# Characteristic Scales of Complexity and Coherence within Interplanetary
Coronal Mass Ejections: Insights from Spacecraft Swarms in Global Heliospheric
Simulations

Camilla Scolini Institute for the Study of Earth, Oceans, and Space,
University of New Hampshire, Durham, NH, USA  Cooperative Programs for the
Advancement of Earth System Science (CPAESS), University Corporation for
Atmospheric Research, Boulder, CO, USA Camilla Scolini<EMAIL_ADDRESS>Réka M. Winslow 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  Stefaan Poedts Department of Mathematics/Centre for mathematical Plasma
Astrophyics, KU Leuven, Belgium  Institute of Physics, University of Maria
Curie-Skłodowska, Lublin, Poland

(Received April 26, 2022)

###### Abstract

Many aspects of the three-dimensional (3-D) structure and evolution of
interplanetary coronal mass ejections (ICMEs) remain unexplained. Here, we
investigate two main topics: (1) the coherence scale of magnetic fields inside
ICMEs, and (2) the dynamic nature of ICME magnetic complexity. We simulate
ICMEs interacting with different solar winds using the linear force-free
spheromak model incorporated into the EUHFORIA model. We place a swarm of
$\sim 20000$ spacecraft in the 3-D simulation domain and characterize ICME
magnetic complexity and coherence at each spacecraft based on simulated time
series. Our simulations suggest that ICMEs retain a lower complexity and
higher coherence along their magnetic axis, but that a characterization of
their global complexity requires crossings along both the axial and
perpendicular directions. For an ICME of initial half angular width of
$45^{\circ}$ that does not interact with other large-scale solar wind
structures, global complexity can be characterized by as little as 7–12
spacecraft separated by $25^{\circ}$, but the minimum number of spacecraft
rises to 50–65 (separated by $10^{\circ}$) if interactions occur. Without
interactions, ICME coherence extends for $45^{\circ}$,
$20^{\circ}$–$30^{\circ}$, $15^{\circ}$–$30^{\circ}$, and
$0^{\circ}$–$10^{\circ}$ for $B$, $B_{\phi}$, $B_{\theta}$, and $B_{r}$,
respectively. Coherence is also lower in the ICME west flank compared to the
east flank due to Parker spiral effects. Moreover, coherence is reduced by a
factor of 3–6 by interactions with solar wind structures. Our findings help
constrain some of the critical scales that control the evolution of ICMEs and
aid in the planning of future dedicated multi-spacecraft missions.

Solar coronal mass ejections (310) — Solar wind (1534) — Interplanetary
magnetic fields (824) — Corotating streams (314)

††journal: ApJ

##  1 Introduction

Coronal mass ejections (CMEs; see e.g. Webb & Howard, 2012) are large
eruptions of plasma and magnetic field from the Sun into interplanetary space.
When probed in situ by spacecraft crossing their structures, they are referred
to as interplanetary CMEs (ICMEs; see e.g. Bothmer & Schwenn, 1998; Cane &
Richardson, 2003; Kilpua et al., 2017a). ICMEs are among the largest and most
energetic transients in the heliosphere, where they shape the space
environment and can drive severe space weather disturbances at Earth and other
planets (see e.g. Gosling et al., 1990; Tsurutani et al., 1992; Zhang et al.,
2007; Kilpua et al., 2017b; Lee et al., 2017; Slavin et al., 2019; Winslow et
al., 2020). ICMEs are characterized by a variety of in-situ signatures, with
their core component being the so-called magnetic ejecta (ME; Winslow et al.,
2015), i.e. a region of enhanced magnetic field corresponding to the original
structure erupted at the Sun, which may or may not be preceded by a forward
shock and a turbulent sheath formed during propagation (Kilpua et al., 2017a).
The entire propagating structure, composed of the ME and its associated shock
and sheath (if present) is referred to as the ICME. We use the term ME as a
more general term than magnetic cloud, which follows the strict definition of
Burlaga et al. (1981). Typically, MEs are associated with a low plasma $\beta$
($\ll 1$) near 1 au, indicating that the magnetic pressure dominates over the
thermal pressure and that their internal dynamics and 3-D structure are
controlled primarily by magnetic forces. MEs also often exhibit rotations in
the magnetic field components, which are compatible with a configuration of
magnetic field bundles twisting around a central axis (Burlaga, 1988; Lepping
et al., 1990; Kilpua et al., 2017a). This has led to the proliferation of
flux-rope (FR) models to describe the ME local and global magnetic structure
(see, e.g., Lepping et al., 1990; Mulligan & Russell, 2001; Marubashi &
Lepping, 2007; Hidalgo & Nieves-Chinchilla, 2012; Hidalgo, 2013; Isavnin,
2016; Nieves-Chinchilla et al., 2016, 2018a; Weiss et al., 2021). FR models
have also been successful in describing the CME pre-eruptive and eruptive
configuration, early evolution in the solar corona, and interplanetary
evolution up to Mars and beyond (e.g. Vourlidas, 2014; Patsourakos et al.,
2020; Palmerio et al., 2021; Davies et al., 2022).

ICMEs evolve significantly as they propagate through the solar wind. In
particular, interactions with other large-scale interplanetary structures such
as high speed streams (HSSs; Cranmer et al., 2017), stream interaction regions
(SIRs; Richardson, 2018), or other ICMEs can induce dramatic changes to ICME
structures (Manchester et al., 2017; Lugaz et al., 2017). Therefore,
understanding the effect of different propagation scenarios on ICMEs is
necessary to characterize their evolution across a wide range of spatial and
temporal scales. To this end, recent studies have explored the changes in
magnetic complexity (i.e., the degree of similarity/deviation from an ideal FR
configuration) that MEs can undergo during propagation. In-depth
investigations of MEs observed by radially-aligned spacecraft provided ample
evidence that MEs evolve and tend to increase their magnetic complexity during
propagation through interplanetary space, primarily due to interaction with
other large-scale solar wind structures (Mulligan et al., 1999; Nieves-
Chinchilla et al., 2012; Good & Forsyth, 2016; Winslow et al., 2016, 2021b,
2021a). Scolini et al. (2022) provided further statistical support to these
results through an investigation of 31 MEs observed in radial alignment
between Mercury and 1 au. Furthermore, Scolini et al. (2021b) conducted a
numerical investigation of the probability of detecting a complexity increase
between an inner and outer spacecraft in perfect radial alignment, finding
that the interaction with HSSs and SIRs doubles the probability for an MEs to
undergo complexity changes. To date it has not been possible to completely
characterize the spatial (i.e. longitudinal and latitudinal) magnetic
complexity distribution of MEs and its evolution with heliocentric distance
due to the limited number of spacecraft crossing individual ME structures.

Direct probing of ICMEs relies on in situ spacecraft measurements obtained at
a given location in space (i.e. along a 1-D trajectory). However, single-point
measurements only provide information on the local properties of an ICME at a
particular stage of its evolution, and it remains unclear to what extent they
are indicative of their global structure (e.g. Möstl et al., 2012; Al-Haddad
et al., 2013, 2018; Marubashi & Cho, 2015; Lugaz et al., 2018). Multi-point
observations at the same heliocentric distance but at different locations in
space represent the best opportunity to probe the 3-D structure of ICMEs at a
given stage of their evolution, as they allow for a direct probing of the in
situ ICME characteristics at multiple locations while mitigating the effect of
local variations (e.g. Möstl et al., 2012; Weiss et al., 2021). However, these
remain relatively scarce, as the majority of ICMEs are still probed in situ by
single spacecraft, and the maximum number of spacecraft crossings at a given
heliocentric distance per individual event is limited to three (e.g. Burlaga
et al., 1981; Mulligan & Russell, 2001; Kilpua et al., 2011; Lugaz et al.,
2022). Additionally, such measurements have so far been obtained only near 1
au, either by multiple spacecraft in L1 orbits (at small angular separations
$\leq 1^{\circ}$), or for spacecraft angular separations above $10^{\circ}$,
with other heliocentric distances and critical scales around
$1^{\circ}-10^{\circ}$ remaining unprobed (Lugaz et al., 2018).

Prominent open questions related to the 3-D nature of ICMEs are whether their
MEs are magnetically coherent objects, and at what spatial scales such a
coherence exists. Strictly speaking, magnetic coherence within an ME refers to
its ability to respond to external perturbations in a coherent (i.e. solid-
like) manner (Owens et al., 2017). Coherent behavior requires information to
propagate across an ME structure, and magnetic forces to be sufficiently
strong to resist external deformation forces (Owens et al., 2017). Following
an analytical approach based on the consideration of expansion and Alfvén
speeds within MEs at 1 au, Owens et al. (2017) and Owens (2020) estimated an
upper limit for the angular scale of magnetic coherence to $\sim 26^{\circ}$,
which is significantly smaller than the typical angular width of MEs (around
$40^{\circ}-60^{\circ}$; e.g. Yashiro et al., 2004; Kilpua et al., 2011; Good
& Forsyth, 2016). From an observational standpoint, the correlation of
magnetic field components among different locations within an ME has been
regarded as a proxy for magnetic coherence (Matsui et al., 2002; Farrugia et
al., 2005; Lugaz et al., 2018; Ala-Lahti et al., 2020), although it remains
unclear how the ability of information to propagate across an ME affects its
local properties, and how this relates to the ability of the same ME to use
such information in order to resist deformation by external factors and behave
as a coherent structure (Owens, 2020). The investigation by Lugaz et al.
(2018) across various angular separations pointed towards the potential
existence of two characteristic scales of correlation within MEs near 1 au:
one related to the magnetic field components (around $4^{\circ}–7^{\circ}$)
and one for the total magnetic field (around $14^{\circ}–20^{\circ}$). Yet,
limitations on the data available prevented the exploration of the whole
parameter space, particularly at scales between $1^{\circ}$ and $10^{\circ}$.

It remains extremely challenging to answer some of the most fundamental
questions on the nature of MEs, particularly with respect to their magnetic
complexity and coherence and their evolution with heliocentric distance. In
this paper, we address these topics by turning to numerical simulations and
exploiting their 3-D capabilities to investigate the following science
questions: (1) What is the spatial distribution of magnetic complexity within
MEs? (2) Across what spatial scales could MEs behave as magnetically coherent
objects? (3) How does all of the above depend on the heliocentric distance and
the specific propagation history of an ME, such as its interaction with other
large-scale solar wind structures? We address the aforementioned questions by
means of global heliospheric simulations performed with the EUropean
Heliospheric FORecasting Information Asset (EUHFORIA; Pomoell & Poedts, 2018),
using idealized solar wind set-up as in Scolini et al. (2021b) and simulating
CMEs/MEs using a linear force-free spheromak model (Verbeke et al., 2019).
This work represents the first-ever attempt to specifically address the
problem of magnetic coherence by analyzing numerical simulation results using
techniques typically employed in the analysis of in situ observations.

The paper is structured as follows. In Section 2, we introduce the numerical
model and the methods used to investigate the magnetic complexity and
coherence of MEs using global heliospheric simulations. In Section 3, the
analysis of two simulations considering different interaction scenarios
between an ME and the surrounding solar wind is presented, and results are
discussed with respect to baseline values obtained for a stand-alone ideal
spheromak magnetic structure, and to the propagation history of each ME. In
Section 4, we examine possible implications of our results for future
dedicated multi-spacecraft missions orbiting close to the ecliptic plane.
Finally, in Section 5, we summarize our conclusions and outlook.

##  2 Methods

###  2.1 Numerical Set-Up

In this work, we build on the analysis of the numerical simulations previously
presented by Scolini et al. (2021b), that were performed using the EUHFORIA
3-D magnetohydrodynamic (MHD) model of the inner heliosphere (Pomoell &
Poedts, 2018). To evaluate the interaction of ICMEs with different solar wind
structures, two background solar wind configurations are simulated: the first
one (hereafter “run A”) includes a uniform solar wind of speed $450$ km s-1
except for a low-inclination heliospheric current/plasma sheet (HCS/HPS). The
second one (hereafter “run B”) differs from the one above by having a more
highly inclined HCS/HPS, and by the inclusion of a HSS of radial speed equal
to $675$ km s-1 and circular cross-section of half angular width equal to
$30^{\circ}$. In both cases, the HPS meridional profile is parametrized using
the description in Odstrčil et al. (1996), resulting in a solar wind speed as
low as 300 km s-1 near the HCS. This choice has been made to ensure full
control over the ICME propagation and interaction with solar wind structures,
and the comparability between different runs. In the two runs, CMEs are
initialized as linear force-free spheromak structures as described by Verbeke
et al. (2019), with the following set of geometric, kinematic, and magnetic
parameters: radial speed equal to $v_{R}=800$ km s-1; initial half angular
width of $\omega/2=45^{\circ}$; positive chirality (${H=+1}$) with axial tilt
${\gamma}=90^{\circ}$ with respect to the northward direction (corresponding
to a SWN flux-rope type; see Bothmer & Schwenn, 1998); toroidal magnetic flux
(${\varphi_{t}}$) equal to $10^{14}$ Wb (corresponding to a magnetic field
strength of $\sim 25$ nT at 1 au). Because of the pressure imbalance between
the CME and the surrounding solar wind upon insertion in the heliospheric
domain (leading to an expansion of the CME structure, as shown by Scolini et
al., 2019, 2021a), the effective initial CME front speed is $\sim 1100$ km
s-1, which results in a fast CME that drives an interplanetary shock and
sheath. Such a combination of initial parameters is representative of those of
a typical fast CME with a reconnected flux of the order of $10^{14}$ Wb (Pal
et al., 2018). The CME initial direction is chosen to reproduce two end-member
scenarios of interaction with different solar wind structures: in run A, the
CME is inserted across the HCS/HPS at $(\theta,\phi)=(0^{\circ},0^{\circ})$;
in run B, the CME is inserted across the HSS at
$(\theta,\phi)=(5^{\circ},0^{\circ})$, in a configuration similar to that of
CMEs originated from “anemone” active regions (e.g. Lugaz et al., 2011; Sharma
& Cid, 2020). The CME insertion time is arbitrarily set on January 1, 2020 at
00:00 UT. A detailed discussion of the model set-up is available in Scolini et
al. (2021b).

Our simulation domain expands from 0.1 to 2.0 au in the radial direction
($r$), covering $\pm 80^{\circ}$ in the latitudinal direction ($\theta$), and
$\pm 180^{\circ}$ in longitude ($\phi$), with a uniform grid composed of
$512(r)\times 80(\theta)\times 180(\phi)$ cells. In our simulation domain, we
place a set of virtual spacecraft where the time-dependent MHD parameters are
extracted in a similar way to that done by real spacecraft monitoring the in
situ plasma conditions in space. Virtual spacecraft are placed at longitudinal
and latitudinal separations of $\Delta\lambda=5^{\circ}$, spanning $\pm
90^{\circ}$ in longitude from the CME initial direction, and covering the full
range of latitudes in the simulation domain. They are also uniformly
distributed in the radial direction between 0.11 and 1.61 au (i.e., from the
model inner boundary to the orbit of Mars) with a 0.1 au radial separation.
Overall, a swarm of 18944 virtual spacecraft (1184 per heliocentric distance)
is placed in the model domain in each simulation. Each virtual spacecraft
records time profiles of the standard MHD output in EUHFORIA: the velocity
vector, magnetic field vector, density, and thermal pressure in Heliocentric
Earth Equatorial (HEEQ) coordinates. Figure 1 (a) provides a global view of
the locations of the virtual spacecraft considered, while a detailed
discussion about their use in this paper’s analysis is provided in the
following sections.

Figure 1:  Spatial distribution of the spacecraft swarm used in EUHFORIA
simulations. (a): Angled view showing the global distribution of virtual
spacecraft in 3-D space (in blue). Red dots indicate the virtual spacecraft
located at 1 au. Black dots mark the virtual spacecraft located along the CME
initial direction. (b): Front view showing only the virtual spacecraft at 1
au. For a reference spacecraft serving as seed location
$(\theta_{s},\phi_{s})$, the re-binning of ME types with nearby spacecraft for
two different $\Delta\lambda$ values is indicated by the dashed polygons. For
the same reference spacecraft, first neighbors considered under the rook
criterion employed to determine the Moran’s $I$ coefficient, for two different
$\Delta\lambda$, are indicated as black and blue colored dots.

###  2.2 Metrics of Magnetic Complexity

Time-dependent information on the MHD variables (i.e. velocity vector,
magnetic field vector, density, and thermal pressure in HEEQ coordinates) at
each simulated spacecraft are used to investigate the spatial and temporal
evolution of the magnetic complexity characterizing the simulated ICMEs, and
particularly their ME structures, as described below.

####  2.2.1 Frequency of Magnetic Ejecta Types

At each virtual spacecraft located at coordinates $(r,\theta,\phi)$ in the
simulation domain, we identify whether an ME signature is present in the
simulated time series using the identification scheme provided in Scolini et
al. (2021b). This classification scheme considers the magnetic field and
plasma $\beta$ to identify the presence or absence of an ME following a
preceding interplanetary shock, and to determine its front and rear
boundaries. The preceding shock is identified based on jump conditions on the
radial speed, density, and magnetic field. If an ME is detected, we
characterize its complexity based on the maximum rotation detected in the
magnetic field components by using the classification scheme previously
introduced by Scolini et al. (2021b) and based on previous observational works
by Nieves-Chinchilla et al. (2018b, 2019). Our classification differs from the
one by Nieves-Chinchilla et al. (2018b, 2019) in the fact that it is adapted
to reflect the full range of rotation angles expected to be observed when
crossing a spheromak structure, ranging between $0^{\circ}$ and $360^{\circ}$
as detailed by Scolini et al. (2021b). In summary, each ME signature is
classified as $F_{270}$, $F_{180}$, $F_{90}$, $F_{30}$ or $E$ when at least
one component (i.e. $B_{R}$, $B_{T}$, or $B_{N}$) rotates by $\geq
270^{\circ}$, $\geq 180^{\circ}$, $\geq 90^{\circ}$, $\geq 30^{\circ}$, or
$<30^{\circ}$, respectively. Additionally, we assign a numerical index,
defined as “complexity index” ($\mathbb{C}$), to each ME class detected in
situ, in order to rank the level of complexity of the detected structure:
$\mathbb{C}=0$ for $F_{270}$ class, $\mathbb{C}=1$ for $F_{180}$ class,
$\mathbb{C}=2$ for $F_{90}$ class, $\mathbb{C}=3$ for $F_{30}$ class, and
$\mathbb{C}=4$ for $E$ class. As a result, a complexity scale between 0 and 4
is defined, with higher complexity values indicating higher deviations from
the FR signatures expected for an ideal spheromak, which exhibit rotations
larger than $270^{\circ}$ for most crossings near the spheromak central axis
(see Figure 6 in Scolini et al., 2021b). Additionally, a value of
$\mathbb{C}=5$ is assigned to shock-only ICME signatures, while $\mathbb{C}=6$
is assigned to all spacecraft locations that did not detect any ICME-related
signature. Example signatures for each ME class are provided in the appendix
of Scolini et al. (2021b). We note this definition of complexity is solely
based on the magnetic field signatures associated with a given ME, and it does
not take into account any plasma information nor the properties of its
preceding shock and sheath. This choice was made to be consistent with
observational works in the past $\sim 15$ years, which could only make use of
magnetic field data at distances other than 1 au (Winslow et al., 2016, 2021b;
Scolini et al., 2022; Davies et al., 2022). However, it will be important for
future investigations of ICME complexity to asses them holistically, i.e., by
looking at changes in all aspects of the ICME as opposed to just the magnetic
parameters within the ME, to understand complexity changes from a more
fundamental standpoint.

In our simulation runs A and B, the frequency of ME types detected at
different spacecraft is then investigated as a function of the heliocentric
distance, and of the angular separation $\Delta\lambda$ of the simulated
spacecraft crossing through the ME structure. To explore different spatial
scales, we resample the initial spacecraft distribution (uniformly separated
in longitude and latitude by $\Delta\lambda=5^{\circ}$) to progressively
larger $\Delta\lambda$, covering $\Delta\lambda$ between $5^{\circ}$ and
$45^{\circ}$. After defining a “seed” location $(\theta_{s},\phi_{s})$
corresponding to the reference location for the resampling of the spacecraft
distribution, spacecraft located between
$(\theta_{s}+n\Delta\lambda,\phi_{s}+m\Delta\lambda)$ and
$(\theta_{s}+(n+1)\Delta\lambda,\phi_{s}+(m+1)\Delta\lambda)$ are binned into
the $n$-th ($m$-th) bin of the new spatial distribution in latitude
(longitude). A representation of such a re-binning process is provided in
Figure 1 (b). Information from individual spacecraft binned within the same
$\Delta A=\Delta\lambda\times\Delta\lambda$ areal unit is then combined by
taking the mode of the ME type distribution across $\Delta A$, which
corresponds to the most probable ME class that would be observed by a
spacecraft crossing anywhere within $\Delta A$.

Because results expressed in terms of $\Delta\lambda$ also depend on the
specific CME half angular width ($\omega/2$) prescribed at insertion through
the domain inner boundary, we also express angular separations as fractions of
the CME initial half angular width ($F_{\omega/2}$, ranging between 0 and 1),
which are independent of the prescribed CME half angular width and can
therefore be extrapolated to CMEs of different sizes. When relevant, the
associated number of virtual spacecraft crossing the ME structure is also
reported.

In the following, we consider the occurrence of ME types detected at different
spacecraft to provide a global measure of ME magnetic complexity: the lowest
the $\mathbb{C}$ values detected within an ME, the lowest its global
complexity. Furthermore, we identify the characteristic scale of magnetic
complexity as the maximum angular separation $\Delta\lambda$ at which
spacecraft are able to detect the same two most frequent ME classes detected
by spacecraft with minimum $\Delta\lambda=5^{\circ}$ separation, at all
heliocentric distances. We clarify this choice in the identification of the
characteristic scale of magnetic complexity in Sections 3.1 and 3.2, when
discussing the results for an ideal spheromak structure and for a spheromak
propagating through a quiet solar wind.

####  2.2.2 Spatial Autocorrelation of Magnetic Ejecta Types: the Moran’s $I$

To evaluate how ordered is the spatial distribution of ME complexity at each
heliocentric distance $r$ considered, we use the Moran’s $I$ (Moran, 1950) as
a measure to quantify the spatial autocorrelation (i.e. the presence of
systematic spatial variation in a certain variable) of different ME types
detected within the simulated ICMEs. The Moran’s $I$ at a given heliocentric
distance $r$ is defined as

$I(r)=\frac{N}{W}\frac{\sum^{N}_{i=1}\sum^{N}_{j=1}w_{ij}(x_{i}-\bar{x})(x_{j}-\bar{x})}{\sum^{N}_{k=1}(x_{k}-\bar{x})^{2}}{,}$
(1)

where $N$ is the total number of spacecraft observations at a given
heliocentric distance $r$; $x_{i}=\mathbb{C}_{i}$ is the value of the ME
complexity index at a given location $i$ corresponding to coordinates
$(r,\theta_{i},\phi_{i})$; $x_{j}=\mathbb{C}_{j}$ is the value of the ME
complexity index at a location $j$ corresponding to coordinates
$(r,\theta_{j},\phi_{j})$; and $\bar{x}=\bar{\mathbb{C}}$ is the mean of all
$x_{i}=\mathbb{C}_{i}$ values, for $i=1,...,N$. The contiguity between two
spatial units is assessed using the rook criterion, which takes as neighbors
any pair of spacecraft that are aligned in longitude or latitude (i.e. whose
location cells on a grid share an edge). For the specific spacecraft swarm
configuration used in this work’s simulations, the neighbors of a given
spacecraft using the rook criterion are indicated by the blue and black
colored dots in Figure 1 (b) for two different $\Delta\lambda$ values.
$w_{ij}$ is the weight of location $i$ relative to $j$, and it assumes a value
of 1 for pairs of spacecraft that are aligned in longitude or latitude, and a
value of 0 for pairs of spacecraft that are diagonally aligned. $W$ is the sum
of all weights $w_{ij}$ over the set of $N$ locations considered. Under such a
prescription, a Moran’s $I(r)$ close to 1 indicates a well-ordered, highly-
clustered spatial distribution of ME types and is taken as indication of the
lowest complexity state possible for an ME magnetic structure; a Moran’s
$I(r)$ approaching 0 indicates a random arrangement of ME types in space, and
therefore indicates the highest complexity state. In the following, we
consider $I>0.5$ as indicative of a low-complexity, highly clustered spatial
distribution of ME classes within an ME, and $I\leq 0.5$ as indicative of a
high-complexity, randomly-arranged distribution.

Similarly to the occurrence of ME types, we compute the Moran’s $I$ for
different spacecraft separations $\Delta\lambda$, and we evaluate its changes
as a function of the heliocentric distance to assess the role of interactions
between ICMEs and solar wind structures in randomizing the internal magnetic
field of MEs. The Moran’s $I$ for different spatial scales is obtained from
the resampled spatial distribution of the complexity index $\mathbb{C}$, as
described in Section 2.2.1. In addition to the occurrence of ME classes
discussed in Section 2.2.1, we use the Moran’s $I$ as a complementary global
metric to quantify the degree of complexity within an ME.

####  2.2.3 Spatial Autocorrelation of Magnetic Ejecta Types: the Local
Moran’s $I$

To gain information on the local properties of ME complexity, we further
inspect the spatial distribution of magnetic complexity through the Local
Moran’s $I$ (also known as Anselin’s Local Indicator of Spatial Association;
Anselin, 1995). This metric corresponds to the local formulation of the
Moran’s $I$, which is calculated locally for each areal unit in the data
spatial distribution. Specifically, the Local Moran’s $I$ at a given location
$(r,\theta_{i},\phi_{i})$ is defined as

$I_{i}(r)=(x_{i}-\bar{x})\sum^{N_{i}}_{j=1}w_{ij}(x_{j}-\bar{x}){,}$ (2)

where $N_{i}$ is the number of neighboring spacecraft observations to the
reference location $i$ corresponding to coordinates $(r,\theta_{i},\phi_{i})$;
$x_{i}=\mathbb{C}_{i}$ is the ME complexity index at location $i$;
$x_{j}=\mathbb{C}_{j}$ is the ME complexity index at location $j$
corresponding to coordinates $(r,\theta_{j},\phi_{j})$ neighboring location
$i$; and $\bar{x}=\bar{\mathbb{C}}$ is the mean of all $x_{j}=\mathbb{C}_{j}$
neighboring values of location $i$, for $j=1,...,N_{i}$. As for Equation 1,
$w_{ij}$ is the weight of location $i$ relative to $j$, as determined by the
rook criterion.

With respect to the specific problem considered in this work, the Local
Moran’s $I$ proves particularly useful to investigate the spatial distribution
of ME complexity as a function of the heliocentric distance, and to quantify
to what degree interactions with interplanetary structures affect the magnetic
complexity in different regions of an ME. In particular, to locate the source
of complexity changes, we track the $(\theta,\phi)$ location of the maximum of
the Local Moran’s $I$ ($\mathrm{max}(I_{i})$) with heliocentric distance, and
relate it to the conditions of the surrounding solar wind along the
corresponding propagation direction.

###  2.3 Metrics of Magnetic Coherence

As previously done for magnetic complexity, we consider time-dependent
information on the MHD plasma variables at each simulated spacecraft to
investigate the spatial and temporal evolution of the magnetic coherence
characterizing the simulated ICMEs, and particularly their ME structures, as
described below. Our final aim is to characterize the spatial properties and
scale of ME magnetic coherence and its evolution depending on the
characteristics of the solar wind encountered by an ICME along its journey
through the heliosphere.

####  2.3.1 Scale of Magnetic Coherence: Correlation of Magnetic Field
Profiles at Different Spacecraft

In the following, we investigate ME magnetic coherence in a similar manner as
previous observational investigations, i.e. by quantifying the spatial scales
at which the time series of the components and magnitude of the magnetic field
within simulated MEs exhibit high levels of correlation. To achieve our goal,
for each heliocentric distance $r$, we assess the correlation of the magnetic
field profile measured within the ME at each spacecraft location
$(r,\theta_{i},\phi_{i})$ with respect to selected reference positions
$(r,\theta_{\mathrm{ref}},\phi_{\mathrm{ref}})$. We then compute average
correlations across all spacecraft located at a given angular separation from
the reference spacecraft ($\Delta\lambda$), i.e. by taking the average along
all directions around the reference location, for different angular
separations. By comparing the average correlations obtained at different
angular separations, we determine the maximum angular separation at which two
spacecraft are likely to observe correlated time profiles for a given magnetic
field component. As correlation scales around a given reference spacecraft may
be anisotropic along different directions, the correlation scale for a given
magnetic field component is estimated by averaging across different directions
around the reference observer and chosen as the maximum angular separation at
which correlation coefficients are $\geq 0.6$ on average. Cartoons describing
its derivation by averaging across different directions with respect to two
different reference observers, one crossing near the center of the ME and one
near its west flank, for three given angular separations $\Delta\lambda$, are
provided in Figure 2 (a) and (b), respectively. We note that our definition of
correlation scale is different from those used by Lugaz et al. (2018) and
Wicks et al. (2010) to investigate the ME and solar wind correlation lengths
from in situ measurements. A justification on the different definition chosen
in this study compared to previous studies on spatial correlations is provided
in Section 3.4, when discussing the results for spheromaks propagating through
different solar winds with respect to the baseline model consisting of an
ideal spheromak structure.

Figure 2:  Sketch of the methodology used to determine the average correlation
scale of the magnetic field profiles around a reference observer crossing
through the ME center (a) and west (b) region. Different angular scales
($\Delta\lambda_{i}$) are indicated in different colors. The ICME is plotted
as propagating heads on towards the observer.

##  3 Results for Spheromak Structures Propagating in the Heliosphere

In this section, we explore the evolution of magnetic complexity and coherence
within MEs propagating through different solar winds in EUHFORIA, and we
discuss how deviations from the patterns retrieved for an ideal spheromak
structure depend on the characteristics of the interaction with the
surrounding solar wind. In light of the results by Scolini et al. (2021a, b),
we focus our discussion on heliocentric distances between 0.3 au and 1.6 au
and neglect results between 0.1 au and 0.3 au, where ICME over-expansion and
challenges in the determination of the ME boundaries from simulated time
series in the early stage of the ICME propagation might have otherwise
influenced our results and conclusions. As a caveat to the interpretation of
the results, we note that the chosen numerical set-up ensures more control
over the various propagation effects affecting different ICME regions during
propagation compared to performing simulations of real ICME events propagating
through a magnetogram-extrapolated solar wind. The insertion of a perfectly-
known global CME magnetic structure at the inner boundary of the heliospheric
domain (located at 0.1 au in our simulations) also ensures enhanced control
over the evolution of its magnetic structure, facilitating our interpretation
of propagation effects. However, it is important to acknowledge that such an
approach will likely result in a more regular/smooth magnetic configuration in
simulated ICMEs compared to real ICMEs at the same heliocentric distance. To
avoid drawing inappropriate conclusions, in the following we put care into the
interpretation of EUHFORIA results with respect to a baseline model (i.e. a
stand-alone ideal spheromak model).

###  3.1 Baseline Results: Complexity and Coherence in an Ideal Spheromak
Structure

We start discussing the results of our investigation by summarizing the
baseline values obtained from the consideration of an ideal, stand alone
spheromak structure. Characterizing the baseline level of complexity and
coherence in such an idealized condition represents a first but necessary step
to later be able to conduct a meaningful interpretation of the results
obtained for spheromak structures propagating through different solar wind
conditions.

As a first global measure of ME magnetic complexity considered, we report the
frequency of detection of different ME classes as a function of the angular
separation of the spacecraft crossings in the ideal spheromak. We find that
for spacecraft separations of $\Delta\lambda=5^{\circ}$, about 80% of the
detections are $F_{270}$ (corresponding to $\mathbb{C}=0$) and $F_{30}$
(corresponding to $\mathbb{C}=3$), which therefore contribute the most to the
overall complexity of the ME. This result justifies our choice of considering
only the two most frequently detected ME types as metrics of global ME
complexity, and it applies also to simulated spheromaks propagating through
different ambient solar winds, as discussed in Section 3.2. On the other hand,
less represented ME classes (i.e. $F_{180}$, $F_{90}$, and $E$ types,
corresponding to $\mathbb{C}=1,2$, and $4$, respectively) are observed by less
than 10% of the spacecraft and quickly disappear as soon as the spacecraft
angular separation becomes $\Delta\lambda\sim 15^{\circ}$ ($F_{\omega/2}\sim
0.33$) or larger. For increasing spacecraft angular separations, we find that
$\Delta\lambda=25^{\circ}$ (corresponding to $F_{\omega/2}\sim 0.55$ of the
CME half angular width) is the maximum separation that guarantees a
characterization of the magnetic complexity of the ideal spheromak as good as
a $\Delta\lambda=5^{\circ}$ separation in terms of the two most frequently
observed ME classes (i.e. $F_{270}$ and $F_{30}$ types).

The second global measure of ME magnetic complexity considered in this work is
the Moran’s $I$. The Moran’s $I$ is around 0.95 for angular separations
$\Delta\lambda=5^{\circ}$ ($F_{\omega/2}=0.11$), and it decreases at larger
angular scales. The decrease of $I$ for increasing $\Delta\lambda$ reflects
the uncorrelated nature of ME signatures detected by spacecraft crossing the
ME trajectories far apart. Overall, a spheromak structure exhibits a highly
clustered spatial distribution of ME classes ($I\geq 0.5$, indicating a low
complexity) when observed at scales $\Delta\lambda\lesssim 25^{\circ}$
(corresponding to $F_{\omega/2}\lesssim 0.55$), and a more randomly-arranged
ME type distribution ($I<0.5$, indicating a high complexity) at scales
$\Delta\lambda=25^{\circ}-45^{\circ}$ (corresponding to
$F_{\omega/2}=0.55-1$).

Finally, as a proxy for the coherence of an ME, we consider the average
correlation scales of the magnetic field profiles around three reference
observers crossing near the east, center, and west portion of the ME
structure. We find that $B$, $B_{\theta}$ and $B_{\phi}$ show high
correlations throughout the spheromak structure for all the observers
considered, while the spatial correlation pattern for $B_{r}$ appears highly
variable depending on the reference location considered, resulting in much
smaller correlation scales ($\Delta\lambda\sim 0^{\circ}-25^{\circ}$ depending
on the observer). While the exact characteristics of such a pattern are
specific to the spheromak model considered in this study, a lower correlation
in $B_{r}$ than in the remaining magnetic field components has also been
reported in actual observations (Lugaz et al., 2018) and by simple geometrical
considerations, is expected to affect other flux rope models as well for
observers crossing along different trajectories.

###  3.2 Characteristic Scales of Magnetic Complexity during Propagation

Figure 3:  Normalized frequency of ME types detected as a function of the
heliocentric distance and of the spacecraft angular separation, for the two
CMEs simulated in EUHFORIA. (a)–(f): results for run A. (g)–(l): results for
run B. Different panels show the results for different spacecraft separation
considered. In all panels, the continuous lines indicate the results for a
seed location at $(\theta_{s},\phi_{s})=(0^{\circ},0^{\circ})$, while the
shaded areas indicate the uncertainty (i.e. the spread, corresponding to the
range of maximum and minimum values) as obtained by picking 20 random seed
locations.

In what follows, we discuss the simulation results for EUHFORIA runs A and B
with regards to scales of magnetic complexity as the ICME evolves during
propagation. Figure 3 shows the frequency of detection of different ME classes
recovered by virtual spacecraft crossing the MEs through different
trajectories in EUHFORIA runs A and B, as a function of the heliocentric
distance. Figure 3 (a) shows that for an ICME propagating through a quiet,
quasi-uniform solar wind (run A) and observed by spacecraft at $5^{\circ}$
separation, about 70% of all detections are $F_{270}$ or $F_{30}$ ME classes
(corresponding to $\mathbb{C}=0$ and $3$, respectively), which therefore
contribute the most to the overall complexity of the ME. This remains true for
all heliocentric distances sampled. This result applies not only to a
spheromak propagating through a quasi-uniform solar wind, but also to a stand-
alone ideal spheromak structure (see Section 3.1), and it justifies our choice
of considering only the two most frequently detected ME types as metrics of
global ME complexity. Based on this result, we consider a given angular scale
$\Delta\lambda$ (or the corresponding $F_{\omega/2}$) to provide a good
representation of the global magnetic complexity of an ME structure when the
two ME classes most frequently detected are well represented for all
heliocentric distances.

The remaining three classes, i.e. $F_{180}$, $F_{90}$ and $E$ (corresponding
to $\mathbb{C}=1,2$ and $4$, respectively), are more rarely detected (each
between 0% and 20%, depending on the heliocentric distance). This ranking is
well represented up to $\Delta\lambda=25^{\circ}$ (Figure 3 (b)–(e)), while at
larger separations, the ranking between most and least frequently observed
classes is significantly altered depending on the seed location chosen (Figure
3 (f)). We therefore conclude that the maximum $\Delta\lambda$ guaranteeing a
good representation of the ME global magnetic complexity and its evolution
with heliocentric distance is about $25^{\circ}$ for the case without
interactions with other structures. This corresponds to about 7 to 12
observing spacecraft, and to spatial scales equal to $F_{\omega/2}=0.55$ of
the initial CME half angular width. Such scales, as well as the occurrence of
different ME types detected within the ME, are both comparable to the results
retrieved from the analysis of an ideal spheromak structure (as discussed in
Section 3.1), indicating that the magnetic complexity of an ME within an ICME
propagating through a quasi-uniform solar wind remains approximately constant
during propagation, and at all times similar to the one of a pristine
structure.

In contrast, the distribution of ME classes is highly variable with
heliocentric distance when interactions occur. Figure 3 (g) shows that in the
case of an ICME interacting with a preceding HSS/SIR (run B) and observed by
spacecraft at $5^{\circ}$ separation, results are similar to run A only up
until 0.9 au, with the two dominant ME classes being $F_{270}$
($\mathbb{C}=0$) and $F_{30}$ ($\mathbb{C}=3$). At larger heliocentric
distances, the second most frequently detected ME type changes from $F_{270}$
to the more complex $F_{90}$ type ($\mathbb{C}=2$). This indicates that the ME
increases its global complexity during propagation as a consequence of the
interaction with the HSS/SIR (consistent with previous findings by e.g.
Winslow et al., 2016, 2021b; Scolini et al., 2021b, 2022). Additionally, the
ranking between most and least frequently observed classes is well represented
only up to spacecraft separations of $\Delta\lambda=10^{\circ}$ (Figure 3
(h)), while at larger separations it varies depending on the seed location
chosen (Figure 3 (i) to (l)). We conclude that the distribution of ME types,
and therefore of ME complexity as a whole, requires observations by about 50
to 65 spacecraft, separated by about $F_{\omega/2}=0.22$ of the initial CME
half angular width. In this case, the characteristic scale necessary to
capture ME complexity is less than half of the scale retrieved from the
analysis of an ideal spheromak structure (see Section 3.1), and of an ME not
undergoing any interactions. This highlights that smaller angular separations
must be used when investigating magnetic complexity within an ME that
interacts with other large-scale structures compared to ICMEs that propagate
through a quasi-uniform, quiet solar wind.

###  3.3 Spatial Autocorrelation of Magnetic Complexity: Evolution of the
Moran’s $I$ during Propagation

Next, we investigate the spatial distribution of magnetic complexity within
the modeled MEs by considering the Moran’s $I$ evolution with heliocentric
distance.

Figure 4:  Moran’s $I$ as a function of the spacecraft separation
$\Delta\lambda$ and heliocentric distance $r$, for the two ICMEs simulated in
EUHFORIA. (a): Moran’s $I$ resulting from run A. (b): Moran’s $I$ resulting
from run B. The color code and values overplotted in each cell indicate the
Moran’s $I$ obtained for a seed location at
$(\theta_{s},\phi_{s})=(0^{\circ},0^{\circ})$, while the $\uparrow$ and
$\downarrow$ symbols indicate the uncertainty (i.e. the spread, corresponding
to the range of maximum and minimum values) as obtained by picking 20 random
seed locations. The dotted white lines distinguish between regions where
$I>0.5$ (considered regions with well-ordered distribution of ME types and low
magnetic complexity) and regions where $I\leq 0.5$ (considered regions with
randomly-ordered distribution of ME types and high magnetic complexity).

In run A (non-interacting ICME), the Moran’s $I$ decreases with increasing
$\Delta\lambda$ but is essentially unchanged during propagation towards larger
heliocentric distances (Figure 4 (a)). At 0.3 au, the range covered by the
Moran’s $I$ lies between $0.93^{+0.00}_{-0.00}$ for $\Delta\lambda=5^{\circ}$
and $\sim 0.30^{+0.01}_{-0.22}$ for $\Delta\lambda=45^{\circ}$, which is very
similar to values at 1.6 au (between $0.90^{+0.00}_{-0.00}$ for
$\Delta\lambda=5^{\circ}$ and $\sim 0.30^{+0.06}_{-0.19}$ for
$\Delta\lambda=45^{\circ}$). Here, error bars indicate the uncertainty arising
from the re-binning process after picking 20 random seed locations, and it
therefore only affect angular scales larger than $5^{\circ}$. Also, the $I$
values obtained for $\Delta\lambda=5^{\circ}$ are extremely similar to the
values retrieved for an ideal spheromak structure observed at the same angular
scale (as discussed in Section 3.1). The maximum separation at which the
Moran’s $I$ remains above 0.5 for all distances and seed locations considered
is $\Delta\lambda=25^{\circ}$ (corresponding to $F_{\omega/2}=0.55$ of the
initial CME half angular width), as marked by the white dotted line in Figure
4 (a). This also indicates similar levels of clustering in the spatial
arrangement of the ME class between the ME in run A and the ideal spheromak
structure. From these results we conclude that spacecraft swarms composed of
at least 7–12 evenly-spaced spacecraft crossing the same ME structure at
different impact angles from the center to the flanks measure a highly-
clustered distribution of ME types and generally a low level of ME magnetic
complexity, while swarms that are more widely separated observe a high-
complexity structure with a randomly-arranged ME type distribution.

The spatial autocorrelation of ME classes within an ME behaves differently
when interactions with other structures affect its propagation. In run B
(Figure 4 (b)), the ME initially has a similar complexity than in run A, as
indicated by the Moran’s $I$ ranging between $0.94^{+0.00}_{-0.00}$ for
$\Delta\lambda=5^{\circ}$ and $0.28^{+0.00}_{-0.27}$ for
$\Delta\lambda=45^{\circ}$ at 0.3 au. Small separations also report higher
Moran’s $I$ values than large separations: at this stage, we observe a low
complexity ($I>0.5$) up to $\Delta\lambda=25^{\circ}$ of separation
(corresponding to $F_{\omega/2}=0.55$ of the initial CME half angular width),
and a high complexity at larger separations. However, in contrast with run A,
the characteristic scale of transition from a low ($I>0.5$) to a high ($I\leq
0.5$) complexity decreases with heliocentric distance, likely due to the
interaction of the ICME with the HSS/SIR. Specifically, we observe a
transition in the location of the $I=0.5$ threshold from
$\Delta\lambda=25^{\circ}$ (corresponding to $F_{\omega/2}=0.55$ of the
initial CME half angular width) to $\Delta\lambda=20^{\circ}$ of separation
(corresponding to $F_{\omega/2}=0.44$) near 0.6 au. By the time the ME reaches
1.6 au, the range covered by the Moran’s $I$ is between $0.90^{+0.00}_{-0.00}$
for $\Delta\lambda=5^{\circ}$ and $\sim 0.32^{+0.00}_{-0.38}$ for
$\Delta\lambda=45^{\circ}$, i.e. similar to the values retrieved in run A, but
we observe a clear decreasing trend with respect to the heliocentric distance
at all scales $\lesssim 20^{\circ}$, where the Moran’s $I$ values at 1.6 au
are significantly smaller than the initial values retrieved at 0.3 au. This
can be seen in the fact that the error bars inferred from the consideration of
20 random seed locations do not overlap. No clear trend with respect to the
heliocentric distance is apparent at larger scales (where the values at 0.3 au
and 1.6 au are consistent within the error bars inferred from the
consideration of 20 random seed locations), indicating at these scales no
changes to the global ME magnetic complexity are detected during propagation.
Overall, these results indicate that the magnetic complexity quantified in
terms of the spatial autocorrelation of ME types observed across different
directions is affected by the interaction with the HSS/SIR structure only at
separation scales as large as $\Delta\lambda=20^{\circ}$ (at spatial scales
$F_{\omega/2}\leq 0.44$ of the initial CME half angular width). This
corresponds to having a minimum of 13–16 spacecraft crossings through the same
ME to capture the evolutionary trends in the clustering of ME types, as swarms
separated by larger separations would consistently observe a highly-randomized
spatial distribution of ME types and complexity regardless of the specific ME
propagation history.

To confirm the role of the interaction with the HSS/SIR as source of the
Moran’s $I$ increase with heliocentric distance in run B, we consider the
spatial distribution of the Local Moran’s $I$ as a function of the
heliocentric distance in both EUHFORIA runs. Figures 5 (a) and (d) show the
spatial distribution of ME types for runs A and B near 1 au which we use to
derive the Local Moran’s $I$ distributions reported in Figures 5 (b) and (e).
To determine the complexity increase affecting different ME regions, we track
the $(\theta,\phi)$ position of its maximum value (marking the lowest
complexity region) as a function of the heliocentric distance. In run A we
find that the $\phi_{\mathrm{max}(Ii)}$ remains between $-5^{\circ}$ and
$-10^{\circ}$ at all heliocentric distances considered (Figure 5 (c)). The
fact that the location of lowest complexity within the ME does not change with
heliocentric distance is consistent with the lack of interactions altering the
ME structure. In run B, on the other hand, we observe that the maximum Local
Moran’s $I$ progressively drifts towards the east with respect to the
Sun–Earth line, with $\phi_{\mathrm{max}(Ii)}$ shifting from $0^{\circ}$ at
0.3 au to $-23^{\circ}$ at 1.6 au (Figure 5 (f)). In other words, during its
propagation the ME retains a low complexity only on the easternmost region,
while the westernmost region progressively increases its complexity. This is
consistent with the westernmost ME portion being more heavily affected by the
interaction with the HSS/SIR, which may be the prime source of increased
complexity in that region of the ME. Interestingly, we also note that even in
run A, $\phi_{\mathrm{max}(Ii)}$ is slightly offset with respect to the
Sun–Earth line. In the absence of any interplanetary structure in the
simulated ambient solar wind that could have induced such a longitudinal
asymmetry in the spatial distribution of ME complexity, we consider this
result to be intrinsic to the propagation of the ICME through a Parker
magnetic field spiral, which presents an intrinsic asymmetry with respect to
the ICME propagation direction. Such an interpretation is further corroborated
by the results obtained from the spatial analysis of magnetic coherence
presented below in Section 3.4.

Figure 5:  Analysis of the Local Moran’s $I$ near 1 au for the two ICMEs
simulated in EUHFORIA. (a), (d): ME type distribution at 1 au (color coded),
for run A and B, respectively. (b), (e): distribution of the Local Moran’s $I$
($I_{i}$) at 1 au (color coded), for run A and B, respectively. The locations
of the maximum of the Local Moran’s $I$ ($\mathrm{max}(I_{i})$) are indicated
by the magenta diamonds, while contours denote the
$[0.9,0.75,0.6]\,\,\mathrm{max}(I_{i})$ levels. (c), (f): longitude of the
maximum of the Local Moran’s $I$, as a function of the heliocentric distance.

###  3.4 Characteristic Scales of Magnetic Coherence during Propagation

In this section, we investigate the scale at which an ME may be able to
respond to external perturbations in a coherent manner. For this purpose, we
consider the correlation of magnetic field components between different
observers as a proxy for the scale of magnetic coherence, similarly to
previous observational investigations (Wicks et al., 2010; Lugaz et al., 2018;
Ala-Lahti et al., 2020). Determining across which spatial scales MEs exhibit
coherent characteristics is a first, necessary step towards a better
understanding the global response of ME structures to external perturbations
induced by the surrounding environment. As reference directions to investigate
the scales of magnetic coherence, we consider hypothetical observers crossing
the ME structures in regions differently affected by the interaction of the
ICME with the surrounding wind and, in run B, with the large-scale structures
therein. Such reference directions are located along the equatorial plane of
the ME ($\theta_{\mathrm{ref}}=\theta_{\mathrm{CME}}$) and at
$\phi_{\mathrm{ref}}$ angles of $-30^{\circ}$, $0^{\circ}$, and $30^{\circ}$,
i.e. corresponding to observers crossing the ME magnetic structure on the
east, center, and west regions near its central magnetic field axis.

Figure 6:  Average correlation profiles for the magnetic field magnitude and
components at 1 au, for run A (top) and run B (bottom) in EUHFORIA. Three
reference observers are considered: one on the east (left), one near the
center (center), and one on the west (right) of the ME. In all panels, the
solid lines indicate the correlations calculated by averaging across all
possible directions with respect to the reference observer, while the shaded
areas represent the error bars estimated from the standard errors, i.e.
$\pm\mathrm{SE}_{\mathrm{corr}B}$, $\pm\mathrm{SE}_{\mathrm{corr}Br}$,
$\pm\mathrm{SE}_{\mathrm{corr}B\theta}$,
$\pm\mathrm{SE}_{\mathrm{corr}B\phi}$, obtained from considering different
directions around the reference observer. The average radii of correlation,
measures of the characteristic correlation scales within the ME, are marked by
the dashed arrows.

Figure 6 shows the results of the analysis for the average correlation scale
of the magnetic field magnitude and the $B_{r}$, $B_{\theta}$ and $B_{\phi}$
components for run A (panels (a) to (c)) and B (panels (d) to (f)) at 1 au,
for these three reference observers.

By comparing the results from runs A (Figure 6 (a)–(c)) and B (Figure 6
(d)–(f)), we find that all magnetic field components (except for $B_{r}$) have
smaller average correlation scales in run B than in run A, indicating
interactions may significantly reduce the ability of MEs to behave as
magnetically coherent structures. For an observer crossing near the center of
the ME, $B$ remains highly correlated throughout the whole magnetic structure
($\Delta\lambda=45^{\circ}$, corresponding to $F_{\omega/2}=1.00$ of the
initial CME half angular width) for both runs A (panel (b)) and B (panel (e)).
These scales are slightly higher than those reported for an ideal spheromak
structure ($\Delta\lambda=40^{\circ}$, $F_{\omega/2}=0.88$, not shown). We
speculate that expansion/deformation mechanisms (such as those reported within
0.3 au by Scolini et al. (2021a); see Figure 12 therein) may be acting across
the whole ME cross-section during early propagation phases (from 0.1 to 0.3
au) leveling out spatial asymmetries in the total magnetic field profile. The
average correlation scales for $B_{\theta}$ and $B_{\phi}$ drop from
$\Delta\lambda\sim 25^{\circ}-30^{\circ}$ ($F_{\omega/2}\sim 0.55-0.66$) in
run A, to $\Delta\lambda\sim 10^{\circ}-15^{\circ}$ ($F_{\omega/2}\sim
0.22-0.33$) in run B. In run A the correlation scale is already significantly
reduced compared to the ideal spheromak case (where
$\Delta\lambda=40^{\circ}$, $F_{\omega/2}=0.88$, not shown), and even more so
in run B. Only the average correlation scale for $B_{r}$ slightly increases
from $\Delta\lambda=5^{\circ}$ ($F_{\omega/2}=0.11$) to
$\Delta\lambda=15^{\circ}$ ($F_{\omega/2}=0.33$), and both runs report higher
correlation scales than in the ideal spheromak case (where the correlation was
lost only a few degrees away from the reference observer). However,
$\mathrm{SE}_{\mathrm{corr}Br}$ is generally larger than for the other
components in both EUHFORIA runs, reflecting the fact that the actual average
correlation scale is highly asymmetric around the reference observer, and
therefore is affected by large error bars.

The results are qualitatively the same for an observer crossing through the
east and west of the ME. For both observers, $B$ remains highly correlated
throughout the whole magnetic structure in both runs A and B
($\Delta\lambda=65^{\circ}-70^{\circ}$, corresponding to
$F_{\omega/2}=1.44-1.55$). The average correlation scales for $B_{\theta}$ and
$B_{\phi}$ reduce by a factor $\sim 2$ or more between run A and run B. For an
east observer, the correlations reduce from $\Delta\lambda\sim
35^{\circ}-45^{\circ}$ ($F_{\omega/2}\sim 0.77-1.00$) in run A to
$\Delta\lambda\sim 20^{\circ}-25^{\circ}$ ($F_{\omega/2}\sim 0.44-0.55$) in
run B. For a west observer, the average correlation scale for $B_{\theta}$
decreases from $\Delta\lambda=15^{\circ}$ ($F_{\omega/2}=0.33$) to
$\Delta\lambda=5^{\circ}$ ($F_{\omega/2}=0.11$), while that of $B_{\phi}$
decreases from $\Delta\lambda=20^{\circ}$ ($F_{\omega/2}=0.44$) to
$\Delta\lambda=15^{\circ}$ ($F_{\omega/2}=0.33$) between run A and B. These
results clearly demonstrate how correlations for a west observer in
$B_{\theta}$ and $B_{\phi}$ are up to a factor 4 smaller than correlations for
an east observer. The average correlation scale for $B_{r}$ remains roughly
constant around $\Delta\lambda\sim 10^{\circ}-15^{\circ}$ ($F_{\omega/2}\sim
0.22-0.33$) in both runs.

It is worth mentioning that even in the case without interactions, smaller
correlation scales are found for a west observer compared to an east observer.
Due to the lack of structures in the simulated ambient solar wind in run A, we
consider this result to be linked to the propagation of the ICME through a
magnetic field characterized by a Parker spiral configuration, which presents
an intrinsic asymmetry with respect to the ICME propagation direction that
leads to different ICME–solar wind interaction behaviors at the two ICME
flanks (e.g. in terms of pressure balance; see Wang et al., 2004). Differences
between the east and west observers are even more pronounced in run B, and
this is interpreted as due to the superposition of the effects related to the
east-west Parker spiral asymmetry, and the interaction with the HSS/SIR mainly
affecting the western portion of the ICME.

In light of these results, we clarify why we use a different definition of
correlation scale compared to observational works (Lugaz et al., 2018; Wicks
et al., 2010). First of all, in both the ideal spheromak structure and
EUHFORIA simulations we found no scale at which correlations in the $B$,
$B_{\theta}$ and $B_{\phi}$ components reach values near zero. This is more
likely due to the very idealized ME structure considered, while actual MEs may
have more complex configurations already following their eruption in the low
solar corona and before reaching 0.1 au. Secondly, contrary to the ME and
solar wind profiles reported by Lugaz et al. (2018) and Wicks et al. (2010),
our correlation does not scale in a quasi-linear or quasi-exponential manner,
so we decided to refrain from fitting curves with specific functions and
rather defined a measure of coherence that was more appropriate for our
specific problem.

Despite these methodological differences, the results retrieved in our
numerical simulations have several elements in common with observational
estimates. Specifically, the ranking of the average correlation scales for the
different magnetic field components (with $B$ and $B_{r}$ being the most and
least correlated, respectively) is consistent with what was found within an
ideal spheromak structure and with estimates obtained from in situ
observations of real MEs by Lugaz et al. (2018). Furthermore, the range of
average correlation scales for $B$ (estimated around $45^{\circ}$ for a
central observer, and $15^{\circ}$ to $70^{\circ}$ for a flank observer)
encompasses previous constraints on the characteristic scale of magnetic
coherence at 1 au by Lugaz et al. (2018) ($12^{\circ}-20^{\circ}$) and Owens
(2020) ($\sim 26^{\circ}$). The range of average correlation scales for
$B_{\theta}$ and $B_{\phi}$ (estimated around $10^{\circ}$ to $30^{\circ}$ for
a central observer, and between $5^{\circ}$ and $45^{\circ}$ for a flank
observer) also encompasses previous estimates on the characteristic scale of
magnetic coherence at 1 au for the magnetic field components, reported in
range $4^{\circ}-7^{\circ}$ by Lugaz et al. (2018). The larger upper limit
found in our study is somewhat expected: the global, ideal MHD nature of our
numerical model, as well as the simplified set-up used in this study
(including the insertion of a perfectly-known global CME magnetic structure at
the inner boundary of the heliospheric domain, located at 0.1 au in our
simulations) likely resulted in a more regular/smooth magnetic configuration
in simulated MEs compared to real MEs at the same heliocentric distance.
Additionally, we note the diverse nature of previous studies (in terms, e.g.,
of working assumptions, data sets, range of parameters explored) when compared
to each other and to this investigation, which may have also led to different
coherence scale estimates across these studies.

We conclude the analysis of magnetic coherence by considering the evolution of
the ME average correlation scales with heliocentric distance. In Figure 7, we
show the ME average correlation scales for the total magnetic field and the
three magnetic field components, as a function of the heliocentric distance,
for the two EUHFORIA simulations and a reference observer crossing near the ME
center. We find that the characteristic correlation scales for an ICME
propagating through a homogeneous solar wind are unchanged during its
propagation (Figure 7 (a)), remaining around $F_{\omega/2}=1$ for $B$,
$F_{\omega/2}=0.44-0.66$ for $B_{\theta}$ and $B_{\phi}$, and below
$F_{\omega/2}=0.22$ for $B_{r}$ at all heliocentric distances. On the other
hand, an interaction with a HSS/SIR induces a clear decrease in the ME average
correlation scales during propagation, particularly for $B_{\theta}$ and
$B_{\phi}$ (Figure 7 (b)). The decrease is more rapid between 0.3 au and 1 au,
while it continues at smaller rates up to 1.6 au. Overall, the maximum scale
of correlation reduces from $F_{\omega/2}=1$ to $0.44$ for $B$, from
$F_{\omega/2}=0.66$ to 0.11 for $B_{\theta}$, from $F_{\omega/2}=0.44$ to 0
for $B_{\phi}$, and from $F_{\omega/2}=0.33$ to 0 for $B_{r}$ between 0.3 au
and 1.6 au. Interactions with other large-scale structures may therefore
induce a decrease by a factor of 3 to 6 in the ME magnetic field correlation
scales.

Figure 7:  Characteristic scale of correlation for the total magnetic field
and the three magnetic field components, as a function of the heliocentric
distance, for the two spheromak CMEs simulated in EUHFORIA. The reference
location is taken at $(\theta,\phi)=(0^{\circ},0^{\circ})$ for run A (a), and
$(\theta,\phi)=(5^{\circ},0^{\circ})$ for run B (b), corresponding to the CME
initial direction in the two simulations. In all panels, the continuous lines
indicate the average correlation scale with respect to the reference observer,
while the shaded areas represent the error bars on the average correlation
scale estimated from the standard errors, i.e.
$\pm\mathrm{SE}_{\mathrm{corr}B}$, $\pm\mathrm{SE}_{\mathrm{corr}Br}$,
$\pm\mathrm{SE}_{\mathrm{corr}B\theta}$,
$\pm\mathrm{SE}_{\mathrm{corr}B\phi}$, obtained from considering different
directions around the reference observer.

##  4 Implications for Ecliptic Missions

Figure 8:  Analysis of the ME type distribution and Moran’s $I$ as a function
of the heliocentric distance, at $\theta=\theta_{\mathrm{CME}}$, for the two
ICMEs simulated in EUHFORIA. (a), (b): distribution of ME types (color coded)
as a function of the heliocentric distance and longitude for runs A and B,
respectively. (c), (d): Moran’s $I$ (also color coded) as a function of the
heliocentric distance and spacecraft separation $\Delta\lambda$, for runs A
and B, respectively. The dotted white lines distinguish between regions where
$I>0.5$ (considered regions with well-ordered distribution of ME types and low
magnetic complexity) and regions where $I\leq 0.5$ (considered regions with
randomly-ordered distribution of ME types and high magnetic complexity).

In this section, we focus on spacecraft near the CME equatorial plane (which
is close to the solar equatorial and ecliptic planes in our particular case)
in order to support the planning and interpretation of observations by future
multi-spacecraft missions exploring ICME structures in the ecliptic plane. The
simulation set-up considered here specifically applies to spacecraft in the
near-ecliptic region that cross through low-inclination FRs, which constitute
about half of the ME FR configurations detected within 1 au (e.g. Lepping et
al., 1990; Bothmer & Schwenn, 1998; Lynch et al., 2003), and the majority of
cases observed between 1 au and 2.2 au (Davies et al., 2022).

In order to determine the main implications for ecliptic missions, we focus on
spacecraft crossings in our EUHFORIA-simulated ICMEs close to the ecliptic
plane and near the spheromak central magnetic axis, i.e. at
$\theta=\theta_{\mathrm{CME}}$. $\theta_{\mathrm{CME}}$ corresponds to
$\theta=0^{\circ}$ in run A and to $\theta=5^{\circ}$ in run B, and we chose
these latitudinal values to more fairly compare crossings across the ICMEs
between the two runs. The main results are displayed in Figure 8 (a) and (b),
which show the spatial distribution of ME types detected at different
heliocentric distances by simulated spacecraft located at
$\theta=\theta_{\mathrm{CME}}$ in runs A and B. In the case of run A, i.e.
ICME with no interaction, Figure 8 (a) shows a relatively well-ordered
distribution of ME types, with spacecraft aligned in different radial
directions recording similar ME types as the ME propagates to larger
heliocentric distances. The dominance of $F_{270}$ and $F_{180}$ types
(associated with $\mathbb{C}=0$ and $1$, respectively) suggests the ME is less
complex in its equatorial region than it is globally. This result is
consistent with the spatial distribution of ME types obtained for an ideal
spheromak structure (as shown in Figure 6 (d) in Scolini et al. (2021b)).
Interestingly, we find that even without interactions with large-scale
structures, a change in the most frequently observed ME type is observed
around 0.8 au, where $F_{270}$ types are taken over by $F_{180}$. $F_{270}$
types are initially dominant and spread across different directions, but
progressively concentrate only in the eastern part of the ME, while $F_{180}$
types become dominant in the western part. As discussed above, this is likely
a reflection of Parker spiral effects induced by propagation through an
asymmetric interplanetary magnetic field. $F_{90}$, $F_{30}$, and $E$ types
(corresponding to $\mathbb{C}=2,3$ and $4$, respectively) are all very rarely
observed in the ecliptic plane, making up less than 20% of the observed cases.
The analysis of the results considering different angular separations (not
shown) reveals that only spacecraft separated by $\Delta\lambda\lesssim
15^{\circ}$ ($F_{\omega/2}\lesssim 0.33$, corresponding to $\sim 6$ spacecraft
crossing the ME in different directions near its central axis) can
characterize the ecliptic complexity of the ME observed by spacecraft at
$\Delta\lambda\lesssim 5^{\circ}$ ($F_{\omega/2}\lesssim 0.11$) across all
heliocentric distances. However, we emphasize that regardless of their angular
separation, observations in the ecliptic plane alone are not sufficient to
characterize the global complexity of the ME (where $F_{270}$ and $F_{30}$
were the two dominant ME types; see Section 3.2), and off-ecliptic
observations are therefore required.

In run B (ICME with interaction), we observe a much more irregular
distribution of the ME types in both the longitudinal and radial directions
(Figure 8 (b)). On the ME equatorial plane we find that the most common ME
types are $F_{270}$ up until 1 au, after which $F_{180}$, $F_{90}$ and $E$
types become comparable. Just as for run A, we therefore conclude that the ME
is less complex near its central axis than it is globally. Results at
different angular separations also indicate that a swarm of at least $\gtrsim
9$ near-equatorial spacecraft separated by $\Delta\lambda\lesssim 10^{\circ}$
($F_{\omega/2}\lesssim 0.22$) would be required to capture the ecliptic ME
spatial complexity and its evolution across all heliocentric distances.
Furthermore, similarly to the case of run A, off-ecliptic observations are
required to characterize the global complexity of the ME as discussed in
Section 3.2.

To calculate the Moran’s $I$, we have to consider a 2-D spacecraft array
distributed in longitude and latitude (see Equation 1). Therefore, we compute
the Moran’s $I$ at each heliocentric distance from near-equatorial regions in
EUHFORIA simulations by considering a near-equatorial array of spacecraft
within $\pm 15^{\circ}$ from $\theta=\theta_{\mathrm{CME}}$. This array
consists of $\Delta\phi=\Delta\theta=5^{\circ}$ separated spacecraft arranged
in $3\times 2+1=7$ latitudinal rows. Figure 8 (c) and (d) report the results
of the analysis. In general, we observe higher values of $I(r)$ compared to
Figure 4, indicating that the ME exhibits a higher spatial autocorrelation in
near-equatorial regions than it does in the perpendicular direction. We obtain
a similar result in the case of an ideal spheromak structure (not shown),
suggesting that an ideal or weakly-perturbed spheromak structure is
particularly ordered near its magnetic axis. For run A (Figure 8 (c)), we find
that at all heliocentric distances, $I(r)$ decreases with $\Delta\lambda$ and
remains above 0.5 up until $\Delta\lambda=30^{\circ}$ ($F_{\omega/2}=0.66$),
which is slightly larger than the separation scale obtained when considering
the whole ME structure (Figure 4 (a)). This further confirms the higher
spatial clustering of ME types within the ME in the near-equatorial regions
compared to other directions. Results are also slightly decreasing with
heliocentric distance at scales $\Delta\lambda\leq 20^{\circ}$
($F_{\omega/2}=0.44$), while they are independent on the propagation phase at
larger angular separations. For run B (Figure 8 (d)), we obtain similar
results to run A in terms of the scale at which $I(r)$ becomes smaller than
0.5 ($\Delta\lambda=30^{\circ}$, corresponding to $F_{\omega/2}=0.66$).
However, we also notice that $I(r)$ shows a more clear dependence on the
heliocentric distance up to scales $\Delta\lambda\leq 40^{\circ}$
($F_{\omega/2}=0.88$), suggesting that the perturbation to the ME induced by
the interaction with the HSS/SIR extends across a wider range of angular
scales in the near-equatorial plane compared to other directions (Figure 4
(b)).

Figure 9:  Correlation maps (color coded) for the total magnetic field and the
three magnetic field components with respect to a central reference observer
($\phi_{\mathrm{ref}}=0^{\circ}$), as a function of the heliocentric distance
and longitude at $\theta=\theta_{\mathrm{CME}}$, for run A (top) and run B
(bottom) in EUHFORIA. (a), (e): $B$. (b), (f): $B_{r}$. (c), (g): $B_{\phi}$.
(d), (h): $B_{\theta}$. In each panel, black contours mark regions were the
correlation is $+0.6$ (continuous lines) or $-0.6$ (dashed lines).

The correlation with respect to a central reference observer tends to be
maximum along the ME magnetic axis (i.e. near the equatorial plane): focusing
on the results at $\theta=\theta_{\mathrm{CME}}$, we find that for both an
ideal spheromak structure (not shown) and for run A (as shown in Figure 9 (a)
to (d)), the correlation extends throughout the whole ME width. These results
confirm that an ideal or weakly-perturbed spheromak structure exhibits
coherent signatures across wider scales particularly along its principal
magnetic axis. In run B (panels (e) to (h)), the correlation scale remains
generally large for $B$ up to 1.3 au, and it then reaches a minimum of
$5^{\circ}$ ($F_{\omega/2}=0.11$) in the west direction and $20^{\circ}$
($F_{\omega/2}=0.44$) in the east direction around 1.5 au. For $B_{\theta}$
and $B_{\phi}$, the ME westernmost region becomes progressively more perturbed
than the easternmost one during propagation, up to the point of losing all
correlations with a central reference observer. Near 1.5 au, the correlation
scale is between $\lesssim 5^{\circ}$ ($F_{\omega/2}\lesssim 0.11$) in the
west direction and $35^{\circ}$ ($F_{\omega/2}=0.77$) in the east direction
for $B_{\phi}$; and between $5^{\circ}$ ($F_{\omega/2}=0.11$) in the west
direction and $45^{\circ}$ ($F_{\omega/2}=1$) in the east direction for
$B_{\theta}$.

Finally, we note that for spacecraft crossings near the spheromak magnetic
field axis (either for an ideal spheromak structure, or in EUHFORIA
simulations), ME signatures are expected to be similar to the ones encountered
by crossing near the axis of cylindrical FR geometries, i.e. with bipolar
$B_{\theta}$ signatures and nearly-unipolar signatures in $B_{\phi}$ (see
Figure 6 in Scolini et al., 2021b). Results along the direction
$\theta=\theta_{\mathrm{CME}}$ in each EUHFORIA simulation can therefore be
expected to be less dependent on the specific FR model used than in other
directions, and we hypothesize they could be potentially extrapolated to other
FR models as well. For the opposite reason, the most severe difference between
the spheromak model and other cylindrical FR models is expected to be found in
the direction perpendicular to the magnetic axis (e.g. as in the case of
ecliptic crossings through high-inclination FR types), but we leave the
consideration of this alternative scenario for future studies.

##  5 Conclusions

| Whole structure | Equatorial plane
---|---|---
| Run A | Run B | Run A | Run B
Complexity: ME types |  $25^{\circ}$(0.55) |  $10^{\circ}$(0.22) |  $15^{\circ}$(0.33) |  $10^{\circ}$(0.22)
Complexity: autocorrelation |  $25^{\circ}$(0.55) |  $20^{\circ}$(0.44) |  $30^{\circ}$(0.66) |  $30^{\circ}$(0.66)
Coherence: $B$ |  45∘(1.00) |  20–45∘(0.44–1.00) |  35–40∘(0.77–0.88) |  5–20∘(0.11–0.44)
Coherence: $B_{r}$ |  0–10∘(0.00–0.22) |  0–5∘(0.00–0.11) |  10–25∘(0.22–0.55) |  5–10∘(0.11–0.22)
Coherence: $B_{\theta}$ |  15–30∘(0.33–0.66) |  0–5∘(0.00–0.11) |  35–40∘(0.77–0.88) |  5–45∘(0.11–1.00)
Coherence: $B_{\phi}$ |  20–30∘(0.44–0.66) |  5–10∘(0.11–0.22) |  30–35∘(0.66–0.77) |  5–35∘(0.11–0.77)
Table 1:  Summary of the characteristic scales of complexity and coherence
within MEs for a central observer (between 0.3 and 1.6 au) retrieved in this
work. Values are expressed in terms of the spacecraft angular separation
$\Delta\lambda$ and of the fraction of CME initial half angular width
$F_{\omega/2}$ (in brackets).

In this study, we presented an in-depth investigation of the spatial scales of
magnetic complexity and coherence within MEs as inferred from spacecraft
swarms in global heliospheric simulations based on the consideration of
various ICME-solar wind interaction scenarios and different complexity and
coherence metrics. This work is also the first-ever attempt to investigate ME
magnetic coherence scales by analyzing the results of global numerical
simulations in an observationally-consistent fashion. Specifically, we
evaluated the magnetic complexity of the simulated MEs based on the frequency
of ME types detected by virtual spacecraft crossing the MEs along different
directions during their propagation through interplanetary space. We further
explored the spatial distribution of magnetic complexity within the simulated
MEs using the Moran’s $I$ spatial autocorrelation index. We determined the
characteristic scale of magnetic coherence within the MEs by evaluating the
maximum angular separation at which virtual spacecraft crossing the same ME
observed magnetic field signatures highly correlated to those detected by a
reference observer. A summary of the characteristic scales derived from
numerical simulation between 0.3 and 1.6 au is provided in Table 1, and the
main conclusions are summarized as follows:

  1. 1.

On the one hand, analyses of the frequency and spatial clustering of ME types
within MEs indicate that their global magnetic complexity does not change with
heliocentric distance if no interaction with other large-scale solar wind
structures occurs during their propagation. On the other hand, the ME magnetic
complexity is likely to increase during its propagation if interactions with
other large-scale solar wind structures occur, confirming previous results by
Winslow et al. (2016, 2021b) and Scolini et al. (2021b, 2022).

  2. 2.

The two global magnetic complexity metrics considered (frequency of ME types,
and the Moran’s $I$ spatial autocorrelation index) both indicate that the
consideration of smaller angular scales is required to capture global
complexity changes in the presence of interactions. As demonstrated in Section
3.2, for an ICME with initial half angular width of ${45}^{\circ}$ as the one
simulated in this work, the spatial characterization of the occurrence of ME
types requires in situ observations at separations no larger than
$\Delta\lambda=25^{\circ}$, but separations no larger than
$\Delta\lambda=10^{\circ}$ are needed in the case of ICME–solar wind
interacting scenarios. Section 3.3 proved that the analysis of the spatial
autocorrelation of ME types via the Moran’s $I$ coefficient requires
observations at separations up to $\Delta\lambda=25^{\circ}$ for a non-
interacting ICME, which reduces to $\Delta\lambda=20^{\circ}$ or less for
ICME–HSS/SIR interacting scenarios. The angular separations above are more
conveniently expressed in terms of scale-invariant quantities, particularly as
fractions of the CME initial half angular width ($F_{\omega/2}$) and in terms
of the associated number of virtual spacecraft required to cross the ME
structure, which are independent of the prescribed initial CME half angular
width and can therefore be extrapolated to CMEs of different sizes. In this
respect, we found that the spatial characterization of the occurrence of ME
types within an ME requires in situ observations at separations
$F_{\omega/2}\leq 0.55$, corresponding to a minimum of 7 to 12 spacecraft
crossings. Separations $F_{\omega/2}\leq 0.22$, corresponding to a minimum of
50 to 65 spacecraft crossings, are needed in the case of ICME–solar wind
interacting scenarios. The analysis of the spatial autocorrelation of ME types
via the Moran’s $I$ coefficient requires observations at separations up to
$F_{\omega/2}=0.55$, corresponding to 7 to 12 spacecraft crossings for a non-
interacting ICME, which reduces to $F_{\omega/2}\leq 0.44$, corresponding to
13 to 16 spacecraft crossings or less for ICME–HSS/SIR interacting scenarios.

  3. 3.

In our numerical simulations, the scale of correlation of the magnetic field
components within MEs was different for different components, and tended to be
the largest for $B$, followed by $B_{\theta}$ and $B_{\phi}$, and ultimately
by $B_{r}$. As discussed in Section 3.4, the characteristic scale of
correlation reduced by a factor of 3–6 for MEs within ICMEs interacting with
preceding HSSs/SIRs ($\Delta\lambda=5^{\circ}$, corresponding to
$F_{\omega/2}=0.11$) compared to ICMEs that do not interact with other large-
scale structures ($\Delta\lambda=15^{\circ}-30^{\circ}$, corresponding to
$F_{\omega/2}=0.33-0.66$). Results at different heliocentric distances also
indicated that as ICMEs propagate to larger heliocentric distances,
intervening interactions may efficiently destroy any pre-existing correlation
scale within their MEs. Overall, such results suggest that the scale of
magnetic coherence of ME structures would be optimally investigated by having
spacecraft crossing through them at angular separations between
$\Delta\lambda=15^{\circ}$ and $30^{\circ}$ ($F_{\omega/2}=0.33-0.66$) in
cases where no significant interactions with other large-scale interplanetary
structures occur. However, angular separations down to
$\Delta\lambda=5^{\circ}$ ($F_{\omega/2}=0.11$) and below, are needed in cases
where significant interactions with other large-scale interplanetary
structures do occur.

Similar conclusions were reached by Farrugia et al. (2011) when analyzing an
ICME interacting with a following HSS/SIR and observed at 1 au by three
spacecraft at $\sim 20^{\circ}$ separation. At the time, the authors
emphasized the need for numerical simulations and/or more in situ observations
for a complete elucidation of ICMEs interacting with SIRs, and the present
work helps identify the most appropriate scales to be targeted in future
studies to fill in current observation and knowledge gaps. Additionally, our
results indicate that achieving small spacecraft angular separations would be
particularly important at larger distances from the Sun, as the correlation
scale is found to progressively shrink during propagation, the longer the
interaction with large-scale solar wind structures occurs. For the opposite
reason, when considering multi-spacecraft observations obtained in radial
alignment, making sure that the angular offset among the different spacecraft
involved is smaller than the aforementioned expected scales of coherence would
be most appropriate to distinguish changes in magnetic signatures that are
actually due to evolutionary effects, rather than to the intrinsic lack of
correlation affecting observations acquired at larger angular separation
scales.

  4. 4.

At least for fast ICMEs as the ones simulated in this work, the magnetic
complexity inferred from the spatial autocorrelation of ME types via the Local
Moran’s $I$ coefficient was found to be higher in the western portion of MEs
than on the eastern one regardless of the presence or absence of interactions
with other large-scale solar wind structures. Additionally, smaller
correlation scales were found around a reference observer crossing on the
western portion of the ME, compared to one on crossing the eastern portion.
East-west differences were more prominent when an interaction with a preceding
HSS/SIR was involved, but were present also in the ICME propagating through a
quasi-homogeneous solar wind. These results are found to be due to a
combination of two effects: a Parker spiral contribution that affects both
interacting and non-interacting ICMEs; and a contribution from the actual
interaction of ICMEs with preceding large-scale solar wind structures, which
is localized in the region most influenced by the interaction.

  5. 5.

Analyses of the magnetic complexity and coherence in the near-ecliptic plane
(approximately corresponding to the direction of the ME magnetic axis) in
Section 4 indicated that MEs may exhibit a lower complexity, higher spatial
autocorrelation, and higher coherence along their magnetic axis compared to
perpendicular directions. In this case, a minimum of $\sim 6$ spacecraft
crossings separated by $F_{\omega/2}\lesssim 0.33$ (corresponding to
$\Delta\lambda\lesssim 15^{\circ}$) is necessary to characterize the evolution
of ME magnetic complexity in the near-equatorial plane, which increases to
$\sim 9$ spacecraft ($\Delta\lambda\lesssim 10^{\circ}$, corresponding to
$F_{\omega/2}\lesssim 0.22$) when interaction with preceding HSSs/SIRs are
involved. However, also in the case of interactions with preceding large-scale
solar wind structures, near-ecliptic observations alone were not able to
characterize the ME global, off-ecliptic complexity.

Overall, our estimates for the scale of magnetic correlation of $B_{\theta}$
and $B_{\phi}$ encompass previous constraints on the characteristic scale of
magnetic coherence at 1 au, reported in range $4^{\circ}-7^{\circ}$ by Lugaz
et al. (2018). The range of correlation scales obtained for $B$ is also
consistent with previous estimates by Lugaz et al. (2018)
($12^{\circ}-20^{\circ}$) and Owens (2020) ($\sim 26^{\circ}$). However, due
to the numerical model and set-up’s specificities, and the multi-scale nature
of the magnetic coherence problem, larger upper limits found in our study are
somewhat expected. Additionally, while from an observational standpoint the
correlation of magnetic field profiles among different locations within an ME
has been regarded as a proxy for magnetic coherence (Lugaz et al., 2018), it
also remains unclear how the ability of information to propagate across an ME
affects its local properties, and how this relates to the ability of the same
ME to use such information in order to resist deformation by external factors
and behave as a coherent structure (Owens, 2020). While this work helps
constrain some of the critical scales controlling the evolution of ME
structures, and establishes new pathways for the investigation of ME magnetic
coherence by analyzing the results of global numerical simulations in an
observationally-consistent fashion (i.e. using spacecraft swarms), in the
future, we envisage more sophisticated numerical tools in combination with new
data from recently-launched missions (e.g. Parker Solar Probe and Solar
Orbiter) will help pinpoint the scale and meaning of magnetic coherence within
ICMEs more precisely.

C.S. acknowledges the NASA Living With a Star Jack Eddy Postdoctoral
Fellowship Program, administered by UCAR’s Cooperative Programs for the
Advancement of Earth System Science (CPAESS) under award no. NNX16AK22G.
R.M.W. acknowledges support from NASA grant 80NSSC19K0914 as well as partial
support from the NASA STEREO Transition grant. N.L. was supported by NASA
grants 80NSSC20K0700 and 80NSSC19K0831. S.P. acknowledges support from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No 870405 (EUHFORIA 2.0), the ESA project “Heliospheric modelling
techniques” (Contract No. 4000133080/20/NL/CRS), and the projects C14/19/089
(C1 project Internal Funds KU Leuven), G.0D07.19N (FWO-Vlaanderen), SIDC Data
Exploitation (ESA Prodex-12), and B2/191/P1/SWiM (Belspo). EUHFORIA is
developed as a joint effort between the University of Helsinki and KU Leuven.
Simulations were carried out at KU Leuven and the VSC – Flemish Supercomputer
Center (funded by the Hercules foundation and the Flemish Government –
Department EWI).

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