Evolution of Interplanetary Coronal Mass Ejection Complexity: A Numerical Study through a Swarm of Simulated Spacecraft

We perform three-dimensional (3D) magnetohydrodynamical simulations of the inner heliosphere using the EUropean Heliospheric FORecasting Information Asset (EUHFORIA; Pomoell & Poedts 2018) model. Our simulation domain covers heliocentric distances (r) between 0.1 and 2 au, ±80° in the latitudinal (θ) direction, and ±180° in the longitudinal (ϕ) direction, employing a uniform grid composed of 512(r) × 80 (θ) × 180 (ϕ) cells.

To evaluate the interaction of CMEs with different solar wind structures, we simulate two background solar wind configurations (Figure 1): the first one (hereafter "run A") includes a low-inclination HCS/HPS reaching up to θ = ±15°, and is characterized by a uniform solar wind of speed 450 km s⁻¹ (intermediate between slow and fast solar wind; Cranmer et al. 2017) everywhere else. The second one (hereafter "run B") differs from the one above by a HCS/HPS reaching up to θ = ±30°, and by the inclusion of a HSS with circular cross section of half-width (ω/2) equal to 30° and radial speed (v_R) equal to 675 km s⁻¹, located just above the HCS/HPS at longitude ϕ = 0°. In both cases, the HPS meridional profile is parametrized using the description in Odstrčil et al. (1996), which results in a solar wind speed as low as 300 km s⁻¹ near the HCS. At low latitudes, these idealized configurations mimic the solar wind originating from an equatorial streamer belt (in both runs) and a coronal hole (in run B). However, they do not include any latitudinal dependence, and are in this respect different from the latitudinal profile observed in the real solar wind (McComas et al. 2008). This choice has been made to ensure full control over the CME propagation and interaction with solar wind structures, and the comparability between different runs.

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In both runs, CMEs are initialized at 0.1 au and modeled using the linear force-free spheromak model (Verbeke et al. 2019). The following initial parameters are used: radial speed equal to 800 km s⁻¹; initial half-width of 45°; positive chirality (H = +1) with axial tilt (γ) of 90° with respect to the northward direction (corresponding to a SWN flux-rope type; see Bothmer & Schwenn 1998); toroidal magnetic flux (φ_t ) equal to 10¹⁴ Wb (corresponding to a magnetic field strength of ∼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, 2021), the effective initial CME speed is ∼1100 km s⁻¹, which results in a fast CME that drives an interplanetary shock and sheath, as discussed in Section 3. Such a combination of initial parameters is representative of those of a typical fast CME with a reconnected flux of the order of 10¹⁴ Wb (Pal et al. 2018). The CME initial direction is chosen to reproduce two end-member scenarios of interaction with different solar wind structures (shown in panels (b) and (d) in Figure 1): in run A, the CME is inserted across the HCS/HPS at (θ, ϕ) = (0°, 0°); in run B, the CME is inserted across the HSS at (θ, ϕ) = (5°, 0°), 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 2020 January 1 at 00:00 UT. A summary of the solar wind and CME parameters at 0.1 au used in this works EUHFORIA simulations is provided in Table 1.

Table 1. Summary of the Solar Wind and CME Parameters at 0.1 au Used in this Work's EUHFORIA Simulations

Solar Wind Parameters—Runs A and B
vR,sw = 450 km s−1	nsw = 675 cm−3
BR,sw = 200 nT	Tsw = 3.5 × 105 K
HSS Parameters—Run B Only
vR,HSS = 675 km s−1	nHSS = 300 cm−3
BR,HSS = 300 nT	THSS = 0.8 × 106 K
θHSS = 5°, ϕHSS = 0°	ωHSS/2 = 30°
HCS/HPS Parameters—Run A (B)
${B}_{{\rm{R}},\mathrm{HPS}}(\theta )={B}_{{\rm{R}},\mathrm{HSS}}\displaystyle \frac{{v}_{{\rm{R}},\mathrm{sw}}}{{v}_{{\rm{R}},\mathrm{HSS}}}\tfrac{\theta +{\theta }_{\mathrm{HCS}}(\phi )}{{\rm{\Delta }}{\theta }_{\mathrm{HPS}}}$
${\theta }_{\mathrm{HCS}}(\phi )={\theta }_{\max }\cos \phi$	${\theta }_{\max }=15^\circ (30^\circ )$
ΔθHPS/2 = 4°	${({{nv}}_{{\rm{R}}}^{2})}_{\mathrm{HPS}}=1.4\,\times \,{10}^{20}$ m−1 s−2
Ptot,HPS = 19.2 nPa	THPS = 3.5 × 105 K
CME Parameters—Run A (B)
θCME = 0°(5°), ϕCME = 0°	ωCME/2 = 45°
vR,CME = 800 km s−1	ρCME = 10−18 kg m−3
φt,CME = 1014 Wb	TCME = 0.8 × 106 K
HCME = + 1	γCME = 90°

Note. v_R —radial speed; B_R —radial magnetic field; n—number density; ρ—mass density; T—temperature; P_tot—total (thermal+magnetic) pressure; θ—latitude; ϕ—longitude; ω/2—half-width; Δθ/2—latitudinal half-width; φ_t —toroidal magnetic field; H—chirality; γ—axial tilt.

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In each simulation, we place virtual spacecraft spanning ±90° in longitude from the CME initial direction, and covering the full range of latitudes in the domain. The virtual spacecraft are equally distributed with longitudinal and latitudinal separations of 5°, and are 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 separation. Overall, a swarm of 18944 virtual spacecraft (1184 per heliocentric distance) is placed in the model domain in each simulation.

At each virtual spacecraft, the start of the CME-driven perturbation (i.e., a shock-like discontinuity) is determined through our algorithm by scanning the radial speed, density, and magnetic field time series forward in time and applying conditions similar to those typically used to detect fast-forward interplanetary shocks at 1 au (e.g., Vorotnikov et al. 2008; Kilpua et al. 2015). The detailed identification criteria are presented in Appendix B.1.

At locations where a CME-driven perturbation is detected, time series are scanned in order to assess whether there is a magnetic ejecta (ME) after the shock-like discontinuity. The exact criteria used to determine the ME start and end times vary with heliocentric distance and for different solar wind regimes (as discussed in Appendix B.2), but are overall based on two typical characteristics of MEs (Burlaga et al. 1981; Kilpua et al.2017a): an enhanced magnetic field strength, and a low plasma β compared to the surrounding solar wind.

After having identified the nominal boundaries of the in situ CME substructures (i.e., sheath, ME) at each virtual spacecraft, we classify the ME signature using a classification scheme inspired by Nieves-Chinchilla et al. (2019) and based on the amount of rotation of the magnetic field components, i.e., B_R , B_T , and B_N , in the local radial-tangential-normal coordinate system. This analysis provides information on the ME structure that is later used in Section 3 to investigate how CME complexity varies with distance, for various propagation scenarios. Because our simulations employ a spheromak magnetic structure for which rotations ≥180° are expected for a large variety of spacecraft crossings (as shown in Appendix A), we have adapted the original classification to better distinguish rotations up to 360°. We also further assign a numerical index (${\mathbb{C}}$) to each ME class in order to rank the level of complexity of the detected structure. If an ME is detected at a spacecraft located at coordinates (r, θ, ϕ), the following classification scheme is applied.

• 1.  
  F₂₇₀: ME signature with at least one component (i.e., B_R , B_T , or B_N ) rotating ≥270°; complexity index ${\mathbb{C}}(r,\theta ,\phi )=0$, corresponding to the least complex state.
• 2.  
  F₁₈₀: ME signature with at least one component rotating ≥180° and <270°; complexity index ${\mathbb{C}}(r,\theta ,\phi )=1$.
• 3.  
  F₉₀: ME signature with at least one component rotating ≥90° and <180°; complexity index ${\mathbb{C}}(r,\theta ,\phi )=2$.
• 4.  
  F₃₀: ME signature with at least one component rotating ≥30° and <90°; complexity index ${\mathbb{C}}(r,\theta ,\phi )=3$.
• 5.  
  E: ME signature with no component rotating ≥30°; complexity index ${\mathbb{C}}(r,\theta ,\phi )=4$, corresponding to the most complex state.

Starting from the ME nominal boundaries, the ME start and end times are varied by ±25% of the total ME duration, in order to assess the variability of the classification with respect to slight variations of the boundaries (reflecting the uncertainties in the boundary identification; Riley et al. 2004; Al-Haddad et al. 2013). The final ME classification is chosen as the most probable classification obtained among all possible combinations of boundaries. An example time series for a CME classified as having an F₂₇₀ ME signature, and the corresponding magnetic hodograms, are shown in Figure 2. Additional examples are provided in Appendix C.

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We include two additional categories representing the non-detection of an interplanetary shock-like perturbation (N ), and the detection of an interplanetary shock-like perturbation, which was not followed by a ME (S). In these cases, we do not assign a complexity index to the observed signatures because of their intrinsically different nature compared to ME signatures.

We tackle the known limitations of the spheromak model in reproducing CME global magnetic structures (particularly with respect to stretched "legs" rooted to the Sun; e.g., Scolini et al. 2019) by focusing the investigation of CME magnetic complexity to central regions only (Sections 3.1 and 3.2). Furthermore, we note that the primary aim of this exploratory work is that of uncovering the complexity trends affecting CME structures during propagation through different solar wind structures, and that conclusive evidence for the applicability of our results to real events and possibly other flux-rope configurations will have to be provided in future studies.

The results of the identification and classification analysis introduced in Section 2.2 are provided in Figure 4, shown as longitude–latitude maps colored by classification type, for selected heliocentric distances.

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As expected, immediately after insertion at 0.1 au, the extent of the CME front in the two simulations is very similar, and it matches well the nominal CME cross section expected from the initial half angular width of 45° (Figure 4, top row). At this very early stage, the spatial distribution of the shock-like and ME classifications is also comparable between runs A and B, and is qualitatively consistent with the results expected for a spheromak structure (Figure 6).

At 0.3 au (Figure 4, second row) the CME cross section has expanded to an effective half-width of ∼60°, indicating an over-expansion in the early stage of the propagation (Scolini et al. 2019, 2021). By the time the CME reaches 0.3 au, significant differences are visible in the CME cross section and in the spatial distribution of the classification types between the two runs. While run A preserves a relatively symmetric classification distribution with respect to the spheromak main axis (lying parallel to the ecliptic plane and the local HCS direction), dominated by F₃₀ and F₂₇₀ types, run B exhibits a distorted cross section and increased E ejecta types. At this stage, an increased detection of shock-only (S) signatures close to the CME flanks indicates the formation of a CME-driven shock and sheath region that is more extended than its driver (Kilpua et al. 2017a).

In both runs, the CME cross section at 1 and 1.6 au (Figure 4, third and fourth rows) remains similar to that at 0.3 au, meaning the angular expansion of the CME is almost negligible beyond Mercury's orbit. On the contrary, beyond 0.3 au the spatial distribution of the ME classification types becomes visibly less regular with heliocentric distance, especially for run B. The differences between runs A and B at this late stage of propagation are remarkable: in run A, the CME cross section remains quasi-circular, despite a shrinking in the detection of shock-like signatures around its flanks. The classification in the CME core region also remains largely unchanged (Figure 4(c) and (d)). In run B (Figures 4(g) and (h)), a similar shrinking in the detection of shock-like signatures is visible around the CME flanks, particularly in the regions affected by the interaction with the HSS (northeast) and HCS/HPS (southeast). However, a decrease of the less complex ME types (F₂₇₀ and F₁₈₀ classes) in favor of more complex ones (F₉₀, F₃₀, and E classes) is observed in the core CME region. The irregular spatial distribution of ME types in this region is also associated with a high probability for an inner and outer spacecraft along a given (θ, ϕ) direction to detect different ME types, as further discussed in Section 3.2. We note that this is partly due to the formation of a CME–SIR merged interaction region (Rouillard et al. 2010), which affects the efficiency of the ME detection algorithm at larger distances. Similar difficulties are likely to affect the identification of MEs from actual in situ data, thereby making this limitation particularly instructive also with respect to future observational applications.

The different CME evolutionary behaviors identified above result in different probabilities to detect the various ME classes and additional S and N signatures as a function of the heliocentric distance in the two simulations performed. Considering only spacecraft crossings within 45° from the CME initial direction (i.e., well within the CME effective half angular width of ∼60° reached at 0.3 au, corresponding to relatively central impact locations), we find that run A is dominated by the detection of F₂₇₀ and F₃₀ ejecta types, and more than 90% of all spacecraft detect the passage of an F type. Notably, these probabilities are mostly independent from heliocentric distance, and remain consistent with those expected for a spheromak structure (i.e., not interacting with the solar wind, as shown in Figure 6). In run B, F₃₀ detections dominate all heliocentric distances, while the second most-detected ejecta type passes from F₂₇₀ to F₉₀ beyond 0.8 au. A more-than-doubled fraction of E, S, and N complex types (>20%) is also observed. The probabilities for run B also exhibit a strong dependence on the heliocentric distance: all signature types are almost equally represented by the time the CME reaches 1.6 au, as opposed to run A where a clearly bi-modal distribution is preserved during propagation.

To quantify the overall change in CME complexity with heliocentric distance, we first consider a generic pair of heliocentric distances, i.e., r₁ and r₂, with r₁ < r₂. Then, we consider all pairs of virtual spacecraft in radial alignment located within 45° from the CME initial direction (justified by the CME initial half-width and the need to restrict ourselves to central CME regions, so to limit the impact of spheromak limitations around the flanks), and that detected an ME signature at both distances. Moving along fixed (θ^*, ϕ^*) directions satisfying the above criteria, we compute the changes in CME complexity between the inner spacecraft at r₁ and the outer spacecraft at r₂, as ${\rm{\Delta }}{\mathbb{C}}({r}_{1},{r}_{2},{\theta }^{* },{\phi }^{* })\,={\mathbb{C}}({r}_{2},{\theta }^{* },{\phi }^{* })-{\mathbb{C}}({r}_{1},{\theta }^{* },{\phi }^{* })$. ${\rm{\Delta }}{\mathbb{C}}({r}_{1},{r}_{2},{\theta }^{* },{\phi }^{* })\gt ,=,$ or < 0 indicates directions where an increased, unchanged, or decreased CME complexity with heliocentric distance was detected. The results of this procedure are provided in Figure 5 (top row) for the notable case of radially aligned spacecraft at 0.3 au (consistent with the orbit of Mercury and Solar Orbiter's perihelion), and at 1 au (consistent with Earth's orbit).

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By counting how many directions detected a increased, unchanged, or decreased complexity, we determine the overall probability (normalized between 0 and 1) to detect complexity changes between two spacecraft in radial alignment at distances r₁ and r₂. Finally, by applying the same process to all distance pairs (r₁, r₂), we construct global matrix plots, as shown in Figure 5 (second and third rows).

The top row in Figure 5 exemplifies how the CME in run B develops a higher complexity as it propagates, compared to run A. In run A (panel (a)), a predominance of ${\rm{\Delta }}{\mathbb{C}}=0$ white squares indicating a stable complexity is visible within 45° from the initial CME direction. A quasi-symmetric distribution of the classification changes beyond 45° is also visible, where spacecraft pairs detecting an ME at the outer spacecraft only (in green) reflect the slight ME expansion between 0.3 and 1 au. Cyan squares are likely the results of spurious ME detection at 0.3 au. Run B (panel (b)), on the other hand, shows a predominance of ${\rm{\Delta }}{\mathbb{C}}\gt 0$ red squares within the nominal (45°) initial CME cross section. The presence of green squares within a 60° range from the CME center indicates a slight CME expansion in the northern hemisphere compared to 0.3 au, while the presence of cyan squares immediately south of the CME and HCS/HPS suggests some erosion of the ME flanks during propagation from 0.3 to 1 au.

Overall, run A supports the idea that CMEs propagating through a quiet heliosphere tend to maintain their complexity relatively unchanged. The most likely scenario is that the ME classification made at an inner spacecraft remains the same at a radially aligned outer spacecraft (Figure 5(b)), regardless of their radial separation (average probability of 69%, minimum probability of 48%). All the most notable alignment configurations (i.e., involving Mercury, Venus, spacecraft at 1 au, and Mars) have a probability between 58% (for a Mercury–Mars conjunction) and 75% (for a 1 au–Mars conjunction) to detect exactly the same ME type at both distances. Unsurprisingly, the larger the radial separation between the spacecraft, the lower this probability. Furthermore, the larger the heliocentric distance of the inner spacecraft, the higher the probability to detect the same complexity level at outer distances, indicating that changes are more likely to occur closer to the Sun. The opposite trend is observed in Figure 5(c), showing that although the probability to detect a complexity increase for run A becomes larger with larger radial separations, it remains quite modest (average probability of 15%, maximum probability of 25%). Complexity decreases (not shown) are similarly unlikely (average probability of 16%, maximum probability of 34%).

Conversely, in run B the indicators used to evaluate CME complexity changes show that, on average, CMEs propagating through a structured solar wind still tend to preserve their complexity (average probability of 54%; Figure 5(e)), but their probability to transition to a more complex configuration is more than doubled compared to run A (average probability of 31%; Figure 5(f)). Results vary with distance, as indicated by a minimum (maximum) probability of 33% (51%) to detect an unchanged (increased) magnetic complexity. Furthermore, the probability to detect the same ME type at two different distances is >50% for only three of the notable radial alignments considered: Mercury–Venus, Venus–1 au, and 1 au–Mars. In all other cases, the most likely scenario is that two locations detect different ME types. Specifically, the probability to detect a complexity increase ranges from 27% for a Venus–1 au conjunction, to 48% for a Mercury–Mars conjunction. Furthermore, in the case of run B, complexity increases are more prominent when considering the alignment of a spacecraft located within 0.4 au from the Sun, with one beyond 1 au, reflecting the interaction of the CME with the preceding SIR over a longer distance range, as shown in Figure 3(e), (f). Finally, the probability to detect complexity decreases in run B remains similar to run A.

Based on our numerical investigation, we conclude that the interaction with solar wind structures, and particularly SIRs, can double the probability for a CME to increase its magnetic complexity. Most importantly (as illustrated in Figures 4 and 5), this result does not depend on the distance of the spacecraft crossing from the ME center. As such, changes of magnetic complexity detected by spacecraft that are in exact radial alignment are likely signs of interaction with other structures, rather than inherent to the CME evolution even if crossed far from the center.

At each virtual spacecraft in the simulation domain, we determine the arrival time of the shock-like CME-driven perturbation by scanning the radial speed, density, and magnetic field time series (having a cadence of Δt = 10 minutes) forward in time, and applying the following conditions:

$$\begin{eqnarray}&&\begin{array}{l}\overline{{v}_{R}}({t}_{i}:{t}_{i}+1\,\mathrm{hr})-\overline{{v}_{R}}({t}_{i}-1\,\mathrm{hr}:{t}_{i})\\ \quad \geqslant \,20\,\mathrm{km}\,{{\rm{s}}}^{-1}\,\,\wedge \,\,\overline{n}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\\ \ \ \geqslant \,\overline{n}({t}_{i}-1\,\mathrm{hr}:{t}_{i})\,\wedge \,\,\overline{B}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\geqslant \overline{B}({t}_{i}-1\,\mathrm{hr}:{t}_{i}),\end{array}\end{eqnarray} \tag{ B1 }$$

where t_i is a generic time in the time series, and $\overline{X}({t}_{i}:{t}_{j})$ is the average of quantity X calculated between time t_i and time t_j . The arrival time of the CME-driven perturbation, t_sh, is determined as the first time t_i at which the system of Equation (B1) is satisfied. These conditions, which we have verified visually for selected spacecraft at different heliocentric distances from the Sun, are adapted versions of the speed, density, and magnetic field conditions used to detect fast-forward interplanetary shocks from in situ solar wind measurements at 1 au employed by the Database of Heliospheric Shock Waves (http://www.ipshocks.fi; Kilpua et al. 2015) and by the ACE Real-Time Shock database (http://www.srl.caltech.edu/ACE/ASC/DATA/Shocks/shocks.html; Vorotnikov et al. 2008).

At locations where a CME-driven perturbation is detected, time series are scanned in order to assess the presence of a ME after the shock-like discontinuity. To do so, we first define the average interplanetary magnetic field (B_sw) and solar wind plasma β (β_sw) in the 6 hr prior to the arrival time of the shock-like CME-driven perturbation, as

$$\begin{eqnarray}\begin{array}{rcl}{B}_{\mathrm{sw}} & = & \overline{B}({t}_{\mathrm{sh}}-6\,\mathrm{hr}:{t}_{\mathrm{sh}}),\\ {\beta }_{\mathrm{sw}} & = & \overline{\beta }({t}_{\mathrm{sh}}-6\,\mathrm{hr}:{t}_{\mathrm{sh}}).\end{array}\end{eqnarray} \tag{ B2 }$$

Depending on the β_sw recovered at a given spacecraft position (which depends on its heliocentric distance and on the local solar wind conditions), we then consider separately the cases of a magnetically dominated (β_sw ≤ 1) and a plasma-dominated (β_sw > 1) environment.

Low-β solar wind—For a magnetically dominated β_sw ≤ 1 solar wind, we scan the magnetic field and plasma β time series forward in time starting from t_sh, and apply the following conditions:

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\\ \ \ \geqslant 1.5\,{B}_{\mathrm{sw}}\ \ \wedge \,\,\overline{\beta }({t}_{i}:{t}_{i}+1\,\mathrm{hr})\leqslant {\beta }_{\mathrm{sw}}.\end{array}\end{eqnarray} \tag{ B3 }$$

The start time of the ME, t_start, is determined as the first time t_i at which Equation (B3) is satisfied. Only if an ME start is detected at a given location, do we continue with the determination of the ME end time. To do so, we scan the magnetic field time series forward in time starting from t_start, and apply the following conditions:

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}-1\,\mathrm{hr}:{t}_{i})\\ \ \ \geqslant 1.5\,{B}_{\mathrm{sw}}\,\,\wedge \,\,\overline{B}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\leqslant 1.5\,{B}_{\mathrm{sw}},\end{array}\end{eqnarray} \tag{ B4 }$$

to detect when the magnetic field drops below 1.5 B_sw, which we have taken as threshold condition to characterize the boundary of the ME in both Equations (B3) and (B4). This threshold value was chosen after having tested values between 1.2 B_sw and 1.6 B_sw, and having verified visually that it provided the best compromise, i.e., minimizing the number of false positive/negative ME detections. The end time of the ME, t_end, is determined as the first time t_i at which Equation (B4) is satisfied. We verified visually that Equations (B3) and (B4) gave reasonable and consistent results throughout the whole range of heliocentric distances sampled by the virtual spacecraft in the model domain. A low β condition to determine the end of the ME was also tested, but was found to perform less reliably than Equation (B4), which is uniquely based on the magnetic field strength.

High-β solar wind—For a plasma-dominated β_sw > 1 solar wind, we scan the magnetic field and plasma β time series forward in time starting from t_sh, and impose high magnetic field and low β conditions to identify the start of the ME:

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\\ \geqslant 1.5\,{B}_{\mathrm{sw}}\,\,\wedge \,\,\overline{\beta }({t}_{i}:{t}_{i}+1\,\mathrm{hr})\leqslant 1.\end{array}\end{eqnarray} \tag{ B5 }$$

The start time of the ME, t_start, is determined as the first time t_i at which Equation (B5) is satisfied. Only if an ME start is detected at a given location via Equation (B5), do we continue with the determination of the ME end time. As the determination of the end time proved to be a more complex task than the identification of the start time, two alternative conditions based on the magnetic field and plasma β are applied, to account for the variety of plasma properties encountered. In particular, we scan the magnetic field time series forward in time starting from t_start, and apply the following conditions:

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\\ \ \ \lt 1.5\,{B}_{\mathrm{sw}}\,\,\wedge \,\,\overline{\beta }({t}_{i}-1\,\mathrm{hr}:{t}_{i})\leqslant 1,\end{array}\end{eqnarray} \tag{ B6 }$$

or, alternatively,

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}-1\,\mathrm{hr}:{t}_{i})\\ \ \ \geqslant 1.5\,{B}_{\mathrm{sw}}\,\,\wedge \,\,\overline{\beta }({t}_{i}:{t}_{i}+1\,\mathrm{hr})\gt 1.\end{array}\end{eqnarray} \tag{ B7 }$$

Equation (B6) identifies the end boundary of the ME based on a low magnetic field condition, while Equation (B7) is based on a high β condition. The end time of the ME, t_end, is determined as the first time t_i at which Equations (B6) or (B7) are satisfied. We verified visually that Equations (B5)–(B7) gave reasonable and consistent results throughout the whole range of heliocentric distances sampled by the virtual spacecraft in the model domain.

A further visual inspection of the results assessed there were cases where the ejecta could be recognized to cross a virtual spacecraft by eye, but which the conditions in Equations (B5)–(B7) failed to identify due to the CME plasma β being lower than β_sw, but higher than 1. To account for these additional cases, a secondary identification of the ME start time is performed by applying the following criteria:

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\\ \ \ \geqslant 1.5\,{B}_{\mathrm{sw}}\,\,\wedge \,\,\overline{\beta }({t}_{i}:{t}_{i}+1\,\mathrm{hr})\leqslant {\beta }_{\mathrm{sw}}.\end{array}\end{eqnarray} \tag{ B8 }$$

Only if an ME start is detected at a given location via Equation (B8), do we continue with the determination of the ME end time. Similarly to the case above, two alternative conditions based on the magnetic field and plasma β, are applied to detect the trailing edge of the ME:

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}:{t}_{i}+1\,\mathrm{hr})\\ \ \ \lt 1.5\,{B}_{\mathrm{sw}}\,\,\wedge \,\,\overline{\beta }({t}_{i}-1\,\mathrm{hr}:{t}_{i})\leqslant {\beta }_{\mathrm{sw}},\end{array}\end{eqnarray} \tag{ B9 }$$

or, alternatively,

$$\begin{eqnarray}&&\begin{array}{l}\overline{B}({t}_{i}-1\,\mathrm{hr}:{t}_{i})\\ \ \ \geqslant 1.5\,{B}_{\mathrm{sw}}\,\,\wedge \,\,\overline{\beta }({t}_{i}:{t}_{i}+1\,\mathrm{hr})\gt {\beta }_{\mathrm{sw}}.\end{array}\end{eqnarray} \tag{ B10 }$$

Also in this case we visually inspected the classification resulting from the application of Equations (B8)–(B10) verifying their reliability and consistency throughout the whole range of heliocentric distances sampled by the virtual spacecraft in the model domain.