structure at time $t$. The quantification of Alfvénicity at time $t$
through the Alfvénicity parameter $p_{A}(t)$ assumes that AFs have a
predominant direction of propagation, so that
$\mathrm{median}_{k}(|\sigma_{c}(k,t)|)$ is close to 1.

These three quantities ($\sigma_{r}$, $\sigma_{c}$, $p_{A}$) provide
complementary metrics to evaluate the Alfvénicity of fluctuations within
plasma. While $\sigma_{r}$ enables the characterization of Alfvénicity through
the identification of all AFs (both uni-directional and counter-propagating),
the consideration of $p_{A}(t)$ specifically targets Alfvénic periods with
predominantly uni-directional AFs. Finally, $\sigma_{c}$ identifies uni-
directional AFs while not considering that AFs may actually provide only a
negligible contribution compared to other wave modes. Considering $\sigma_{r}$
and $p_{A}$ (i.e. a combination of $\sigma_{r}$ and $\sigma_{c}$) therefore
guarantees a more accurate representation of the actual contribution of uni-
directional vs counter-propagating AFs within MEs.

####  2.2.2 Correlation of multi-point magnetic field measurements

In order to investigate the relation between a coherent behavior and the
correlation of ICME signatures measured at different locations, we compute the
correlation between the time profiles of the magnetic field strength and
magnetic field components within the MEs measured at Wind and ACE. To do so,
for each event we take the shock time and ME boundaries at Wind as references.
The sheath and ME time series portions at ACE are each shifted and stretched
to match the sheath and ME start and end times at Wind. We then analyze the
correlation in two ways: (1) first, we compute the global correlation of the
magnetic field strength and magnetic field components within the ICME between
Wind and ACE. In this case, both Wind and ACE data sets are rebinned to 30-min
averages following the same approach of Lugaz et al. (2018). The correlation
is then computed separately for sheath and ME periods, for the magnetic field
strength and the three magnetic field components, and is provided in the form
of global Pearson correlation coefficients
$\vec{cc}=(cc_{B},cc_{B_{R}},cc_{B_{T}},cc_{B_{N}})$. This approach enables us
to measure the global synchronicity between measurements taken at Wind and
ACE, and reduces the relation between the two signals to a single value. (2)
To gain insight into how the magnetic field correlation is distributed
throughout the different ICME sub-structures, we further explore the
instantaneous (i.e. time-dependent) Pearson correlation between ACE and Wind
time series as a function of different time scales. We do so in both the
sheath and ME regions by measuring the Pearson correlation between the Wind
and ACE starting from a small portion of the signal, and then repeating the
process along a rolling window until the entire structure is covered. To be
consistent with the time scales explored in the study of Alfvénicity in
Section 2.2.1, we consider time windows $\Delta t_{i}$ equally spaced between
5 minutes and 12 hours with increments of 5 minutes, and integrate the results
across these time scales by computing their median values
($\mathrm{median}_{\Delta t}$) across the various $\Delta t_{i}$ considered.
The correlation of the magnetic field strength and three magnetic field
components across the ME is provided in the form of time-dependent Pearson
correlation coefficients $\vec{cc}(t)=(\mathrm{median}_{\Delta
t}(cc_{B}(t)),\mathrm{median}_{\Delta
t}(cc_{B_{R}}(t)),\mathrm{median}_{\Delta
t}(cc_{B_{T}}(t)),\mathrm{median}_{\Delta t}(cc_{B_{N}}(t)))$.

####  2.2.3 Superposed epoch analysis of Alfvénicity and magnetic field
correlation

To determine the general profiles of Alfvénicity and time-dependent magnetic
field correlation observed within ICME sheaths and MEs at 1 au, we make use of
the superposed epoch analysis (SEA; Chree, 1913) technique. This technique
allows the superposition of the time profile for a given quantity observed for
different events, and the calculation of its averaged time profile. In the
case of structures with multiple well-defined boundaries such as ICMEs (i.e.,
shock time, ME start, ME end), the time series are normalized in time for each
sub-structure. Such three-bound SEAs have previously been performed to
determine the average magnetic field and plasma profiles of ICMEs (e.g.,
Masías-Meza et al., 2016; Regnault et al., 2020; Janvier et al., 2021) and the
average Alfvénicity profile at inertial scales in terms of $\sigma_{r}(t)$ and
$\sigma_{c}(t)$ (e.g., Good et al., 2022), but to the best of our knowledge,
they have never been applied to multi-point correlation profiles within ICMEs.

In this work, we investigate the sheath and ME profiles of $\sigma_{r}(t)$,
$|\sigma_{c}(t)|$, $p_{A}(t)$, and of $\vec{cc}(t)$ between ACE and Wind using
a three-bound SEA with the sheath start, ME start, and ME end as reference
times. The normalized time unit is set to be between 0 and 1 for the sheath
region. Then, an average scaling factor is calculated as the ratio of the
average ME duration to the average sheath duration, across all events
considered. This average scaling factor is 2.18, which is used to determine
the duration of the normalized time for the SEA ME profile. As a result, the
normalized time runs from 0 to 1 for the sheath, and from 1 to 3.18 for MEs.
From the normalized time series, the $\sigma_{r}(t)$, $|\sigma_{c}(t)|$,
$p_{A}(t)$, and $\vec{cc}(t)$ data for each event are averaged into 22 bins
within the sheath region, and 50 bins within the ME region, corresponding to
bins of about 30 minutes in both the sheath and ME regions. We set bins of 30
minutes to ensure a sufficient resolution to resolve the temporal variation of
the various SEA quantities across both sheaths and MEs, similarly to Good et
al. (2022). The mean and median values for each bin across all events are then
calculated in order to build average profiles of $\sigma_{r}(t)$,
$|\sigma_{c}(t)|$, $p_{A}(t)$, and $\vec{cc}(t)$ within ICMEs at 1 au when
observed at longitudinal separations of $0.5^{\circ}-0.7^{\circ}$.

##  3 Results

###  3.1 Average Alfvénic Content

Figure 1:  Comparison between Wind and ACE observations for an example event
(event no. 7 in Table 1). (a): Wind/MFI magnetic field and plasma data from
Wind/3DP. (b): Wind/MFI magnetic field and plasma data from Wind/SWE. (c):
ACE/MAG magnetic field and plasma data from ACE/SWEPAM. From top to bottom:
magnetic field components; radial velocity; proton density; magnetic, thermal,
and total pressures; $\sigma_{r}$; $\sigma_{c}$; median of $|\sigma_{r}|$ and
$|\sigma_{c}|$ across scales $k_{i}$ between 5 min and 12 hours. The vertical
magenta lines indicated the ICME shock time and the ME start and end times.

As a starting point, we aim to characterize how many ICMEs, and what fraction
of their sheath and ME exhibit a high Alfvénicity at Wind and ACE.

We start by evaluating the average Alfvénicity of ICMEs by computing the mean
$\sigma_{r}$, $|\sigma_{c}|$, and $p_{A}$ and their standard error (SE) across
the different events for the Wind/3DP, Wind/SWE, and ACE data sets. Results
are provided in Table 2. We observe that the results from the different data
sets are consistent within their estimated uncertainties, so overall, we find
a good agreement across different instruments and spacecraft near 1 au.
Differences among the different data sets can be traced back to instrumental
and processing effects entering the measurement and extraction of ion moments
(primarily the proton velocity and density in this case) which enter the
calculation of the plasma Alfvénicity (e.g. King & Papitashvili, 2005). The
comparison between Wind/3DP, Wind/SWE and ACE data for an example event is
provided in Figure 1. Despite having resampled all data sets to a common
resolution, and despite our consideration of time scales larger than 5 min in
this study, we point out that the different (intrinsic) time resolution
between Wind/3DP and Wind/SWE measurements as well as data gaps that may be
present in one of the two data sets may additionally contribute to slight
differences in the resulting Alfvénicity calculation.

Good et al. (2022) performed a similar analysis of sheaths and MEs for the
frequency range $10^{-3}-10^{-2}$ Hz (equivalent to wave periods of 16.7 –
1.67 minutes, falling in the inertial range of the power spectrum; Good et
al., 2020), reporting a mean $\sigma_{r}$ of $-0.36$ for MEs ($-0.35$ across
sheaths) and a mean $\sigma_{c}$ (corrected for the sector magnetic polarity,
so approximately corresponding to $|\sigma_{c}|$) of 0.18 for MEs (0.24 for
sheaths). In the solar wind, Chen et al. (2013) found a mean
$\sigma_{r}=-0.19$ and a mean $\sigma_{c}=0.40$. The negative values found for
$\sigma_{r}$ in this work is therefore consistent with previous works in
finding that ME and sheath fluctuations exhibit a higher deviation from energy
equipartition (in favor of magnetic fluctuations) compared to the ambient
solar wind. The higher $|\sigma_{c}|$ retrieved in both sheaths and MEs
compared to Good et al. (2022) also shows that Alfvénic fluctuations within
ICMEs tend to be less balanced in their propagation direction at injection
scales compared to inertial scales investigated by Good et al. (2022). We
chose to consider $|\sigma_{c}|$ in order to compare ICME events that have
positive and negative $\sigma_{c}$ values. However, $|\sigma_{c}|$ only allows
to distinguish between AFs that are uni-directional (high $|\sigma_{c}|$) vs
counter-propagating (low $|\sigma_{c}|$), but does not allow us to draw
conclusions on the specific direction of propagation of AFs with respect to
the magnetic field background (parallel for $\sigma_{c}<0$ and anti-parallel
for $\sigma_{c}>0$). The usefulness of $\sigma_{c}$ to infer information on
the AF origin and propagation for individual ICMEs will be demonstrated in an
upcoming study currently in preparation.

|  | Wind/3DP | Wind/SWE | ACE
---|---|---|---|---
Sheath: | $\sigma_{r}\pm\mathrm{SE}_{\sigma_{r}}$ | $-0.35\pm 0.04$ | $-0.22\pm 0.04$ | $-0.30\pm 0.03$
| $|\sigma_{c}|\pm\mathrm{SE}_{|\sigma_{c}|}$ | $0.50\pm 0.03$ | $0.35\pm 0.04$ | $0.44\pm 0.03$
| $p_{A}\pm\mathrm{SE}_{p_{A}}$ | $0.31\pm 0.02$ | $0.23\pm 0.02$ | $0.29\pm 0.02$
ME: | $\sigma_{r}\pm\mathrm{SE}_{\sigma_{r}}$ | $-0.32\pm 0.04$ | $-0.23\pm 0.03$ | $-0.25\pm 0.03$
| $|\sigma_{c}|\pm\mathrm{SE}_{|\sigma_{c}|}$ | $0.45\pm 0.03$ | $0.34\pm 0.03$ | $0.40\pm 0.02$
| $p_{A}\pm\mathrm{SE}_{p_{A}}$ | $0.30\pm 0.03$ | $0.24\pm 0.03$ | $0.27\pm 0.02$
Table 2: Mean $\sigma_{r}$, $|\sigma_{c}|$, and $p_{A}$ across the sheaths and
MEs in our sample, for different data sets.

Next, we aim to establish to what extent ICMEs are Alfvénic. Answering this
question requires a formal definition for the identification of AFs,
particularly in terms of threshold values for $\sigma_{r}$, $\sigma_{c}$, and
$p_{A}$, in order to discriminate between (highly) Alfvénic and non-Alfvénic
periods. Since AFs show $\sigma_{r}\sim 0$ and uni-directional AFs are marked
by $|\sigma_{c}|\sim 1$, strong AFs exhibiting a predominant direction of
propagation are expected to be associated with $p_{A}\sim 1$. Because the
thresholds to distinguish between Alfvénic and non-Alfvénic plasma are
unknown, we explore different thresholds for $\sigma_{r}$, $\sigma_{c}$, and
$p_{A}$, and evaluate how many MEs meet each of their respective threshold
conditions. Specifically, we consider different threshold values
$\sigma_{r}^{*}$, $\sigma_{c}^{*}$, and $p_{A}^{*}$ and evaluate how many
events fulfill $|\sigma_{r}|$ lower than (or $|\sigma_{c}|,\,\,p_{A}$ greater
than) each of these thresholds. We compute the number of MEs satisfying these
conditions for different fractions of the ME duration, i.e. from 10% to 100%.
Our results are shown in Figure 2 for both sheaths (panels (a) to (c)) and MEs
(panels (d) to (f)) for a combination of Wind/SWE and ACE observations. As
highlighted above, the results from Wind/3DP (not shown) are slightly
different but they do agree with those from Wind/SWE and ACE within the
uncertainties.

Figure 2:  Alfvénic content of sheaths and MEs for different threshold values
$\sigma_{r}^{*}$, $\sigma_{c}^{*}$, and $p_{A}^{*}$. (a), (b), (c): combined
results for sheaths from ACE and Wind/SWE. (d), (e), (f): combined results for
MEs from ACE and Wind/SWE. The red dashed lines show the mean values reported
in Table 2.

From Figure 2 (d), we observe that the minimum value threshold satisfied by
all MEs is $|\sigma_{r}|\leq 0.2$, but this happens in all events for only 10%
of the ME duration. 70% of the ME observations (8 at Wind, and 6 at ACE)
satisfy the condition $|\sigma_{r}|\leq 0.2$ for at least 30% of the ME
duration. 90% of them (9 at Wind, and 9 at ACE) exhibit $|\sigma_{r}|\leq 0.2$
for at least 20% of the ME duration. Imposing a stricter Alfvénicity
condition, we find that 65% of the ME observations (8 at Wind and 4 at ACE)
present levels of $|\sigma_{r}|\leq 0.1$ for at least 20% of the ME duration.
These results prove that the majority of ICMEs considered exhibit highly
Alfvénic conditions for a significant portion of their MEs, indicating AFs
within MEs at 1 au may be more common than initially estimated (e.g. by Marsch
et al., 2009; Yao et al., 2010). Sheaths are less Alfvénic than MEs (Figure 2
(a)), as indicated by the slightly higher mean $|\sigma_{r}|$ and by the lower
number of events (6 for Wind, and 3 for ACE) exhibiting AFs ($|\sigma_{r}|\leq
0.1$) for more than 20% of the total duration.

Results for the normalized cross helicity (Figure 2 (e)) show that the maximum
value threshold satisfied by all MEs is $|\sigma_{c}|\geq 0.4$, but this is
sustained in all events for only 10% of the ME duration. Periods of
$|\sigma_{c}|\geq 0.5$ are observed in 90% of the cases, with a variable
duration between 10% and 50% of the ME. For comparison with the $|\sigma_{r}|$
signatures discussed above, we focus our attention to the maximum
$|\sigma_{c}|$ found in most MEs for at least 20% of their duration: this is
measured to be $|\sigma_{c}|\geq 0.5$, and it applies to 55% of the MEs. The
extent of the white areas in Figure 2 also provide evidence that while AFs are
rather common in MEs (panel (d)), they often occur without a predominant
direction of propagation (as indicated by the smaller white area in panels (e)
and (f)). Similar conclusions hold for sheaths. However, they typically
feature a higher $|\sigma_{c}|$ than MEs, both from an average standpoint and
in terms of duration, indicating that slightly more unidirectional AFs are
present in sheaths.

We combine signatures of low residual energy and high cross helicity together
into the Alfvénicity parameter in Figure 2 (f). We observe that a smaller
(though considerable) number of MEs satisfy the threshold condition $p_{A}\geq
p_{A}^{*}$ than for the residual energy and cross helicity taken
independently, as indicated by the smaller white area in panel (f) compared to
panels (d) and (e). We can further quantify the contribution from uni-
directional vs. counter-propagating AFs by considering that for
$\sigma_{r}^{*}=0.1$ and $\sigma_{c}^{*}=0.5$ (corresponding to the limit
thresholds holding for at least 20% of the ME duration in most of the events),
we expect $p_{A}^{*}=\sigma_{c}^{*}(1-\sigma_{r}^{*})=0.45$. We obtain that
30% to 40% of MEs exhibit $p_{A}\geq 0.45$ for at least 20% of their duration,
compared to 65% for $|\sigma_{r}|\leq 0.1$, and 55% for $|\sigma_{c}|\geq
0.5$. Based on these fractions, we estimate that between two thirds and one
half of all AFs are uni-directional, while one third to one half are likely
counter-propagating. The lower detection retrieved from the consideration of
$p_{A}$ over $\sigma_{c}$ also implies that a significant fraction of all
Alfvénic periods identified through the cross helicity calculation may not
actually be Alfvénic. Sheaths tend to present slightly higher values of
$p_{A}$ than MEs, but just as for MEs, this parameter shows that only a small
fraction of AFs within sheaths propagate in a predominant direction within
truly Alfvénic periods. We conclude that while the cross helicity may be used
to reliably define Alfvénicity in the solar wind (e.g. Stansby et al., 2019),
it falls short in accurately representing Alfvénicity in ICME sheaths and MEs.
In contrast, the Alfvénicity parameter enables a more accurate representation
of the actual contribution of uni-directional AFs within sheaths and MEs.

To the best of our knowledge, the only statistical survey making a
quantitative estimation of the frequency and duration of AFs within MEs was
performed by Li et al. (2016b). The authors investigated 33 MEs observed by
Voyager 2 between 1 and 6 au, and in contrast to the previous literature
generally identifying a scarcity of AFs within MEs, Li et al. (2016b) also
reported abundant AFs within MEs as we do in this study. Specifically, the
authors considered scales from $5\times 10^{-4}$ Hz to $10^{-2}$ Hz
(corresponding to 33 to 1.7 minutes) and reported a highly Alfvénic content in
about 90% of the MEs investigated, and found AFs were present for about 17% of
the ME duration at 1.5 au. The fraction was also found to decrease with
heliocentric distance, and could be back-extrapolated to about 20% at 1 au
assuming a linear decay with heliocentric distance. Their estimate at 1 au,
although retrieved from a different methodology (based on the Walén test; see
Li et al., 2016a), is very consistent with the estimates reported from our
study for a $|\sigma_{r}|\leq 0.2$ threshold, and makes us confident of the
robustness of our results. Li et al. (2016b) also found the duration of AFs
within MEs to reduce to 4% near 6 au. While the results by Li et al. (2016b)
indirectly support a solar origin and interplanetary dissipation of AFs within
ICMEs during propagation through larger heliocentric distances, the actual
origin of AFs within ICMEs remains debated. A definitive conclusion remains
difficult to achieve without contextualization of single-point observations
with respect to the solar wind conditions encountered by individual ICMEs
during propagation, and of the solar environment affecting their early
evolution.

###  3.2 Superposed Epoch Analyses of Alfvénicity and Magnetic Field
Correlation

Figure 3: Results of the SEA analysis. Panels (a), (b), and (c) show the
profiles for the median $\sigma_{r}$, $|\sigma_{c}|$, and $p_{A}$ profiles for
sheaths and MEs at ACE and Wind/SWE. Panels (d), (e), (f), and (g) show the
profiles for the correlation of $B$, $B_{R}$, $B_{T}$, and $B_{N}$ between ACE
and Wind for sheaths and MEs. The mean and median profiles are given by the
black and red lines, respectively, with associated SEs reported as shaded
areas. Single bins spanning 6 hours of solar wind before and after the ICMEs
are also shown. Vertical dashed lines indicate the shock, start and end of the
ME profile.

Next, we are interested in exploring the temporal distribution of Alfvénicity
and magnetic field correlation within sheaths and MEs at 1 au. To do so, we
perform SEAs of the median $\sigma_{r}(t)$, $|\sigma_{c}(t)|$, and $p_{A}(t)$
time profiles at scales $k_{i}$ corresponding to time scales between 5 minutes
and 12 hours. The combined results from ACE and Wind/3DP observations are
shown in Figure 3 (panels (a) to (c)). As from the inspection of global
Alfvénicity metrics discussed above, here we find slightly negative
$\sigma_{r}(t)$ values across both sheaths and MEs, which are consistent with
Good et al. (2022) in indicating an excess of magnetic energy in the
fluctuations within ICMEs. The temporal profile of $\sigma_{r}(t)$ also
indicates that the Alfvénicity is higher immediately downstream of the shock
and rapidly decreases throughout the sheath, suggesting Alfvénic fluctuations
formed in the shock downstream region may efficiently decay into compressive
modes or undergo damping phenomena as they propagate through highly turbulent
sheaths (see, e.g., Ala-Lahti et al., 2019; Farrugia et al., 2020).
Conversely, $|\sigma_{c}(t)|$ and $p_{A}(t)$ are larger in the first half of
the sheath, suggesting that when present, AFs near sheath fronts tend to
propagate along preferential directions. Throughout the ME, $\sigma_{r}(t)$
shows an increase towards zero: 7 out of 10 MEs exhibit a more negative
average $\sigma_{r}$ in the ME first half than in the second half (on average
more negative by $-0.09$ at Wind and by $-0.12$ at ACE). This suggests that
AFs are preferentially located within ME backs than ME fronts.
$|\sigma_{c}(t)|$ and $p_{A}(t)$ present irregular behaviors without a clear
increasing or decreasing trend within MEs. Their values exhibit minimal
differences between the ME first and second half: on average,
$|\sigma_{c}(t)|$ is only 0.01 larger in the ME first half, while $p_{A}(t)$
is equal in the ME first and second half at both ACE and Wind. This implies
that the directionality of AFs within MEs is not related to their location
within ME structures. We also note that the SEA profiles of all the above
parameters present a high temporal variability (both within sheaths and MEs),
underlying a high temporal variability within individual event profiles.

|  | Global (30-min resample) | Global (92-s resample) | Time-dependent
---|---|---|---|---
Sheath: | $cc_{B}\pm\mathrm{SE}_{cc_{B}}$ | $0.56\pm 0.09$ | $0.44\pm 0.08$ | $0.31\pm 0.06$
| $cc_{BR}\pm\mathrm{SE}_{cc_{BR}}$ | $0.33\pm 0.11$ | $0.19\pm 0.05$ | $0.08\pm 0.05$
| $cc_{BT}\pm\mathrm{SE}_{cc_{BT}}$ | $0.64\pm 0.06$ | $0.38\pm 0.11$ | $0.18\pm 0.09$
| $cc_{BN}\pm\mathrm{SE}_{cc_{BN}}$ | $0.50\pm 0.08$ | $0.24\pm 0.07$ | $0.13\pm 0.04$
ME: | $cc_{B}\pm\mathrm{SE}_{cc_{B}}$ | $0.90\pm 0.04$ | $0.88\pm 0.05$ | $0.56\pm 0.08$
| $cc_{BR}\pm\mathrm{SE}_{cc_{BR}}$ | $0.66\pm 0.07$ | $0.59\pm 0.08$ | $0.34\pm 0.06$
| $cc_{BT}\pm\mathrm{SE}_{cc_{BT}}$ | $0.84\pm 0.04$ | $0.79\pm 0.05$ | $0.43\pm 0.07$
| $cc_{BN}\pm\mathrm{SE}_{cc_{BN}}$ | $0.69\pm 0.07$ | $0.64\pm 0.07$ | $0.32\pm 0.04$
Table 3: Mean magnetic field correlations between Wind and ACE for the sheaths
and MEs in our sample, for time scales between 5 min and 12 hours.

Second, we perform SEAs of the time profile of the correlation for each
component of
$\vec{cc}(t)=(cc_{B}(t),cc_{B_{R}}(t),cc_{B_{T}}(t),cc_{B_{N}}(t))$ between
Wind and ACE for the ICMEs in our study. Figure 3 (panels (d) to (g)) shows
the time-dependent distribution of correlation throughout the ICME sub-
structures. We observe that the correlation in all magnetic field components
tends to be higher near the ME front, while it is typically lower near the
back. This is particularly evident for the total magnetic field, and for the
$B_{R}$ and $B_{N}$ components. $B_{T}$ tends to have a more irregular profile
without a clear decreasing or increasing trend throughout the ME. We can
quantify this for individual events by comparing the average correlation in
the first quarter and last quarter of each ME. We find that indeed, higher
correlations are found at the ME front for 8 events in $B$ and $B_{R}$, 6
events in $B_{N}$, and only 3 events in $B_{T}$. On average, the ME front
correlation is higher than the correlation in the ME back by 0.30, 0.21, 0.13,
and 0.03 for $B$, $B_{R}$, $B_{N}$, and $B_{T}$, respectively. Because
different MEs may have different orientations of their flux rope axis, we also
apply the minimum variance analysis (MVA; Sonnerup & Scheible, 1998) technique
to each ME and repeat the analysis after having projected the magnetic field
signatures of each event to that event’s MVA frame. In this frame,
approximately corresponding to the frame of the flux rope, the magnetic field
components are projected in the $min$, $int$, and $max$ directions
corresponding to the directions of minimum, intermediate, and maximum
variance. For a flux rope structure, the $int$ direction corresponds to the
direction of its magnetic axis, $max$ corresponds to the poloidal direction,
and $min$ completes the right-handed triad. In terms of correlations (Figure
4), the MVA frames reveal that the largest correlation is present in $B_{max}$
(panel (d)), followed by $B_{int}$ (panel (c)) and $B_{min}$ (panel (b)). This
trend is expected given that $B_{min}$ is likely to be the most sensitive
component to the specific crossing of the spacecraft with respect to the flux
rope axis, while $B_{max}$ is dominated by a large-scale bipolar signature
that maximizes the correlation at different spacecraft over smaller-scale
fluctuations (see, e.g., Figure 2 in DiBraccio et al., 2015). $B_{int}$ is
mainly uni-polar and presents an intermediate correlation between $B_{min}$
and $B_{max}$. Visually, in the MVA frame the only component exhibiting a
clear difference between the ME front and back is $B_{max}$. As done in the
RTN frame, we can therefore evaluate if this result holds for individual
events by comparing the average correlation in the first quarter and last
quarter of each ME. In this case we find that higher correlations are found at
the ME front for 6 events in $B_{max}$ and $B_{min}$, and 5 events in
$B_{int}$. On average, the ME-front correlation is higher than the ME-back one
by 0.24, 0.10, $-$0.05 for $B_{max}$, $B_{min}$, and $B_{int}$, respectively.

Sheaths appear less correlated than MEs but more correlated than the preceding
and following solar wind in all magnetic field components, confirming the
relative correlation scales identified by previous studies (see Wicks et al.,
2010; Lugaz et al., 2018; Ala-Lahti et al., 2020).

For completeness, Table 3 reports the average (mean) correlations obtained for
sheaths and MEs observed at longitudinal separations of
$0.5^{\circ}-0.7^{\circ}$ using the global (as in Lugaz et al., 2018) and the
time-dependent correlation approaches described in Section 2.2.2. The results
between the two methods are significantly different, with the time-dependent
method consistently providing lower average correlation values than the global
method. We argue that one reason behind this result is that the global method
assumes a 30-min rebinning of the data, while the time-dependent one considers
scales between 5 min and 12 hours. Repeating the global calculation using a
rebinning of 92-s instead of 30-min, we obtain more similar results between
the global and time-dependent calculations, especially within the sheath. In
conclusion, while the time-dependent method represents a powerful and under-
utilized tool to investigate the time evolution of ICME magnetic field
correlations, its results can be significantly different from that of the more
commonly-used global method. Therefore, in the perspective of future studies,
it is critical to interpret the ICME magnetic field correlation coefficients
with care and to only compare those obtained from similar methods.

Figure 4: SEA of the correlation of the ME magnetic field components,
calculated after projecting each event into its MVA frame. Panels (a), (b),
(c), and (d) show the profiles for the correlation of $B$, $B_{R}$, $B_{T}$,
and $B_{N}$ between ACE and Wind. The mean and median profiles are given by
the black and red lines, respectively, with associated SEs reported as shaded
areas. Here the normalized time runs from 0 (start of the ME) to 1 (end of
ME).

###  3.3 Correlation between Alfvénicity and Magnetic Field Correlation

|  | Wind/SWE | ACE
---|---|---|---
Sheath: |  $cc_{B}$ vs $\sigma_{r}$ |  $-0.84$ (p$<$0.05) |  $-0.79$ (p$<$0.05)
|  $cc_{B_{R}}$ vs $\sigma_{r}$ |  $-0.14$ (p=0.54) |  $-0.19$ (p=0.40)
|  $cc_{B_{T}}$ vs $\sigma_{r}$ |  $-0.73$ (p$<$0.05) |  $-0.69$ (p$<$0.05)
|  $cc_{B_{N}}$ vs $\sigma_{r}$ |  $-0.66$ (p$<$0.05) |  $-0.65$ (p$<$0.05)
ME: |  $cc_{B}$ vs $\sigma_{r}$ |  $-0.35$ (p$<$0.05) |  $-0.22$ (p=0.12)
|  $cc_{B_{R}}$ vs $\sigma_{r}$ |  $-0.42$ (p$<$0.05) |  $-0.21$ (p=0.14)
|  $cc_{B_{T}}$ vs $\sigma_{r}$ |  $-0.12$ (p=0.41) |  $-0.13$ (p=0.36)
|  $cc_{B_{N}}$ vs $\sigma_{r}$ |  $-0.13$ (p=0.37) |  $\,\,\,\,0.10$ (p=0.47)
|  $cc_{Bmin}$ vs $\sigma_{r}$ |  $-0.51$ (p$<$0.05) |  $-0.51$ (p$<$0.05)
|  $cc_{Bint}$ vs $\sigma_{r}$ |  $\,\,\,\,0.17$ (p=0.23) |  $\,\,\,\,0.06$ (p=0.67)
|  $cc_{Bmax}$ vs $\sigma_{r}$ |  $-0.33$ (p$<$0.05) |  $-0.12$ (p=0.41)
Table 4: Mean correlation between SEAs of $\sigma_{r}$ and the time-dependent
magnetic field correlations at Wind and ACE for the sheaths and MEs in our
sample, for time scales between 5 min and 12 hours.

After having characterized the Alfvénicity and magnetic field correlation
independently of each other, we want to explore whether these two
characteristics of ICMEs are correlated with each other and can provide
insight into the role of AFs as mediators of coherent behavior across ICME
structures. For this purpose, we calculate the correlation between
$\vec{cc}(t)$ and $\sigma_{r}(t)$ obtained from the SEA profiles in Figure 3
within sheaths and MEs, and report the average values in Table 4.

We find that the correlation of the magnetic field profiles between the two
spacecraft is weakly anti-correlated to the Alfvénicity of the MEs measured at
each spacecraft (in terms of $\sigma_{r}$). The anti-correlation for $B$ is
$-0.35$ and $-0.22$ for Wind/SWE and ACE, respectively. For the magnetic field
components, the anti-correlation ranges between $-0.12$ ($B_{T}$) and $-0.42$
($B_{R}$) at Wind/SWE, and between $0.10$ ($B_{N}$) and $-0.21$ ($B_{R}$) at
ACE. However, as reported in Table 4, some of these correlations are
associated with p-values larger than 5%, indicating we cannot reject the null
hypothesis that the Alfvénicity and magnetic field correlation are not
correlated. The most reliable (anti-)correlations are those for $B$ and
$B_{R}$, which are associated with p-values smaller than 5% at Wind (while at
ACE they present p-values around 10–15%). When comparing the Alfvénicity and
the magnetic field correlations in the MVA frame, we observe a moderate anti-
correlation in $B_{min}$ ($-0.51$ with p-value smaller than 5% at both
Wind/SWE and ACE), corresponding to the direction of minimum variance, and a
weak anti-correlation for $B_{max}$ (from $-0.33$ with p$<5$%, to $-0.12$ with
p$=41$% at Wind/SWE and ACE, respectively), corresponding to the direction of
maximum variance. No statistically significant correlation/anti-correlation is
found for $B_{int}$ ($0.17$ and $0.06$ with p-value larger than 5% for
Wind/SWE and ACE, respectively), corresponding to the direction of the flux
rope axis. Because the magnetic field component aligned with the direction of
propagation of AFs is expected to be the least affected by fluctuations, such
a picture may suggest that AFs are globally propagating primarily along the
flux rope axis (along $B_{int}$, which typically constitutes the dominant
magnetic field component within ICME flux ropes; see e.g. Hu et al., 2015;
Lanabere et al., 2020, 2022), and manifest the larger anti-correlation in the
other two magnetic field components ($B_{min}$ and $B_{max}$).

The anti-correlation is strong within sheaths (in terms of $\sigma_{r}$),
where it ranges from $-0.84$ to $-0.14$ at ACE and $-0.79$ to $-0.19$ at
Wind/SWE (results are similar for Wind/3DP). All correlations except the
weakest ones associated with $B_{R}$ have an associated p-value that is
smaller than 5%, and can be thus considered statistically significant (Table
4). The stronger anti-correlation found within sheaths compared to MEs
suggests that AFs may generate magnetic field differences at smaller scales
within sheaths, while within MEs, the differences may affect larger scales.
This arises from the consideration that fluctuations observed at different
spacecraft may have different properties and be measured during different
phases of their oscillation, and the smaller their scale and the larger the
separation between the spacecraft measurements, the less correlation in the
magnetic field signatures is observed at different spacecraft. Another factor
likely to increase the anti-correlation relates to the mechanism of formation
of ICME sheaths (Kilpua et al., 2017). While the front part of sheaths is
typically composed of shocked solar wind material, inner layers can be
composed of preexisting compressed material of either coronal or
interplanetary origin (Vourlidas et al., 2013; Lugaz et al., 2020), or even
more coherent material originally part of the CME erupted structure which was
later eroded, e.g., via magnetic reconnection processes at the ME front (e.g.,
Dasso et al., 2006; Ruffenach et al., 2012, 2015). At least in the case of
ICME sheaths, the mixing of material of various origin (presenting different
intrinsic Alfvénicity and correlation scales) likely contributes to the
enhancement of the anti-correlation between the Alfvénicity and correlation of
magnetic field components. Whether the strength of this anti-correlation can
be used to infer information about the formation history of individual ICME
sheaths remains unclear, but is certainly worth to be explored in future
studies.

It is also noteworthy that these anti-correlations apply to the average sheath
and ME profiles determined from the consideration of all 20 ICMEs profiles
measured at ACE and Wind and analyzed through the SEA technique. Conversely,
such a behavior is not found when taking the average of the correlations
obtained from individual profiles contributing to the SEA (where we obtain
correlations ranging between $0.00$ and $-0.18$ ($0.01$ to $-0.15$) for
sheaths, and $-0.06$ to $-0.20$ ($0.13$ to $0.00$) for MEs at Wind/SWE (ACE);
results are similar for Wind/3DP). In this respect, the SEA profiles reveal
trends that are not immediately visible in individual events. The reason
behind this is that individual events alternate periods of anti-correlation
between $\sigma_{r}$ and the magnetic field correlation, with periods where
their (anti-)correlation is not well determined. Additionally, the SEA
profiles for $\sigma_{r}$ shown in Figure 3 (a) shows a high temporal
variability, indicating that individual events also present high temporal
variability of this quantity, which likely makes the correlation with the
(smoother) profiles in Figure 3 (right) less evident when events are
considered individually.

As discussed in Section 3.1, highly Alfvénic periods are found in most MEs but
typically cover only 20%–30% of the ME duration. The anti-correlation between
the Alfvénicity and the correlation of the magnetic field from SEA profiles
suggests AFs may contribute to reduce the correlation scale within sheaths and
MEs, but that other factors may play a role as well. This appears particularly
the case for MEs, which exhibit less clear anti-correlation trends than
sheaths.

The slight preference of AFs for ME back regions (Figure 3 (a)), and the fact
that lower magnetic field correlations are also found in that region (Figure 3
(right panels)) may also be explained in terms of AFs being formed, in most
cases, through interaction of MEs with the following solar wind (see for
example the study by Dhamane et al., 2023). Conversely, the interaction with
the preceding solar wind (even if highly Alfvénic) may not lead to an increase
in the Alfvénicity within MEs due to the presence of the sheath, which may act
to protect MEs from the propagation and formation of AFs (this scenario is
consistent with the case presented by Farrugia et al., 2020). Specifically
with respect to the Alfvénicity time profile, every event was quite unique
when considered alone. Nevertheless, we would like to mention that Event 6
(the only one in our set that did not drive a preceding shock and sheath)
presented a relatively high Alfvénicity ($|\sigma_{r}|\sim 0.3-0.4$) at the ME
front, and a lower Alfvénicity at the ME back ($|\sigma_{r}|>0.5$) in ACE
data. Such trend, however, was not clear from Wind observations, where the
Alfvénicity tended to fluctuate around $|\sigma_{r}|\sim 0.5$ across the ME.
Overall, it is hard to say if this is related to the lack of a preceding
shock/sheath or to other factors (such as the eruptive scenario, or the
propagation through interplanetary space), but taken on its own, this event
seems to confirm our interpretation that the location of AFs of interplanetary
formation may be heavily affected by the presence/absence of a preceding shock
and sheath.

##  4 Discussion and Conclusions

In this work, we investigated the in-situ characteristics of 10 ICMEs (9 of
which drove a shock and sheath) observed at 1 au by ACE and Wind while at
longitudinal separations between $0.5^{\circ}$ and $0.7^{\circ}$
(corresponding to $\sim 200-300$ Earth radii, or $\sim 0.009-0.013$ au at 1
au). For each event, we analyzed the Alfvénicity of the sheath and ME in terms
of the residual energy ($\sigma_{r}$) and cross helicity ($\sigma_{c}$) of
fluctuations in the injection range and low-end of the inertial range at
$2.3\times 10^{-5}$ to $3.3\times 10^{-3}$ Hz (corresponding to time scales
between 5 min and 12 hours). We purposely considered such scales in order to
explore the role of AFs in altering the internal structure of ICMEs at large
to intermediate (i.e., “meso”) scales. Additionally, we evaluated the
coherence of ICMEs in terms of the correlation between the magnetic field
signatures measured at Wind and ACE within sheaths and MEs. The analysis of
this set of 10 ICMEs highlighted the following trends:

  * •

The average Alfvénicity of ICME sheaths and MEs is comparable, and is broadly
consistent when investigated using plasma data from the ACE, Wind/SWE, and
Wind/3DP data sets (Figure 1 and Table 2).

  * •

Though less common than in the solar wind, AFs are abundant within ICME
sheaths and MEs (Figure 2). Strongly Alfvénic periods ($\sigma_{r}\leq 0.2$)
lasting at least 20% of a given sub-structure duration are found in about 65%
of sheaths and 90% of MEs.

  * •

Highly Alfvénic periods within sheaths and MEs exhibit a variety of
$\sigma_{c}$ signatures, which provides information about the direction of
propagation of AFs. About half of these highly Alfvénic periods are associated
with AFs propagating either parallel or anti-parallel to the local magnetic
field, while the other half of AFs are characterized by counter-propagating
wave packets (Figure 2).

  * •

From the construction of average time profiles of Alfvénicity within sheaths
and MEs using the SEA method, we find that AFs within sheaths are mainly
located immediately downstream of the shock, while AFs within MEs are
preferentially located near the back of MEs (Figure 3 (a)).

  * •

When measured at longitudinal separations of $0.5^{\circ}-0.7^{\circ}$, the
magnetic field profiles within sheaths are significantly less correlated than
those within MEs (Figure 3 (a) and Table 3). The correlation is uniformly
distributed within sheaths, while the correlation presents a decreasing trend
throughout ME structures: ME backs tend to be less correlated than ME fronts,
both in individual events and in the combined SEA profiles (Figure 3 (b)).

  * •

The comparison of the average profiles constructed through the SEA method
reveals the Alfvénicity (measured in terms of $\sigma_{r}$) is anti-correlated
to the magnetic field correlation. This anti-correlation appears stronger
within sheaths, while it is weaker within MEs. Such an anti-correlation was
not always prominent in individual ICMEs, and was more clearly revealed by the
SEA.

Our first goal was to quantify the frequency and duration of AFs within ICMEs
at 1 au. Having determined that AFs are abundant within sheaths and MEs, our
second goal was to determine whether AFs can be responsible for mediating
coherent behavior across ME structures, assuming such a behavior can be
measured in terms of the correlation of magnetic field profiles measured at a
given longitudinal separation (i.e. the higher the correlation, the higher the
coherence). In other words, we tested if AFs can contribute to making ICMEs
more self-similar along different directions (i.e. whether they increase the
ICME magnetic field correlation scale). Contrary to early studies, but
consistently with a statistical study by Li et al. (2016b) at larger
heliocentric distances, we found that AFs are relatively abundant within ICME
sheaths and MEs at 1 au. Surprisingly, our analysis suggests that instead of
increasing the correlation of the magnetic field components within ICMEs, AFs
may actually decrease it. This is in agreement with the fact that spacecraft
crossing the same ME along different trajectories likely sample AFs in
different oscillation phases, and can be interpreted in two ways. First, as an
indication that the assumption that information mediating coherence across an
ICME propagate at the Alfvén speed (i.e. it is carried by Alfvén waves) may be
inappropriate. This in turn would imply that the correlation length may be
even smaller than predicted by Owens et al. (2017) and Owens (2020), because
information would necessarily propagate at slower speeds than the Alfvén speed
throughout MEs. In this case, a different carrier of information, alternative
to Alfvén waves, has to be identified in the future. Alternatively, in view of
the anti-correlation between AFs and magnetic field correlation, one has to
admit the possibility that the magnetic field correlation may not be a true
measure of coherence, and that a re-thinking of the way we evaluate coherence
based on in-situ data may be necessary. In this scenario, AFs would actually
be mediators of information in the form of anti-coherence, and thus
potentially in competition with mechanisms mediating coherence, (so far) still
to be identified. Large- to meso-scale waves would perturb the quasi-static
(background) ICME structure simply due to their propagation. The larger their
amplitude and the more separation between the spacecraft measurements, the
less coherence will be observed. Changes to the ME structure may be either
permanent or temporary due to the propagation of such waves through the
structure. The contextualization of single-point observations with respect to
the solar wind conditions encountered by individual ICMEs during propagation,
and of the solar environment affecting their early evolution is critical to
further clarify these points. We intend to demonstrate the close physical
relationship between AFs and the correlation of magnetic field signatures of
ICMEs in a following paper investigating a case study of one of the events
from this larger study.

Our analysis also provides indications about the possible evolution of the
magnetic field correlation scales of MEs during propagation. The fact that
despite interacting with the preceding and following solar wind, MEs still
retain a larger correlation scale than the solar wind by the time they reach 1
au implies that MEs most likely have even larger correlation scales during the
eruption/early propagation phases. One possible mechanism to explain such a
degradation, as identified from our study, is given by AFs, which can have two
distinct origins and effects on the magnetic field correlation profile of MEs.
On the one hand, about one half of the identified AFs within MEs present
counter-propagating in situ signatures (i.e. low $\sigma_{r}$ and low $p_{A}$)
consistent with an origin at the Sun before the Alfvén surface. While the
localization of these AFs within ME structures went beyond the capabilities of
our analysis (as the $p_{A}$ SEA profile was designed specifically to locate
uni-directional AFs rather than counter-propagating ones), such AFs are not
expected to have a preferential location within ME cross sections, e.g.,
front, middle, back. On the other hand, the results above indicate about one
half of AFs within MEs propagate in a predominant direction (i.e. low
$\sigma_{r}$ but high $p_{A}$), consistent with an interplanetary origin.
These AFs are mainly found near ME backs. This is a robust result holding for
individual MEs, and suggests that two populations of AFs can co-exist within
the general ME population at 1 au: one population originated at the Sun, i.e.
prior to the CME crossing the Alfvén surface, which most likely had time to
travel across the ME structure by the time it reaches 1 au (Good et al., 2022)
and to reduce the correlation scale uniformly across the whole ME cross
section. And one population originated in the interplanetary space, primarily
at the back of MEs through interaction with the following solar wind, which
had time to propagate only locally within causally-connected ME regions and
contributed to reduce the correlation length mainly near ME backs. This
interpretation is also consistent with the presence of a sheath (found in 90%
of the ICMEs in our set) that may “protect” the front of the ME from being
disrupted by the interaction with the preceding solar wind. Past studies
suggested this may be the case particularly when the ICME propagates through a
preceding highly Alfvénic solar wind (Farrugia et al., 2020). ME backs, on the
other hand, would remain exposed to interactions with the following wind which
may lead to the formation of AFs regardless of the presence/absence of a
sheath ahead. This is especially true for MEs being overtaken by a following
high speed stream. Fast solar wind streams are typically “inundated” by AFs
(Bruno & Carbone, 2013) and can provide an additional reason why AFs in ICMEs
tend to be more common at the back.

We conclude by emphasizing that these new insights have been obtained from the
analysis of a small-population set of ICMEs near 1 au, due to past data
limitations. Despite the relatively small sample size, through our analysis we
were able to pinpoint the existence of significant trends that contribute to
the understanding of the fundamental physical relationships between Alfvén
waves and ICME coherence. However, in the near future it will be of prime
importance to validate these results through statistical studies considering
larger sets of events, and through the examination of different heliocentric
distances, in order to draw general conclusions regarding the relationship
between Alfvénicity and the magnetic field correlation and coherence of ICMEs.
The results presented in this study also serve as an important benchmark for
further investigation and interpretation of individual ICME events, which we
plan to address in an upcoming study currently in preparation. Finally,
determining whether the correlation scales of the solar wind and ICMEs are
larger closer to the Sun, what their relative magnitude is, and how quickly
they drop with heliocentric distance will be important to understand how the
solar wind can degrade the correlation scale of ICMEs in earlier propagation
phases. Both statistical and case studies closer than 1 au are needed in order
to characterize all these factors, and will soon be possible thanks to multi-
point coordinated ICME observations from Parker Solar Probe and Solar Orbiter.

C.S. was supported by the NASA ECIP program (grant no. 80NSSC23K1057). C.S.
and N.L. acknowledge support from NASA grants 80NSSC20K0197 and 80NSSC20K0700.
C.S. and R.M.W. acknowledge support from NASA grant 80NSSC19K0914. C.J.F.
acknowledges support from NASA grant 80NSSC19K1293. N.M. acknowledges Research
Foundation – Flanders (Fonds voor Wetenschappelijk Onderzoek (FWO) -
Vlaanderen) for their support through Postdoctoral Fellowship 12T6521N. F.B.
acknowledges support from the FED-tWIN programme (profile Prf-2020-004,
project “ENERGY”) issued by BELSPO. All data used in this study is publicly
available through the NASA Solar Physics Data Facility (SPDF). The authors
thank the ACE/SWEPAM, ACE/MAG, Wind/3DP, Wind/SWE, and Wind/MAG instrument
teams for providing the necessary data to the public.

## References

  * Al-Haddad et al. (2022) Al-Haddad, N., Galvin, A. B., Lugaz, N., Farrugia, C. J., & Yu, W. 2022, ApJ, 927, 68, doi: 10.3847/1538-4357/ac32e1
  * Ala-Lahti et al. (2019) Ala-Lahti, M., Kilpua, E. K. J., Souček, J., Pulkkinen, T. I., & Dimmock, A. P. 2019, Journal of Geophysical Research (Space Physics), 124, 3893, doi: 10.1029/2019JA026579
  * Ala-Lahti et al. (2020) Ala-Lahti, M., Ruohotie, J., Good, S., Kilpua, E. K. J., & Lugaz, N. 2020, Journal of Geophysical Research (Space Physics), 125, e28002, doi: 10.1029/2020JA028002
  * Alfvén (1942) Alfvén, H. 1942, Nature, 150, 405, doi: 10.1038/150405d0
  * Arregui et al. (2018) Arregui, I., Oliver, R., & Ballester, J. L. 2018, Living Reviews in Solar Physics, 15, 3, doi: 10.1007/s41116-018-0012-6
  * Bale et al. (2016) Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, Space Sci. Rev., 204, 49, doi: 10.1007/s11214-016-0244-5
  * Bavassano et al. (1978) Bavassano, B., Dobrowolny, M., & Moreno, G. 1978, Sol. Phys., 57, 445, doi: 10.1007/BF00160117
  * Belcher & Davis (1971) Belcher, J. W., & Davis, Leverett, J. 1971, J. Geophys. Res., 76, 3534, doi: 10.1029/JA076i016p03534
  * Belcher et al. (1969) Belcher, J. W., Davis, Leverett, J., & Smith, E. J. 1969, J. Geophys. Res., 74, 2302, doi: 10.1029/JA074i009p02302
  * Bothmer & Schwenn (1998) Bothmer, V., & Schwenn, R. 1998, Annales Geophysicae, 16, 1, doi: 10.1007/s00585-997-0001-x
  * Bruno & Carbone (2013) Bruno, R., & Carbone, V. 2013, Living Reviews in Solar Physics, 10, 2, doi: 10.12942/lrsp-2013-2
  * Burlaga et al. (1981) Burlaga, L., Sittler, E., Mariani, F., & Schwenn, R. 1981, J. Geophys. Res., 86, 6673, doi: 10.1029/JA086iA08p06673
  * Cane & Richardson (2003) Cane, H. V., & Richardson, I. G. 2003, Journal of Geophysical Research (Space Physics), 108, 1156, doi: 10.1029/2002JA009817
  * Cane et al. (1997) Cane, H. V., Richardson, I. G., & Wibberenz, G. 1997, J. Geophys. Res., 102, 7075, doi: 10.1029/97JA00149
  * Cartwright & Moldwin (2010) Cartwright, M. L., & Moldwin, M. B. 2010, Journal of Geophysical Research (Space Physics), 115, A08102, doi: 10.1029/2009JA014271
  * Chen (2016) Chen, C. H. K. 2016, Journal of Plasma Physics, 82, 535820602, doi: 10.1017/S0022377816001124
  * Chen et al. (2013) Chen, C. H. K., Bale, S. D., Salem, C. S., & Maruca, B. A. 2013, ApJ, 770, 125, doi: 10.1088/0004-637X/770/2/125
  * Chree (1913) Chree, C. 1913, Philosophical Transactions of the Royal Society of London Series A, 212, 75, doi: 10.1098/rsta.1913.0003
  * Cranmer (2018) Cranmer, S. R. 2018, ApJ, 862, 6, doi: 10.3847/1538-4357/aac953
  * D’Amicis et al. (2021) D’Amicis, R., Bruno, R., Panasenco, O., et al. 2021, A&A, 656, A21, doi: 10.1051/0004-6361/202140938
  * Dasso et al. (2006) Dasso, S., Mandrini, C. H., Démoulin, P., & Luoni, M. L. 2006, A&A, 455, 349, doi: 10.1051/0004-6361:20064806
  * De Pontieu et al. (2007) De Pontieu, B., McIntosh, S. W., Carlsson, M., et al. 2007, Science, 318, 1574, doi: 10.1126/science.1151747
  * Dhamane et al. (2023) Dhamane, O., Raghav, A., Shaikh, Z., et al. 2023, Sol. Phys., 298, 34, doi: 10.1007/s11207-023-02127-4
  * DiBraccio et al. (2015) DiBraccio, G. A., Slavin, J. A., Imber, S. M., et al. 2015, Planet. Space Sci., 115, 77, doi: 10.1016/j.pss.2014.12.016
  * Farrugia et al. (2020) Farrugia, C. J., Lugaz, N., Vasquez, B. J., et al. 2020, Journal of Geophysical Research (Space Physics), 125, e27638, doi: 10.1029/2019JA027638
  * Farrugia et al. (2005) Farrugia, C. J., Matsui, H., Kucharek, H., et al. 2005, Journal of Geophysical Research (Space Physics), 110, A09S13, doi: 10.1029/2004JA010968
  * Good & Forsyth (2016) Good, S. W., & Forsyth, R. J. 2016, Sol. Phys., 291, 239, doi: 10.1007/s11207-015-0828-3
  * Good et al. (2022) Good, S. W., Hatakka, L. M., Ala-Lahti, M., et al. 2022, MNRAS, 514, 2425, doi: 10.1093/mnras/stac1388
  * Good et al. (2020) Good, S. W., Kilpua, E. K. J., Ala-Lahti, M., et al. 2020, ApJ, 900, L32, doi: 10.3847/2041-8213/abb021
  * Gopalswamy et al. (2015) Gopalswamy, N., Yashiro, S., Xie, H., Akiyama, S., & Mäkelä, P. 2015, Journal of Geophysical Research (Space Physics), 120, 9221, doi: 10.1002/2015JA021446
  * Gosling et al. (2005) Gosling, J. T., Skoug, R. M., McComas, D. J., & Smith, C. W. 2005, Geophys. Res. Lett., 32, L05105, doi: 10.1029/2005GL022406
  * Hollweg et al. (1982) Hollweg, J. V., Jackson, S., & Galloway, D. 1982, Sol. Phys., 75, 35, doi: 10.1007/BF00153458
  * Hu et al. (2015) Hu, Q., Qiu, J., & Krucker, S. 2015, Journal of Geophysical Research (Space Physics), 120, 5266, doi: 10.1002/2015JA021133
  * Janvier et al. (2021) Janvier, M., Démoulin, P., Guo, J., et al. 2021, ApJ, 922, 216, doi: 10.3847/1538-4357/ac2b9b
  * Kigure et al. (2010) Kigure, H., Takahashi, K., Shibata, K., Yokoyama, T., & Nozawa, S. 2010, PASJ, 62, 993, doi: 10.1093/pasj/62.4.993
  * Kilpua et al. (2017) Kilpua, E., Koskinen, H. E. J., & Pulkkinen, T. I. 2017, Living Reviews in Solar Physics, 14, 5, doi: 10.1007/s41116-017-0009-6
  * Kilpua et al. (2011) Kilpua, E. K. J., Jian, L. K., Li, Y., Luhmann, J. G., & Russell, C. T. 2011, Journal of Atmospheric and Solar-Terrestrial Physics, 73, 1228, doi: 10.1016/j.jastp.2010.10.012
  * Kilpua et al. (2015) Kilpua, E. K. J., Lumme, E., Andreeova, K., Isavnin, A., & Koskinen, H. E. J. 2015, Journal of Geophysical Research (Space Physics), 120, 4112, doi: 10.1002/2015JA021138
  * King & Papitashvili (2005) King, J. H., & Papitashvili, N. E. 2005, Journal of Geophysical Research (Space Physics), 110, A02104, doi: 10.1029/2004JA010649
  * Kivelson & Russell (1995) Kivelson, M. G., & Russell, C. T. 1995, Introduction to Space Physics
  * Lanabere et al. (2020) Lanabere, V., Dasso, S., Démoulin, P., et al. 2020, A&A, 635, A85, doi: 10.1051/0004-6361/201937404
  * Lanabere et al. (2022) Lanabere, V., Démoulin, P., & Dasso, S. 2022, A&A, 668, A160, doi: 10.1051/0004-6361/202245062
  * Lavraud et al. (2020) Lavraud, B., Fargette, N., Réville, V., et al. 2020, ApJ, 894, L19, doi: 10.3847/2041-8213/ab8d2d
  * Lepping et al. (1995) Lepping, R. P., Acũna, M. H., Burlaga, L. F., et al. 1995, Space Sci. Rev., 71, 207, doi: 10.1007/BF00751330
  * Li et al. (2016a) Li, H., Wang, C., Chao, J. K., & Hsieh, W. C. 2016a, A new approach to identify interplanetary Alfvén waves and to obtain their frequency properties, doi: 10.1002/2015JA021749
  * Li et al. (2016b) Li, H., Wang, C., He, J., et al. 2016b, ApJ, 831, L13, doi: 10.3847/2041-8205/831/2/L13
  * Li et al. (2017) Li, H., Wang, C., Richardson, J. D., & Tu, C. 2017, ApJ, 851, L2, doi: 10.3847/2041-8213/aa9c3f
  * Liang et al. (2012) Liang, H., Xiao, C., Zhou, G., et al. 2012, Plasma Science and Technology, 14, 102, doi: 10.1088/1009-0630/14/2/04
  * Lin et al. (1995) Lin, R. P., Anderson, K. A., Ashford, S., et al. 1995, Space Sci. Rev., 71, 125, doi: 10.1007/BF00751328
  * Longcope & Welsch (2000) Longcope, D. W., & Welsch, B. T. 2000, ApJ, 545, 1089, doi: 10.1086/317846
  * Lugaz et al. (2018) Lugaz, N., Farrugia, C. J., Winslow, R. M., et al. 2018, ApJ, 864, L7, doi: 10.3847/2041-8213/aad9f4
  * Lugaz et al. (2020) Lugaz, N., Winslow, R. M., & Farrugia, C. J. 2020, Journal of Geophysical Research (Space Physics), 125, e27213, doi: 10.1029/2019JA027213
  * Lugaz et al. (2022) Lugaz, N., Salman, T. M., Zhuang, B., et al. 2022, ApJ, 929, 149, doi: 10.3847/1538-4357/ac602f
  * Marsch et al. (2009) Marsch, E., Yao, S., & Tu, C. Y. 2009, Annales Geophysicae, 27, 869, doi: 10.5194/angeo-27-869-2009
  * Masías-Meza et al. (2016) Masías-Meza, J. J., Dasso, S., Démoulin, P., Rodriguez, L., & Janvier, M. 2016, A&A, 592, A118, doi: 10.1051/0004-6361/201628571
  * Matsui et al. (2002) Matsui, H., Farrugia, C. J., & Torbert, R. B. 2002, Journal of Geophysical Research (Space Physics), 107, 1355, doi: 10.1029/2002JA009251
  * McComas et al. (1998) McComas, D. J., Bame, S. J., Barker, P., et al. 1998, Space Sci. Rev., 86, 563, doi: 10.1023/A:1005040232597
  * Moldwin et al. (1995) Moldwin, M. B., Phillips, J. L., Gosling, J. T., et al. 1995, J. Geophys. Res., 100, 19903, doi: 10.1029/95JA01123
  * Möstl et al. (2017) Möstl, C., Isavnin, A., Boakes, P. D., et al. 2017, Space Weather, 15, 955, doi: 10.1002/2017SW001614
  * Nieves-Chinchilla et al. (2018) Nieves-Chinchilla, T., Vourlidas, A., Raymond, J. C., et al. 2018, Sol. Phys., 293, 25, doi: 10.1007/s11207-018-1247-z
  * Ogilvie et al. (1995) Ogilvie, K. W., Chornay, D. J., Fritzenreiter, R. J., et al. 1995, Space Sci. Rev., 71, 55, doi: 10.1007/BF00751326
  * Okamoto et al. (2007) Okamoto, T. J., Tsuneta, S., Berger, T. E., et al. 2007, Science, 318, 1577, doi: 10.1126/science.1145447
  * Owens (2016) Owens, M. J. 2016, ApJ, 818, 197, doi: 10.3847/0004-637X/818/2/197
  * Owens (2020) —. 2020, Sol. Phys., 295, 148, doi: 10.1007/s11207-020-01721-0
  * Owens et al. (2017) Owens, M. J., Lockwood, M., & Barnard, L. A. 2017, Scientific Reports, 7, 4152, doi: 10.1038/s41598-017-04546-3
  * Regnault et al. (2020) Regnault, F., Janvier, M., Démoulin, P., et al. 2020, Journal of Geophysical Research (Space Physics), 125, e28150, doi: 10.1029/2020JA028150
  * Richardson & Cane (2010) Richardson, I. G., & Cane, H. V. 2010, Sol. Phys., 264, 189, doi: 10.1007/s11207-010-9568-6
  * Roberts et al. (1992) Roberts, D. A., Goldstein, M. L., Matthaeus, W. H., & Ghosh, S. 1992, J. Geophys. Res., 97, 17115, doi: 10.1029/92JA01144
  * Ruffenach et al. (2012) Ruffenach, A., Lavraud, B., Owens, M. J., et al. 2012, Journal of Geophysical Research (Space Physics), 117, A09101, doi: 10.1029/2012JA017624
  * Ruffenach et al. (2015) Ruffenach, A., Lavraud, B., Farrugia, C. J., et al. 2015, Journal of Geophysical Research (Space Physics), 120, 43, doi: 10.1002/2014JA020628
  * Sanchez-Diaz et al. (2017) Sanchez-Diaz, E., Rouillard, A. P., Davies, J. A., et al. 2017, ApJ, 851, 32, doi: 10.3847/1538-4357/aa98e2
  * Scolini et al. (2018) Scolini, C., Messerotti, M., Poedts, S., & Rodriguez, L. 2018, Journal of Space Weather and Space Climate, 8, A9, doi: 10.1051/swsc/2017046
  * Scolini et al. (2023) Scolini, C., Winslow, R. M., Lugaz, N., & Poedts, S. 2023, ApJ, 944, 46, doi: 10.3847/1538-4357/aca893
  * Smith et al. (1998) Smith, C. W., L’Heureux, J., Ness, N. F., et al. 1998, Space Sci. Rev., 86, 613, doi: 10.1023/A:1005092216668
  * Sonnerup & Scheible (1998) Sonnerup, B. U. Ö., & Scheible, M. 1998, ISSI Scientific Reports Series, 1, 185
  * St. Cyr & Webb (1991) St. Cyr, O. C., & Webb, D. F. 1991, Sol. Phys., 136, 379, doi: 10.1007/BF00146543
  * Stansby et al. (2019) Stansby, D., Horbury, T. S., & Matteini, L. 2019, MNRAS, 482, 1706, doi: 10.1093/mnras/sty2814
  * Telloni et al. (2012) Telloni, D., Bruno, R., D’Amicis, R., Pietropaolo, E., & Carbone, V. 2012, ApJ, 751, 19, doi: 10.1088/0004-637X/751/1/19
  * Telloni et al. (2013) Telloni, D., Perri, S., Bruno, R., Carbone, V., & Amicis, R. D. 2013, ApJ, 776, 3, doi: 10.1088/0004-637X/776/1/3
  * Telloni et al. (2021) Telloni, D., Scolini, C., Möstl, C., et al. 2021, A&A, 656, A5, doi: 10.1051/0004-6361/202140648
  * Tomczyk et al. (2007) Tomczyk, S., McIntosh, S. W., Keil, S. L., et al. 2007, Science, 317, 1192, doi: 10.1126/science.1143304
  * Torrence & Compo (1998) Torrence, C., & Compo, G. P. 1998, Bulletin of the American Meteorological Society, 79, 61, doi: 10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2
  * Velli & Liewer (1999) Velli, M., & Liewer, P. 1999, Space Sci. Rev., 87, 339, doi: 10.1023/A:1005110315988
  * Vourlidas et al. (2013) Vourlidas, A., Lynch, B. J., Howard, R. A., & Li, Y. 2013, Sol. Phys., 284, 179, doi: 10.1007/s11207-012-0084-8
  * Wang et al. (2012) Wang, X., He, J., Tu, C., et al. 2012, ApJ, 746, 147, doi: 10.1088/0004-637X/746/2/147
  * Wicks et al. (2010) Wicks, R. T., Owens, M. J., & Horbury, T. S. 2010, Sol. Phys., 262, 191, doi: 10.1007/s11207-010-9509-4
  * Winslow et al. (2015) Winslow, R. M., Lugaz, N., Philpott, L. C., et al. 2015, Journal of Geophysical Research (Space Physics), 120, 6101, doi: 10.1002/2015JA021200
  * Yao et al. (2010) Yao, S., Marsch, E., Tu, C.-Y., & Schwenn, R. 2010, Journal of Geophysical Research (Space Physics), 115, A05103, doi: 10.1029/2009JA014914
  * Yu et al. (2014) Yu, W., Farrugia, C. J., Lugaz, N., et al. 2014, Journal of Geophysical Research (Space Physics), 119, 689, doi: 10.1002/2013JA019115
  * Zhao et al. (2021) Zhao, L. L., Zank, G. P., He, J. S., et al. 2021, ApJ, 922, 188, doi: 10.3847/1538-4357/ac28fb
  * Zurbuchen & Richardson (2006) Zurbuchen, T. H., & Richardson, I. G. 2006, Space Sci. Rev., 123, 31, doi: 10.1007/s11214-006-9010-4