# Correlation of ICME Magnetic Fields at Radially Aligned Spacecraft

## Abstract

The magnetic field structures of two interplanetary coronal mass ejections (ICMEs), each observed by a pair of spacecraft close to radial alignment, have been analysed. The ICMEs were observed *in situ* by MESSENGER and STEREO-B in November 2010 and November 2011, while the spacecraft were separated by more than 0.6 AU in heliocentric distance, less than 4° in heliographic longitude, and less than 7° in heliographic latitude. Both ICMEs took approximately two days to travel between the spacecraft. The ICME magnetic field profiles observed at MESSENGER have been mapped to the heliocentric distance of STEREO-B and compared directly to the profiles observed by STEREO-B. Figures that result from this mapping allow for easy qualitative assessment of similarity in the profiles. Macroscale features in the profiles that varied on timescales of one hour, and which corresponded to the underlying flux rope structure of the ICMEs, were well correlated in the solar east–west and north–south directed components, with Pearson’s correlation coefficients of approximately 0.85 and 0.95, respectively; microscale features with timescales of one minute were uncorrelated. Overall correlation values in the profiles of one ICME were increased when an apparent change in the flux rope axis direction between the observing spacecraft was taken into account. The high degree of similarity seen in the magnetic field profiles may be interpreted in two ways. If the spacecraft sampled the same region of each ICME (*i.e.* if the spacecraft angular separations are neglected), the similarity indicates that there was little evolution in the underlying structure of the sampled region during propagation. Alternatively, if the spacecraft observed different, nearby regions within the ICMEs, it indicates that there was spatial homogeneity across those different regions. The field structure similarity observed in these ICMEs points to the value of placing *in situ* space weather monitors well upstream of the Earth.

## Keywords

Interplanetary coronal mass ejections Flux ropes Inner heliosphere Radially aligned spacecraft## 1 Introduction

Interplanetary coronal mass ejections (ICMEs) are discrete, large-scale magnetic field and plasma structures observed in the solar wind (*e.g.* Forsyth and Gosling 2001). ICMEs variously display enhanced magnetic field strengths, depressed proton temperatures, bi-directional electron strahls, as well as a range of other *in situ* signatures (Zurbuchen and Richardson 2006). Originating in the solar corona, ICMEs typically take 1 – 4 days to reach 1 AU, and they form a direct link between the solar and terrestrial environments. Fast ICMEs with large and sustained southward magnetic field components are the primary drivers of adverse space weather at the Earth (Eastwood 2008).

Sustained periods of geoeffective southward field are often associated with the subset of ICMEs that display a flux rope field geometry. Magnetic flux ropes consist of nested, helical field lines wound around a central axis, where the pitch angle of the field relative to the axis direction decreases as the axis is approached. A spacecraft passing through a flux rope will observe a smooth rotation of the magnetic field direction over a wide angle. ICMEs that display flux rope geometries and low plasma-\(\beta \) are known as magnetic clouds (Burlaga *et al.*1981). It has been suggested that all ICMEs contain flux ropes, and that ICMEs observed without flux rope signatures are intersected by the observing spacecraft at the periphery of the ICME, away from the centrally located flux rope (*e.g.* Russell, Shinde, and Jian 2005; Richardson and Cane 2010). All current models of (I)CME initiation in the solar corona incorporate flux ropes, either as pre-existing structures or as by-products of the initiation process (Chen 2011).

ICMEs may undergo non-radial deflections (Wang *et al.*2014), rotations (*e.g.* Nieves-Chinchilla *et al.*2013; Good *et al.*2015; Winslow *et al.*2016), and reconnective erosion (*e.g.* Ruffenach *et al.*2012) during propagation through interplanetary space. The prevalence of these effects within the inner heliosphere, and the degree to which they alter the underlying structure of ICMEs, remain open questions. In this study, the magnetic field structures of two ICMEs observed by a pair of spacecraft close to radial alignment with the Sun and separated by approximately 0.6 AU have been examined. These observations offer snapshots of the ICMEs at different stages in their propagation through the inner heliosphere. The extent to which the magnetic field structures of the ICMEs remained intact has been considered through direct comparison of the field profiles at each spacecraft. It has been found that the two ICMEs examined in this study displayed robust field structures, with at least one of the ICMEs showing evidence of rotation in its flux rope axis.

Both ICMEs displayed a flux rope structure, and both were magnetic clouds. The flux rope profiles have been mapped from the inner spacecraft to the heliocentric distances of the outer spacecraft through the application of a simple technique; the mapping has been performed in a way that factors out the radial expansion and drop in field magnitudes that occurred during propagation to allow direct comparison of the underlying field structure at the different observation points. Inputs required to perform the mapping include the spacecraft separation distance, the arrival times of the flux rope’s leading and trailing edges at each spacecraft, and the magnetic field profiles observed at each spacecraft. Qualitative assessments of similarity in the magnetic field profiles are made from figures of overlapped data that result from the mapping, and similarity is quantified through the calculation of correlation coefficients for each field component.

In this work, we place particular emphasis on drawing conclusions directly from observations while making minimal assumptions about the ICMEs’ global field structure. For example, no assumption is made as to whether the fields are force-free, and no particular cross-sectional shape for the flux ropes is assumed. Given the lack of plasma data at the inner spacecraft for both of the ICMEs studied, we do not attempt an analysis of the kind performed by Nakwacki *et al.* (2011), who considered, amongst other things, the evolution in magnetic flux, helicity, energy and expansion rate of a magnetic cloud observed *in situ* by radially aligned spacecraft at 1 and 5.4 AU.

We also note that this work does not provide any scheme for making predictions of arrival times, speeds or field magnitudes at an outer spacecraft based on *in situ* observations at an inner spacecraft; rather, these variables are obtained from the observations at both spacecraft *a posteriori* to produce best-fit mappings. This contrasts, for example, with the mapping technique recently introduced by Kubicka *et al.* (2016), who used inputs from one imager and an *in situ* spacecraft to predict ICME parameters at a second, radially aligned spacecraft.

In Section 2, the data mapping technique is described, and the results of its application to the two ICMEs are presented. An analysis of the overall correlation in the profiles, and an analysis of differences in correlation of features at microscopic and macroscopic scales, is also included in this section. A full discussion of the results and their interpretation is presented in Section 3.

## 2 Event Analysis

In this work, we examine the magnetic field structure of two ICMEs observed by a pair of radially aligned spacecraft. The two ICMEs have been selected for analysis because they both displayed flux rope structures that were clearly observed by each of the aligned spacecraft, because both ICMEs displayed relatively unambiguous boundaries, and because the observing spacecraft were reasonably well aligned. Moreover, the observing spacecraft were sufficiently well separated in radial distance for any evolution that may have occurred during propagation to become apparent. We note that the available plasma data indicate that both ICMEs were also magnetic clouds.

*MErcury Surface, ENvironment GEochemistry, and Ranging*(MESSENGER; MES) spacecraft and the

*Solar TErrestrial RElations Observatory B*(STEREO-B; STB) in November 2010 while both spacecraft were separated by approximately 83° in heliographic longitude from the Sun–Earth line. This ICME was first described by Good and Forsyth (2016), and has recently been analysed by Amerstorfer

*et al.*(2017) to test the ElEvoHI arrival time forecasting model. The flux rope of ICME 1 arrived at MES at 5 November 2010, 16:52 UT; approximately 58 hours later, the flux rope arrived at STB. During the observation period, the spacecraft were separated by an average of 7.0° in heliographic latitude, 1.0° in heliographic longitude, and 0.618 AU in radial distance. The left-hand panels in Figure 1 show, from top to bottom, the magnetic field magnitude and components at MES, the field magnitude and components at STB, and the bulk plasma speed profile at STB. Vertical dashed lines denote the flux rope boundaries. The three panels all display data across a time span of three days. Magnetic field data are displayed in Spacecraft Equatorial (SCEQ) coordinates, in which \(z\) is parallel to the solar rotation axis, \(y\) points to solar west, and \(x\) completes the right-handed system. SCEQ coordinates are very similar to the RTN system for spacecraft near the solar equatorial plane, and identical to it for a spacecraft in the plane.

Key parameters of the two ICMEs. \(\left\langle r _{\mathrm{H}} \right\rangle \) is the mean heliocentric spacecraft distance, \(\Delta \theta _{\mathrm{HGI}}\) and \(\Delta \varphi _{\mathrm{HGI}}\) are the latitudinal and longitudinal spacecraft separations, \(t_{\mathrm{L}}\) and \(t_{\mathrm{T}}\) are the observation times of the flux rope leading and trailing edges, \(\Delta t_{ \mathrm{MES}}\) and \(\Delta t_{\mathrm{STB}}\) are the ICME crossing times at each spacecraft, \(\Delta t_{\mathrm{L}}\) and \(\Delta t_{ \mathrm{T}}\) are the leading and trailing edge propagation times, \(v_{\mathrm{L}}\) and \(v_{\mathrm{T}}\) are the leading and trailing edge plasma speeds at STB, \(v_{\mathrm{EXP}}\) is the expansion speed at STB, \(\left\langle v_{\mathrm{L}} \right\rangle \) and \(\left\langle v_{ \mathrm{T}} \right\rangle \) are the mean speeds of the leading and trailing edges during propagation, \(\left\langle v_{\mathrm{EXP}} \right\rangle \) is the mean expansion speed during propagation, \(v_{\mathrm{c}}\) is the cruise speed at STB, \(\left\langle v_{ \mathrm{c}} \right\rangle \) is the mean cruise speed during propagation, \(\left\langle B_{\mathrm{STB}} / B_{\mathrm{MES}} \right\rangle \) is the mean ratio of the field magnitude at the outer to the inner spacecraft, \(\theta _{\mathrm{A}}\) and \(\varphi _{\mathrm{A}}\) are the latitude and longitude directions of the flux rope axes, \({\lambda _{1}} / {\lambda _{2}}\) and \({\lambda _{3}} / {\lambda _{2}}\) are the ratios of the maximum and minimum eigenvalues to the intermediate values found in the MVA, \(C\) is the overall correlation coefficient, \(C_{\mathrm{L}}\) is the macroscale correlation, and \(C_{\mathrm{S}}\) is the microscale correlation.

ICME 1 | ICME 2 | |
---|---|---|

| ||

\(\langle r_{\mathrm{H}} \rangle \), MES [AU] | 0.465 | 0.439 |

\(r_{\mathrm{H}}\), STB [AU] | 1.083 | 1.086 |

\(\Delta \theta _{\mathrm{HGI}}\), \(\Delta \varphi _{\mathrm{HGI}}\) | 7.0°, 1.0° | 6.8°, 3.5° |

\(t_{\mathrm{L}}\), MES | 5 Nov 2010 16:52 UT | 5 Nov 2011 00:43 UT |

\(t_{\mathrm{T}}\), MES | 6 Nov 2010 13:08 UT | 5 Nov 2011 17:05 UT |

\(\Delta t_{\mathrm{MES}}\) | 20 hr 16 min | 16 hr 22 min |

\(t_{\mathrm{L}}\), STB | 8 Nov 2010 03:24 UT | 6 Nov 2011 22:57 UT |

\(t_{\mathrm{T}}\), STB | 9 Nov 2010 09:04 UT | 8 Nov 2011 17:48 UT |

\(\Delta t_{\mathrm{STB}}\) | 29 hr 40 min | 42 hr 51 min |

\(\Delta t_{\mathrm{L}}\) | 58 hr 32 min | 46 hr 14 min |

\(\Delta t_{\mathrm{T}}\) | 67 hr 56 min | 72 hr 43 min |

\(v_{\mathrm{L}}\), STB [km s | 402 | 618 |

\(v_{\mathrm{T}}\), STB [km s | 418 | 410 |

\(v_{\mathrm{EXP}}\), STB [km s | −16 | 208 |

\(\langle v_{\mathrm{L}}\rangle \) [km s | 437 | 580 |

\(\langle v_{\mathrm{T}}\rangle \) [km s | 380 | 370 |

\(\langle v_{\mathrm{EXP}}\rangle \) [km s | 57 | 210 |

\(v_{\mathrm{c}}\), STB [km s | 394 | 473 |

\(\langle v_{\mathrm{c}}\rangle \) [km s | 409 | 475 |

\(\langle B_{\mathrm{STB}} / B_{\mathrm{MES}}\rangle \) | 0.39 | 0.23 |

| ||

Axis direction, MES | \(\theta _{\mathrm{A}} = -41\)°, \(\varphi _{\mathrm{A}} = 287\)° | – |

Axis direction, STB | \(\theta _{\mathrm{A}} = -20\)°, \(\varphi _{\mathrm{A}} =271\)° | – |

Axis separation |
| – |

\({\lambda _{1}} / {\lambda _{2}}\), \({\lambda _{3}} / {\lambda _{2}}\), MES | 4.70, 0.14 | – |

\({\lambda _{1}} / {\lambda _{2}}\), \({\lambda _{3}} / {\lambda _{2}}\), STB | 4.13, 0.03 | – |

| ||

| [−0.04,0.82,0.91] | [0.12,0.70,0.91] |

| [−0.31,0.89,0.96] | – |

\(C_{\mathrm{L}}\) | [−0.02,0.85,0.92] | [0.14,0.84,0.94] |

\(C_{\mathrm{S}}\) | [−0.06,0.05,0.13] | [0.03,0.02,−0.07] |

\(C_{\mathrm{L}}\), axis-aligned | [−0.41,0.94,0.96] | – |

\(C_{\mathrm{S}}\), axis-aligned | [−0.09,0.06,0.12] | – |

Data from magnetometers on board MES (MAG; Anderson *et al.*2007) and STB (IMPACT MAG; Acuña *et al.*2007) are used in this study. MES data were obtained from the PDS:PPI archive ( pds-ppi.igpp.ucla.edu ), and STB data from the SPDF CDAWeb archive ( cdaweb.sci.gsfc.nasa.gov ). Only magnetic field data were routinely available at MES since the spacecraft did not carry any dedicated instruments for analysing the solar wind plasma; both magnetic field and plasma data were available at STB. Data in RTN coordinates were obtained from the archives and transformed to the SCEQ system through a rotation about \(T\) by the heliographic latitude of the spacecraft at the observation time, such that \(N\) became aligned with the solar rotation axis direction.

### 2.1 Radial Alignment Mapping

In order to compare directly the magnetic field profiles of the ICMEs at each spacecraft, the magnetic field measurements within the flux rope at MES have been mapped forward in time and heliocentric distance to overlap with the measurements made at STB. Conceptually, we imagine each magnetic field vector measured at the inner spacecraft being frozen-in to a discrete plasma parcel. The collection of parcels that constitute the ICME propagate radially to the heliocentric distance of the outer spacecraft. The mapping involves determining the arrival time of each parcel and its magnetic field vector at the outer spacecraft distance, which requires knowledge of the parcel speeds. It is assumed that the parcel velocities are entirely in the radial direction.

ICMEs tend to expand in radial width as they propagate from the Sun until reaching some equilibrium state with the ambient solar wind. The speed observed *in situ* at the leading edge of an expanding ICME will exceed the speed at the trailing edge, and will typically decline linearly in between. For the two ICMEs, measurements of the speed profiles were only available at the outer spacecraft. ICME 1 displayed a flat speed profile at STB (bottom-left panel, Figure 1), indicating that radial expansion had ceased by the time it arrived at the spacecraft. In contrast, ICME 2 displayed a linearly declining speed profile (bottom-right panel, Figure 1), indicating that the ICME was still expanding. These profiles represent a series of instantaneous speeds measured as the ICMEs passed over the outer spacecraft, and may be different to the speeds within the ICMEs during propagation.

*e.g.*Osherovich, Farrugia, and Burlaga 1993). Note that Figure 2 illustrates the relative spacing of features on arrival at the second spacecraft, and not any changes in magnitude that may have occurred during propagation.

### 2.2 Mapping of ICME 1 and ICME 2 Data

The flux rope orientation at MES was different to the orientation observed at STB for ICME 1. The orientation of a flux rope’s central axis may be estimated through minimum variance analysis (MVA). MVA is a widely used technique that involves finding the eigenvalues and eigenvectors of the covariance matrix of the field data within the flux rope, where the eigenvector associated with the intermediate eigenvalue will correspond ideally to the direction of the rope axis (Goldstein 1983). MVA gives an estimated orientation of [\(\theta _{\mathrm{A}} = -41\)°, \(\varphi _{\mathrm{A}} = 287\)°] for the mapped ICME 1 MES data and an orientation of [\(\theta _{\mathrm{A}} = -27\)°, \(\varphi _{\mathrm{A}} = 271\)°] at STB, where \(\theta _{\mathrm{A}}\) and \(\varphi _{\mathrm{A}}\) are analogous to the field direction angles defined above. Visual inspection of the data suggests an intermediate variance direction close to the \(-y\) direction (solar east) at both spacecraft, in agreement with the MVA. The direction angles found in the mapped MES data are the same as those found in the original data series, to the nearest degree. For the mapped MES data, the ratio of the minimum to intermediate eigenvalue, \({\lambda _{1}} / {\lambda _{2}}\), was equal to 4.70, and the ratio of the maximum to intermediate value, \({\lambda _{3}} / {\lambda _{2}}\), was 0.14. The corresponding values at STB were 4.13 and 0.03, respectively. The ratios in both datasets meet the Siscoe and Suey (1972) conditions, namely \({\lambda _{1}} / {\lambda _{2}}\) > 1.37 and \({\lambda _{3}} / {\lambda _{2}}\) < 0.72, indicating that the variance directions were well defined.

We now consider the effect of transforming the mapped MES flux rope to align its central axis, pointing in direction \(\boldsymbol{a}_{ \mathrm{MES}}\), with the axis found in the STB rope data, pointing in direction \(\boldsymbol{a}_{\mathrm{STB}}\). This transformation has been achieved by defining the plane in which both axis directions are coplanar, determining the angle between the axes in that plane, \(\psi = \tan ^{-1} ( \vert \boldsymbol{a}_{\mathrm{MES}} \times \boldsymbol{a}_{\mathrm{STB}} \vert / \boldsymbol{a}_{\mathrm{MES}} \cdot \boldsymbol{a}_{\mathrm{STB}} )\), and determining the direction normal to that plane, \(\boldsymbol{A} = \boldsymbol{a}_{ \mathrm{MES}} \times \boldsymbol{a}_{\mathrm{STB}}\); rotating all of the magnetic field vectors of the mapped MES data by \(\psi \) about the normal direction \(\boldsymbol{A}\) such that the flux rope axes become aligned gives the required transformation. This transformation cancels out the apparent change in axis orientation between the spacecraft, and cancels out differences between the profiles that are purely due to the different orientations of the flux rope.

This transformation of the mapped MES data for ICME 1 is shown on the right-hand side of Figure 3, overlying the STB rope data. Compared to the left-hand side, there is a marked increase in similarity of the \(\hat{B}_{y}\) and \(\hat{B}_{z}\) profiles, particularly in the front and middle regions of the rope. There is little overall change in similarity for \(\hat{B}_{x}\). The axis separation, \(\psi \), was equal to 19°.

There is a significant gap in the MES data for ICME 2, during which the spacecraft was passing through Mercury’s magnetosphere. The gap spans 30% of the flux rope time series, and obscures the closest approach of the spacecraft to the flux rope axis. These factors are the likely cause of the dubious estimate of the axis orientation obtained from MVA when applied to the normalised field data. We therefore did not attempt the axis-aligned mapping for ICME 2 that was performed for ICME 1. Good *et al.* (2015) analysed the flux rope orientation of this ICME in more detail.

The bottom-centre panels in Figures 2 and 3 show the ratio of the field magnitude at the outer spacecraft to the magnitude at the inner spacecraft for ICMEs 1 and 2, respectively, across the flux rope profiles. Values below unity indicate a drop in field magnitude. In ICME 1, the ratio rose smoothly from approximately 0.3 in the front half of the rope to approximately 0.5 towards the trailing edge. The mean value of the magnitude ratio across the rope was 0.39. In contrast, the ratio for ICME 2 was relatively flat across the profile, with a mean value of 0.23.

#### 2.2.1 ICME Expansion Speeds and Crossing Times

The expansion speed of an ICME, \(v_{\mathrm{EXP}}\), may be defined as the difference between its leading and trailing edge speeds, \(v_{\mathrm{L}} - v_{\mathrm{T}}\). The expansion speeds observed *in situ* at STB were −16 km s^{−1} for ICME 1 and 208 km s^{−1} for ICME 2. These values, and the leading and trailing-edge plasma speeds from which they are calculated, are listed in Table 1. The \(v_{\mathrm{EXP}}\) values indicate that ICME 1 had ceased to expand by the time it arrived at STB and that ICME 2, in contrast, was still rapidly expanding at STB. \(v_{\mathrm{EXP}}\) values could not be calculated at MES given the lack of plasma data.

The mean expansion speeds during propagation between the spacecraft, \(\langle v_{\mathrm{EXP}} \rangle = \langle v_{ \mathrm{L}} \rangle - \langle v_{\mathrm{T}} \rangle \), were 57 km s^{−1} for ICME 1 and 210 km s^{−1} for ICME 2. The mean propagation speeds of the leading edge, \(\langle v_{\mathrm{L}} \rangle \), and trailing edge, \(\langle v_{\mathrm{T}} \rangle \), are as defined by Equations 1a and 1b. The positive, non-zero \(\langle v_{\mathrm{EXP}} \rangle \) values confirm that both ICMEs expanded during propagation, and account for the increased ICME crossing times, \(t_{\mathrm{L}} - t_{\mathrm{T}}\), seen at STB; for ICME 1, the crossing time rose from 20 hr 16 min at MES to 29 hr 40 min at STB, and from 16 hr 22 min at MES to 42 hr 51 min at STB for ICME 2. The mean expansion speed of ICME 2 between MES and STB was very similar to the expansion speed observed at STB, indicating that the ICME expansion speed was roughly constant during propagation between the spacecraft.

For ICME 1, the *in situ* speed of the leading edge at STB, \(v_{\mathrm{L}}\), was approximately 402 km s^{−1}, lower than the \(\langle v_{\mathrm{L}} \rangle \) value of 437 km s^{−1}. In contrast, the *in situ* trailing edge speed, \(v_{\mathrm{T}}\), was around 418 km s^{−1}, higher than the \(\langle v_{ \mathrm{T}} \rangle \) value of 380 km s^{−1}. Thus, the leading edge decelerated, the trailing edge accelerated, and the speed profile across the rope flattened during propagation. For ICME 2, the leading and trailing edge speeds at STB (618 km s^{−1} and 410 km s^{−1}, respectively) were both somewhat higher than their mean propagation values (580 km s^{−1} and 370 km s^{−1}, respectively).

Another characteristic ICME speed of interest is the radial “centre of mass” speed or cruise speed, \(v_{\mathrm{C}}\), which may be defined as the speed of an ICME midway between its leading and trailing edge (Owens *et al.*2005). ICME 1 had an *in situ*\(v_{\mathrm{C}}\) value of approximately 394 km s^{−1} at STB, very similar to the mean value during propagation, \(\langle v_{\mathrm{C}} \rangle = ( \langle v_{\mathrm{L}} \rangle + \langle v_{ \mathrm{T}} \rangle ) /2\), of 409 km s^{−1}; for ICME 2, \(v_{\mathrm{C}} \approx 473\) km s^{−1} and \(\langle v_{ \mathrm{C}} \rangle = 475\) km s^{−1}. The finding that \(v_{\mathrm{C}} \approx \langle v_{\mathrm{C}} \rangle \) indicates that the cruise speed of both ICMEs was approximately constant during propagation, in agreement with the commonly held assumption of \(v_{\mathrm{C}}\) constancy in ICMEs.

The \(B\) ratio profiles in Figures 2 and 3 may be explained in terms of the ICMEs’ expansion and spacecraft crossing times. ICME 1 was likely to be expanding at the location of MES, given the shape of the field magnitude profile observed by the spacecraft (see Figure 1). When MES observed the front half of the rope, the ICME was relatively less expanded, and field magnitudes were consequently relatively high; by the time that the rear half was observed, the ICME had expanded more, and the observed magnitudes were lower. This produces the “ski ramp” profile that is characteristic of expansion (Farrugia *et al.*1993; Osherovich, Farrugia, and Burlaga 1993) and which was seen in ICME 1 at MES. By the time the ICME reached STB, radial expansion across the rope had ceased, the speed profile within the rope was flat, and the field magnitude profile was symmetric (and flat). Thus, the ratio of the observed magnitudes at STB to MES is lower in the front half of the rope than in the rear half. This contrasts with ICME 2, where the ICME was expanding at both MES and STB: magnitudes in the front half were higher than in the rear half by a similar proportion at both spacecraft, hence the ratio of the two profiles is flat. The flat ratio suggests that the expansion rates of ICME 2 were similar to each other at the observing spacecraft. The greater small-scale variability in the ratio for ICME 2 than for ICME 1 is notable. The flatness in the magnitude profile for ICME 1 at STB may indicate that the axial field component of the flux rope (dominant near the midpoint of the profile) dropped faster with propagation distance than the azimuthal component (dominant at the edges of the profile), an effect predicted by analytical models of flux rope evolution (*e.g.* Osherovich, Farrugia, and Burlaga 1993; Démoulin and Dasso 2009).

### 2.3 Correlation of the Magnetic Field Components

#### 2.3.1 Overall Correlation

*i.e.*if \(B_{i}\) were transformed to \(a+b \hat{B}_{i}\) and \(\hat{B}_{i \mathrm{m}}\) to \(c+d \hat{B}_{i \mathrm{m}}\), where \(a\), \(b\), \(c\) and \(d\) are constants, then \(C_{i}\) would be unchanged. The mapped data have been linearly interpolated to the same resolution as the outer spacecraft data in order to determine the coefficients.

The correlation \(C= [ C_{x}, C_{y}, C_{z} ] \) of the three field components for the ICME 1 mapping shown on the left-hand side of Figure 3 has values of \([-0.04, 0.82, 0.91]\), indicating a relatively high correlation in \(\hat{B}_{y}\) and \(\hat{B}_{z}\), and no significant correlation in \(\hat{B}_{x}\). The axis-aligned mapping on the right-hand side of Figure 3 has \(C\) values of \([-0.31, 0.89, 0.96]\), indicating an increased correlation in \(\hat{B}_{y}\) and \(\hat{B}_{z}\), and no significant change in the \(\hat{B}_{x}\) correlation. The mapping for ICME 2 displayed in Figure 4 has a correlation of \([0.12, 0.70, 0.91]\): as in the ICME 1 mappings, there is a high correlation in \(\hat{B} _{y}\) and \(\hat{B}_{z}\), and no correlation in \(\hat{B}_{x}\).

#### 2.3.2 Correlation at Different Temporal Scales

The \(C\) coefficients above give the overall correlation across all temporal scales. Features within the same component that vary with different characteristic timescales may not show the same degree of correlation. Here we define macroscopic features in the profiles to be those that vary with a timescale of one hour, and microscopic features to be those that vary with a timescale of one minute. This choice of timescales is somewhat arbitrary, but does allow the relationship between correlation and timescale to be illustrated effectively. Shocks, tangential discontinuities, and reconnection exhausts are typically observed at timescales of approximately one minute in the solar wind at 1 AU, while large-scale heliospheric structures such as ICMEs vary at timescales of approximately one hour.

To obtain profiles of macroscopic features, robust LOWESS smoothing (Cleveland 1979) was applied to the magnetic field data. This smoothing is similar in effect to a low-pass filter. In outline, the LOWESS technique involves selecting a span of data centred on the point to be smoothed, determining weights for each point within the span, and performing a weighted linear least-squares regression with a first-order polynomial across the span. The robust version of the technique applies an additional weighting, where points with high residual values relative to the initial regression are reduced in weight. The regression is then recalculated with the additional weighting. The additional weighting and regression are performed iteratively five times until a final regression is obtained; the value of the final regression at the point of interest gives the smoothed value. These steps were repeated for all points in the data series. A span width of one hour was used. Further details on the LOWESS technique used may be found at mathworks.com/help/curvefit/smoothing-data.html . Profiles of the microscale features were obtained by subtracting the smoothed values from the original datasets.

The macroscale correlation, \(C_{\mathrm{L}}\), in the initial and axis-aligned mappings of ICME 1 were found to be \([-0.02, 0.85, 0.92]\) and \([-0.41, 0.94, 0.96]\), respectively, indicating strong \(\hat{B}_{y}\) and \(\hat{B}_{z}\) macroscale correlations, and increased correlations relative to the overall values. The \(\hat{B}_{x}\) correlations are again low because of the low variance found in this component. The corresponding microscale correlations, \(C_{\mathrm{S}}\), for the two mappings were \([-0.06, 0.05, 0.13]\) and \([-0.09, 0.06, 0.12]\), indicating that there was no correlation of microscale features in any component for either mapping. The ICME 2 correlations, \(C_{\mathrm{L}} = [0.14, 0.84, 0.94]\) and \(C_{\mathrm{S}} = [0.03, 0.02, -0.07]\), show a similar trend to that found in the ICME 1 correlations.

*e.g.*a microscale feature near the leading edge of ICME 2 in \(\hat{b}_{y}\), at around DoY 311.1).

## 3 Discussion

The mappings displayed in Figures 3 and 4 indicate that there was a high degree of similarity between the ICME magnetic field profiles observed at MES and STB. The correlation analysis derived from these mappings gives a good quantified measure of the similarity in the solar east–west and north–south directions: features at the macroscopic scale in the two events were well correlated, while features at the microscopic scale were uncorrelated. Macroscale features broadly correspond to the underlying flux rope structure of the ICMEs. The degree of similarity that is observed is remarkable given the approximately two-day propagation time and approximately 0.6 AU propagation distance for both ICMEs.

Others who have studied ICMEs observed at spacecraft with very small longitudinal separations (*e.g.* Mulligan *et al.*1999; Nakwacki *et al.*2011) have reported comparable levels of similarity in macroscopic field structure with propagation distance for some ICMEs. The approximate self-similarity in the field profiles is also in agreement with the analytical flux rope modelling of Démoulin and Dasso (2009). Their work indicates that a range of flux rope configurations would expand in the radial direction almost self-similarly with heliocentric distance, \(r_{\mathrm{H}}\), if the ropes are embedded in solar wind plasma with pressure that falls according to an empirically derived \(r_{\mathrm{H}}^{-2.8}\) power law (*e.g.* Gazis *et al.*2006).

The field profile similarity may be interpreted in two ways. If the same region of each ICME was sampled by the observing spacecraft pair – *i.e.* if the spacecraft angular separations can be neglected – then the similarity suggests there was little evolution in the observed region during propagation, and that the field structure of the observed region was robust. Alternatively, if adjacent regions were observed by the two spacecraft – *i.e.* if the spacecraft angular separations cannot be neglected – then the similarity indicates that the adjacent regions were similar, and that there was spatial homogeneity across the angular extent sampled. Different regions may also have been sampled if there had been significant non-radial components to the ICMEs’ propagation velocities.

Determining which of these interpretations is correct is difficult given the available observations. It may be possible to neglect the spacecraft angular separations if they were small relative to the overall latitudinal and longitudinal extents of the ICME: however, these global extents cannot be determined from the *in situ* measurements. Compared to the average CME latitudinal span of 50° to 60° seen in coronagraph images (*e.g.* Yashiro *et al.*2004), the spacecraft longitudinal separations (1.0° for ICME 1 and 3.5° for ICME 2) were indeed relatively small, whereas the latitudinal separations (7.0° for ICME 1 and 6.8° for ICME 2) were somewhat more significant.

In ICME 1, the overall increase in profile similarity and correlation of the axis-aligned fields relative to the initial, unaligned mapping is notable. If the two spacecraft observed the same region of the ICME, the increased correlation indicates that the local flux rope orientation changed during propagation. The increased correlation would suggest that the MVA-determined axis directions are reasonably accurate. Given how well the axis-aligned profiles overlap all along their length, it would also suggest that this apparent rotation did not distort the underlying structure of the flux rope. If different regions of ICME 1 were observed, the MVA analysis suggests that the regions had different local axis orientations. The 7° latitudinal (*i.e.* north–south) separation of the spacecraft corresponded to an arc-length spatial separation of 0.13 AU at the heliocentric distance of STB: in the case of ICME 1, with flux rope axis directions pointing approximately to the west-southwest, the homogeneity across the 0.13 AU separation would thus have been both along and normal to the rope axis directions.

It has been assumed that reconnection did not erode the ICME flux ropes during propagation between the spacecraft by any significant amount. The strong similarities seen in the profiles, and the agreement in field direction at the leading and trailing edges, would support this assumption. Many ICME flux ropes arriving at 1 AU show signs of erosion (Ruffenach *et al.*2015), but much of this erosion is thought to occur within the orbital radius of Mercury (Lavraud *et al.*2014), where the Alfvén speed, and hence the reconnection rate, is considerably higher. If a significant amount of erosion had occurred at the rope edges during propagation, points within the rope interval (*i.e.* not the boundaries) at the inner spacecraft would need to be mapped to the boundaries observed at the outer spacecraft. The upcoming *Solar Orbiter* (Müller *et al.*2013) and *Parker**Solar Probe* (Fox *et al.*2016) missions will travel to within the orbit of Mercury, where ICMEs may appear less eroded than at 1 AU.

The nature of the microscale features displayed in Figures 7 and 8 is not considered in this work. However, we speculate that these features may be substructures that are smaller in angular extent than the angular spacecraft separation (and hence not encountered by both spacecraft), or they may be structures that evolve significantly during propagation. They may also be temporal, transient features such as waves, the observation of which is dependent on the local wave speed. These kinds of features would not be correlated at the different spacecraft.

The correlation is of course sensitive to how the mapping is performed, and it would be worthwhile to consider whether other mappings would produce an increase in the correlation at microscopic scales. Other mappings could involve removing the assumption of linearity in the mean rope velocity profile, for example, or relaxing the condition that the predetermined rope edges at each spacecraft must line up with each other. A least-squares mapping might be attempted that minimises the residuals between the datasets, where axis directions, rope boundaries and the velocity profile are free parameters.

There are limitations to the correlation analysis technique presented in Section 2.3. It does not measure the overall similarity of the vectors, and it is only suited to a component-by-component analysis. In the case of ICME 1, the similarity seen in the \(\hat{B}_{x}\) profiles is not reflected in the correlation coefficient because of the low variance of this component. However, for both ICMEs, the high correlation values for \(\hat{B}_{y}\) and \(\hat{B}_{z}\) do reflect, and give a quantified measure, of the high degree of similarity seen in these components. We intend in future studies to develop the correlation analysis technique introduced here.

*e.g.*Burlaga 1988) that would be observed by the two spacecraft during the passage of the idealised flux rope; the \(\hat{B}_{y}\) and \(\hat{B} _{z}\) profiles are perfectly correlated, and the \(\hat{B}_{x}\) profiles perfectly anti-correlated. Thus, a small deviation from radial alignment that results in the spacecraft traversing the rope either side of the axis could produce the correlations observed in ICME 2. However, it cannot be determined whether the scenario described above arose for this ICME without further analysis of its global structure and the spacecraft impact parameters.

From a space weather perspective, the macroscale structure of an ICME is more significant than any short-duration microscale features. ICMEs are a major source of sustained periods of southward-directed magnetic field at 1 AU. There is now much focus in the scientific community on attempting to forecast \(B_{z}\) in ICMEs incident at the Earth with forecast lead times greater than the approximately 45 minutes provided by ACE, *Wind* and DSCOVR at L1. Many current efforts concentrate on using remote observations to make predictions (*e.g.* Savani *et al.*2015; Möstl *et al.*2017), although some have proposed *in situ* space weather monitors that could be placed well upstream of the Earth (*e.g.* the *Sunjammer* mission concept; Eastwood *et al.*2015); Kubicka *et al.* (2016) have recently proposed a method for forecasting \(B_{z}\) that combines remote and *in situ* observations. If ICME magnetic field structure in the inner heliosphere is generally well correlated along radial lines from the Sun, as in the case of the two ICMEs studied in this work, then the task of predicting ICME magnetic field properties at 1 AU from sub-1 AU *in situ* observations would be made less difficult: a model that accurately predicts arrival times, radial expansion, and magnitude from sub-1 AU observations would suffice. However, if the underlying magnetic topologies of ICMEs are significantly altered during propagation through strong interactions with solar wind structures, as in a case reported by Winslow *et al.* (2016), or through interactions with other ICMEs (Lugaz *et al.*2017, and references therein), then more complex modelling would be required. How commonly such changes in ICME field structure occur remains an open question. Statistical analyses of more ICMEs observed by radially aligned spacecraft pairs are needed to shed light on this matter.

## 4 Conclusion

- i)
Both ICMEs expanded in the radial direction during propagation, and both propagated at approximately constant centre-of-mass cruise speeds.

- ii)
There was a high degree of qualitative similarity in the flux rope profiles.

- iii)
Macroscale features in the profiles, which correspond approximately to the underlying flux rope structure, were well correlated in the \(y\) and \(z\) directions, with Pearson’s correlation coefficients of approximately 0.85 and 0.95, respectively.

- iv)
Microscale features that varied on timescales of approximately one minute were uncorrelated.

- v)
Macroscale correlation in one of the ICMEs was increased when an apparent rotation of 19° by the flux rope axis was considered.

- vi)
The similarity in the field profiles may be interpreted in two ways. If the same region of each ICME was intersected by the observing spacecraft, it indicates that the underlying, large-scale \(B\)-field structure of the observed regions remained intact during propagation. If the same region was not observed in each case, it indicates homogeneity in field structure across the angular extent of the ICME spanned by the spacecraft.

If the similarity in magnetic field structure at different heliocentric distances seen in these ICMEs is common, then the task of predicting \(B_{z}\) in ICMEs arriving at the Earth using an upstream, *in situ* space weather monitor would be much simplified.

## Notes

### Acknowledgements

We wish to thank the MESSENGER and STEREO instrument teams for providing the data used in this work, and the PDS:PPI and SPDF CDAWeb data archives for their distribution of data. We also wish to thank the referees for their comments and suggestions for improvements to the manuscript. This work has been supported with funding provided by the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement No. 606692 [HELCATS]. C.M. thanks the Austrian Science Fund (FWF): [P26174-N27].

### Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

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