© ESO 2021

1. Introduction

The Sun continuously emits charged and neutral particles in the form of solar wind (Parker 1958, 1965; Cranmer et al. 2017). In addition, it also ejects gigantic structures of plasma and magnetic field, known as coronal mass ejections (CMEs; see e.g., Webb & Howard 2012). These structures have been observed by coronagraphs as they propagate away from the Sun (Illing & Hundhausen 1985; Vourlidas et al. 2013) and they have been probed in situ by spacecraft monitoring the conditions of the interplanetary medium (where they are named ICMEs, i.e. Interplanetary CMEs¹; Zurbuchen & Richardson 2006; Kilpua et al. 2017). In situ, CMEs can be distinguished from the ambient solar wind through which they propagate via a combination of plasma and magnetic field parameters (Burlaga et al. 1981; Bothmer & Schwenn 1996; Wimmer-Schweingruber et al. 2006; Zurbuchen & Richardson 2006). Typically, when a fast-enough CME is detected, the first observed signature is a CME-driven interplanetary shock, followed by a sheath region composed of solar wind material accumulated between the shock and the actual CME structure. This region typically presents increased density and a stronger magnetic field, and it exhibits turbulent variations of the magnetic field magnitude and direction (Kilpua et al. 2017). If the configuration of the spacecraft crossing through the CME structure is appropriate (see e.g., Gopalswamy et al. 2009a; Janvier et al. 2014, for examples of different crossing configurations), the sheath is followed by the actual CME body, referred to as a ‘magnetic ejecta’ because of the magnetically dominated plasma and the low level of magnetic field fluctuations that are typically observed in this region.

The evolution of CMEs in the inner heliosphere is the result of the delicate interplay between the internal plasma and magnetic field properties and the conditions of the surrounding solar wind. It is primarily shaped by two effects: (1) the expansion of the magnetic ejecta, which controls its internal magnetic field magnitude and size (Démoulin & Dasso 2009a; Démoulin 2010; Gulisano et al. 2010); and (2) the interaction with the surrounding solar wind, which controls the CME kinematics and is often described in terms of a drag force (Cargill 2004; Vršnak et al. 2010). Furthermore, the erosion of the CME magnetic field (and its opposite, flux injection), as a consequence of magnetic reconnection processes with the surrounding solar wind (Dasso et al. 2006, 2007; Ruffenach et al. 2012, 2015; Lavraud et al. 2014; Manchester et al. 2014; Pal et al. 2020), can also alter the CME magnetic configuration and size. The interaction of CMEs with other transients in the solar wind, such as high-speed streams (e.g., Gopalswamy et al. 2009a; Wood et al. 2012; Rodriguez et al. 2016; Winslow et al. 2021), the heliospheric current sheet (Winslow et al. 2016), and other CMEs (e.g., Dasso et al. 2009; Lugaz et al. 2017; Shen et al. 2017), also contribute in a major way to altering the CME kinematics and internal properties during propagation. The geometry, morphology, and kinematics of CME structure in the solar corona and interplanetary space are all closely related to the complexity of the propagation scenario, which can result in major evolutionary deformations of a various nature (e.g., Isavnin 2016).

Over recent decades, the radial evolution of CMEs in the inner heliosphere has primarily been investigated by means of statistical studies that use observations of a large number of CMEs to derive an average scenario of CME propagation (Liu et al. 2005; Gulisano et al. 2010; Winslow et al. 2015; Janvier et al. 2019; Salman et al. 2020), as well as by means of numerical simulations (see e.g., the review by Manchester et al. 2017). More recently, the increasing number of spacecraft probing the solar wind plasma and magnetic field conditions in different regions of the inner heliosphere has also allowed for more direct observations of individual CMEs at different heliocentric distances, thanks to the higher number of longitudinal conjunction events among different spacecraft (Rodriguez et al. 2008; Nakwacki et al. 2011; Winslow et al. 2016, 2018; Good et al. 2015, 2018; Lugaz et al. 2020; Salman et al. 2020). Despite the increased capabilities for observing the same CME event from multiple spacecraft located at different heliocentric distances, at the time of writing, for most CMEs, in situ observations remain limited to a single spacecraft. Furthermore, in situ CME observations within 0.3 au from the Sun are still extremely rare (e.g., Korreck et al. 2020; Möstl et al. 2020; Nieves-Chinchilla et al. 2020; Weiss et al. 2021), with NASA’s Parker Solar Probe (PSP; Fox et al. 2016) having so far operated during a low solar activity phase when just a few CMEs were observed. Because of these limitations, numerical simulations still represent an invaluable tool for filling in observational gaps and providing context to the sparse observations available. Despite their potential, however, the performances of numerical solar wind and CME propagation models for the inner heliosphere developed for space weather research and forecasting purposes remain largely untested for heliocentric distances other than that of the Earth (for a recent exception, see Al-Haddad et al. 2019). For instance, the effect of the presence of the model inner boundary on the early CME propagation and evolution modelled is a critical, but thus far uninvestigated, issue. This is relevant to the potential application of such models in predicting the CME properties close the Sun in the near future (Odstrcil et al. 2020).

In this work, we aim to bridge the gap between observational and modelling studies by borrowing techniques that are traditionally employed to analyse in situ CME observations (e.g., Gulisano et al. 2010; Janvier et al. 2019) and applying them to modelled in situ CME signatures. The double purpose of the study is: (1) to validate the numerical modelling of interplanetary CMEs performed by the EUropean Heliospheric FORecasting Information Asset (EUHFORIA; Pomoell & Poedts 2018) integrated with the linear force-free spheromak CME model (Verbeke et al. 2019), over a broad range of heliocentric distances; and (2) to perform a comprehensive analysis of the radial evolution of interplanetary CMEs exploiting the potential of available observational studies and techniques in combination with numerical models. We focus on addressing a number of fundamental questions on the interplanetary evolution of CME structures that have not been targeted by past studies using the EUHFORIA model, considering, in particular, how realistically the radial evolution of the ambient solar wind is modelled in EUHFORIA and how it affects the radial propagation of magnetised CME structures in numerical simulations. We also consider how the average CME properties and the size of the various CME structures evolve with radial distance and how expansion affects the magnetic field and kinematics of CMEs at various heliocentric distances. Futhermore, we explore how long it takes for the solar wind to recover to pre-event values after the passage of a CME.

To serve our purposes, we take as our case study the Earth-directed fast halo CME observed at the Sun on 12 July 2012. This event was previously analysed in detail using EUHFORIA simulations by Scolini et al. (2019), hereafter referred to as ‘Paper I’. In the present work, we further analyse the simulations presented in Paper I, focusing in particular on Run 03 therein, corresponding to an initial CME configuration that has been optimised based on physical arguments and a thorough comparison with observations.

This paper is structured as follows. In Sect. 2, we provide an overview of the observational CME characteristics of the event and introduce the modelling tool used to investigate its evolution in interplanetary space. In Sects. 3 and 4, we present the results obtained from the analysis of the radial evolution of the ambient solar wind and CME structure. In Sect. 5, we discuss our main findings and provide a physical interpretation of the results. Finally, in Sect. 6, we present our conclusions.

2. Observations and modelling tools

2.1. Remote-sensing and in situ observations of the 12 July 2012 CME

In this work, we investigate the radial evolution of an Earth-directed fast halo CME observed on 12 July 2012. On the day of the eruption, its source region (NOAA AR 11520) was located close to the solar disk centre, and the CME was first observed in LASCO coronagraphs (Brueckner et al. 1995) on board the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995) at 16:48 UT, appearing as a fast halo propagating with an average projected speed in the plane of the sky of 885 km s⁻¹ (from the CDAW CME catalog²). An intense X1.4-class flare was also detected by the GOES satellites, starting from 15:37 UT.

After propagating through the inner heliosphere, the CME was observed to arrive at the Sun–Earth Lagrangian point 1 (L1) a few days later, where its preceding shock was detected by the Wind spacecraft (Ogilvie et al. 1995) on 14 July 2012 at 17:39 UT (from the Heliospheric Shock Database³; Kilpua et al. 2015). According to the Richardson & Cane list (hereafter R&C list⁴; Cane & Richardson 2003; Richardson & Cane 2010), a magnetic ejecta presenting the characteristics of a magnetic cloud (i.e. an enhanced magnetic field, a smooth rotation of the magnetic field vector, and low proton temperature; e.g., Burlaga et al. 1981; Klein & Burlaga 1982) was detected, starting around 06:00 UT on 15 July and ending at about 05:00 UT on 17 July. The orientation of the magnetic field within the magnetic ejecta was characterised by a strongly-negative B_z component in Geocentric Solar Ecliptic (GSE) coordinates and, therefore, it was mostly anti-parallel with respect to the magnetic field of the Earth, triggering an intense geomagnetic storm as indicated by the Disturbance Storm Time (Dst) index, which reached a minimum of −139 nT on 15 July. Figure 1 shows the 1 min averaged measurements from the Wind spacecraft taken during the passage of the CME. The region between the CME-driven shock and the starting of the magnetic ejecta (i.e. the CME sheath) is indicated by the yellow shaded area, while the duration of the magnetic ejecta is bounded by the blue shaded area.

Fig. 1. 1 min averaged solar wind magnetic field and plasma parameters as detected by the Wind spacecraft at L1. From top to bottom: bulk speed (v), proton number density (np), magnetic field magnitude (B), magnetic field components in GSE coordinates (Bx, By, Bz), proton temperature (Tp), magnetic field elevation (θB), and proton β (βp). Bottom panel: 1 h Dst index. The duration of the sheath is indicated by the yellow area, while the duration of the magnetic ejecta is bounded by the blue area. The dashed blue line marks the start of the perturbation in the speed profile due to an approaching fast solar wind stream at the back of the CME. The temperature panel shows the comparison of the observed proton temperature, Tp, (in black) with the expected temperature, Texp, calculated using the speed-temperature relation by Lopez & Freeman (1986; in red). In the βp panel, the value βp = 1 is indicated by the dashed grey line.

The remote-sensing and in situ observational signatures of this event were discussed in detail in Paper I. Here, we highlight two characteristics of the in situ speed profile v(t) that are more relevant for the following discussion: (1) v(t) is higher at the front than at the back of the magnetic ejecta, indicating that this was an expanding magnetic structure in the solar wind at 1 au; (2) v(t) is decreasing with a nearly-linear behaviour only in the front part of the ejecta, while a distortion is present in the rear part of the ejecta starting around 14:00 UT on 16 July (marked by the dashed blue line in Fig. 1). This was most probably caused by the presence of an overtaking fast solar wind stream visible at the back of the magnetic ejecta. The CME under study therefore belongs to the class of ‘perturbed’ magnetic ejecta, as defined by Gulisano et al. (2010), since it shows a strongly-distorted velocity profile compared to non-perturbed cases (which show a linear velocity profile for almost the full duration of the magnetic ejecta event). We further discuss how these characteristics were reproduced by numerical simulations and how they affected the CME propagation in the following sections. In addition, we note that the identification of the CME boundaries from in situ data, particularly the rear one, can be a difficult task. In fact, various authors frequently consider different boundaries for the same CME, typically with differences of few hours, due to the application of different identification criteria (e.g., Dasso et al. 2006; Al-Haddad et al. 2013), which can, in turn, significantly affect the reconstruction of the parameters that are meant to describe the internal characteristics of CME.

2.2. EUHFORIA model

As a coronal and heliospheric model recently designed for space weather research and prediction purposes, EUHFORIA is composed of a semi-empirical coronal model and a three-dimensional (3D) magnetohydrodynamics (MHD) model of the inner heliosphere (Pomoell & Poedts 2018). The solar wind conditions at the inner boundary of the heliospheric domain, located at 0.1 au, are derived through a semi-empirical approach based on the Wang-Sheeley-Arge (WSA; see e.g., Arge et al. 2004) coronal model, which takes as its input synoptic maps of the photospheric magnetic field from the Global Oscillation Network Group (GONG) network. A detailed discussion of the coronal model currently used in EUHFORIA is provided in the recent paper by Pomoell & Poedts (2018). The solar wind map generated by the coronal model is then used as inner boundary condition for the heliospheric model, which solves the time-dependent ideal MHD equations in 3D space in the Heliocentric Earth EQuatorial (HEEQ) coordinate system. The heliospheric model in EUHFORIA treats the solar wind plasma as a single fluid, solving the time-dependent MHD equations and providing as output the following eight MHD variables: the plasma number density n, the plasma velocity components v_x, v_y, v_z, the plasma magnetic field components B_x, B_y, B_z, and the plasma thermal pressure P_th. The standard computational domain for the heliospheric model extends from 0.1 au to 2 au in the radial direction (D), spanning ±60° in latitude (θ), and ±180° in longitude (ϕ).

In addition to the modelling of the ambient solar wind, EUHFORIA admits the possibility to model CMEs either using a simplified cone model (Scolini et al. 2018) or via a more realistic linear force-free spheromak (flux-rope) model (Verbeke et al. 2019). In both models, CMEs are inserted in to the heliosphere as time-dependent boundary conditions at the inner radial boundary of the domain, namely, at the heliocentric distance of D = 0.1 au. The initial geometry and kinematics of cone and spheromak CMEs are defined via the following input parameters: the CME insertion start time (t_CME, i), speed (v_CME, i), latitude (θ_CME, i), longitude (ϕ_CME, i), and half width (ω_CME, i/2). These parameters are normally derived from remote-sensing white-light observations of CMEs in the corona. In addition, due to more limited observational inputs, two additional parameters defining the CME mass density and temperature are set to be homogeneous and equal to the following default values: ρ_CME, i = 1 × 10⁻¹⁸ kg m⁻³ and T_CME, i = 0.8 × 10⁶ K (Pomoell & Poedts 2018). A recent analysis on the remote-sensing estimation of the CME density and its comparison with observations at 1 au has been provided by Temmer et al. (2021). In the future, this work will likely become an extremely valuable contribution in helping to constrain the CME initial density for heliospheric models such as EUHFORIA. The magnetic configuration of spheromak CMEs is then specified via three additional parameters: the chirality, or helicity sign (H_CME, i), the tilt (γ_CME, i), and the toroidal magnetic flux (φ_{t, CME, i}), which can often be constrained using several EUV (low-coronal) and magnetic (photospheric) proxies (see Palmerio et al. 2017, and references therein).

The comparison of EUHFORIA results with actual in situ measurements of the solar wind properties requires us to relate the above single-fluid MHD variables to the macroscopic quantities measured by in situ spacecraft. Employing a single-fluid approach, the EUHFORIA heliospheric model makes no distinction between different particle populations (e.g., protons, electrons, α particles, etc.) and it assumes all populations to be characterised by the same macroscopic parameters. In considering the proton and electron populations as the two primary contributors to solar wind plasma (von Steiger et al. 2000; Zurbuchen et al. 2002; Reisenfeld et al. 2007; Heber et al. 2009), we can treat the two species as characterised by the same (isotropic) temperature, T = T_p = T_e. To further ensure the quasi-neutrality of the plasma at all times and spatial locations in the heliosphere, we also assume the two species to be characterised by a uniform number density, n_p = n_e, with the plasma number density of all particle populations being n = n_p + n_e. As a result, the solar wind thermal pressure, P_th, provided by EUHFORIA can be described as P_th = P_p, th + P_e, th = n_pk_BT_p + n_ek_BT_e = 2P_p, th, that is, twice the thermal pressure of the proton population alone. In this picture, the plasma β corresponds to β = P_th/P_mag = 2P_p, th/P_mag = 2β_p, that is, two times the proton β. In the following, we therefore derive the proton thermal pressure and proton β parameter from EUHFORIA outputs by dividing the plasma thermal pressure and plasma β by a factor 2, that is, P_p, th = P_th/2 and β_p = β/2. The plasma total pressure P_tot from both EUHFORIA results and Wind data is calculated as: P_tot = P_mag + P_th = P_mag + P_p, th + P_e, th = P_mag + 2P_p, th, while the plasma (and proton) temperature is calculated as: T = T_p = P/nk_B = P_p/n_pk_B. The speed, v, of the solar wind plasma corresponds to the speed of the centre of mass of the system and can be expressed in terms of the proton and electron properties as: v = (v_pm_p + v_em_e)/(m_p + m_e)∼v_p.

2.3. Modelling the 12 July 2012 CME with EUHFORIA

In this work, we further analyse the simulations of the 12 July 2012 CME presented in Paper I. We place a particular focus on the investigation of the results obtained from Run 03, as this was found to be the most realistic simulation of the CME performed using a magnetised flux-rope model. The ambient solar wind in the heliosphere was obtained using the GONG standard synoptic map on 12 July 2012 at 11:54 UT as its input. The spheromak CME structure was initialised using kinetic and geometric parameters derived from the 3D reconstruction based on the graduated cylindrical shell (GCS; Thernisien et al. 2009; Thernisien 2011) model applied to multi-viewpoint observations of the CME in the corona. The GCS reconstruction provided the following set of CME initial parameters (extrapolated to 0.1 au assuming a self-similar propagation): passage time on 12 July 2012 at 19:24 UT; translation speed of 763 km s⁻¹; longitude of −4°; latitude of −8°; half width of 38°. We refer the reader to Paper I for further information on the analysis of the CME coronal signatures. Here, we only further note that because of their large angular separations, the two STEREO spacecraft were in an optimal configuration to observe the radial propagation of the CME and provide accurate measurements of the CME kinematics in the direction that was most needed for the simulation (i.e. towards Earth). The magnetic parameters of the spheromak CME were initialised based on EUV and photospheric magnetic observations of the source region, which indicated the eruption of a right-handed flux rope magnetic structure with inclination of about 45° with respect to the solar equatorial plane and with the axial magnetic field pointing towards the south-east direction. Again, we refer to Paper I for further details on the analysis of the CME source region signatures. The simulation was performed using EUHFORIA version 1.0.4 with the following resolution in the heliospheric domain: 512 grid cells in the radial direction between 0.1 au and 2.0 au (corresponding to a radial resolution of ΔD = 0.00371 au) and a 2° angular resolution in longitude and latitude. To ensure the CME and its wake had time to propagate beyond 2 au, corresponding to the radial outer boundary in our simulation domain, we ran the model for 19 days after the time of the magnetogram, that is, until 31 July 2012.

To track the heliospheric propagation of the CME in the simulation, we placed a first set of virtual spacecraft along the Sun–Earth line. Such virtual spacecraft were uniformly distributed in the radial direction between 0.2 and 1.9 au, with a separation of 0.1 au among each other. A second set of virtual spacecraft was placed around the Earth position (i.e. at 1 au), at Δσ = 5° and Δσ = 10° longitudinal or latitudinal separation from Earth in order to assess the spatial variability of the results in the vicinity of Earth. Figure 2 shows a schematic representation of the location of the virtual spacecraft in the 3D heliospheric domain of EUHFORIA. An overview of the CME propagation in the EUHFORIA heliospheric domain is provided in Fig. 3. A comparison between EUHFORIA results at (and in the vicinity of) Earth and in situ measurements obtained by the Wind spacecraft is provided in Fig. 4.

Fig. 2. Location of the virtual spacecraft placed in the EUHFORIA heliospheric simulation domain, as seen from two different views. Blue dots: virtual spacecraft along the Sun–Earth line. Red dots: virtual spacecraft near the location of the Earth. Yellow dot: location of the Sun. Green dot: location of the Earth. We focus on the virtual spacecraft in blue to study the radial evolution of the Earth-directed CME event considered.

Fig. 3. Overview of EUHFORIA results in the heliographic equatorial plane. Snapshots of the radial speed, vr (a), scaled number density, n (D/1 au)2 (b), co-latitudinal magnetic field, Bclt = −Bθ (c), and scaled radial magnetic field, Br (D/1 au)2 (d), on 15 July 2012 at 23:53 UT which is the time when the CME was crossing through Earth.

Fig. 4. Comparison of 10 min cadence EUHFORIA predictions at Earth (red solid line) and 1 min averaged Wind proton data (black solid line). The dark red and light red shaded areas show the maximum variation of EUHFORIA predictions at virtual spacecraft separated by Δσ = 5° and Δσ = 10° in longitude and/or latitude from Earth. From top to bottom: speed (v), proton number density (np), magnetic field magnitude (B), magnetic field components in GSE coordinates (Bx, By, Bz), and proton thermal pressure (Pp, th).

3. Results: Radial evolution of the ambient solar wind

As a first step towards the understanding of the heliospheric propagation of the CME studied here, we start by characterising the conditions of the ambient solar wind through which the CME propagates.

3.1. Theoretical and observational expectations

To date, the most extensive database of solar wind measurements at different heliocentric distances on the ecliptic plane dates back to in situ observations carried out in the 1970s and 1980s by the Helios 1 and Helios 2 missions (Schwenn & Marsch 1990), which covered heliocentric distances between 0.29 and 0.98 au. Assuming a radial solar wind outflow, the radial dependence of the solar wind parameters extracted from the Helios data can be typically described through a power law scaling. Their radial behaviour can be therefore written in the form of a power law function:

(1)

where X is the given solar wind parameter considered, D is a generic heliocentric distance in astronomical units (au), D₀ is a reference heliocentric distance in au, a_x is the power law exponent for the solar wind parameter X, and X₀ is the magnitude of the solar wind parameter at the reference distance D₀ (in au). Typically, the reference heliocentric distance is taken at 1 au, where continuous measurements of the solar wind parameters are available. Numerous existing studies constrained the exponents of the various parameters based on Helios observations.

Speed. The Parker’s solar wind model predicts the solar wind speed v in interplanetary space to be approximately constant with respect to the heliocentric distance (Parker 1958, 1965). Observational studies agree in their consideration of the solar wind speed as essentially radial and independent on the heliocentric distance (i.e. v ∝ D⁰). Schwenn (1983, 1990) derived radial dependencies of v ∝ D^0.083 and v ∝ D^0.036 for Helios 1 and Helios 2, respectively, while Venzmer & Bothmer (2018) reported a velocity exponent v ∝ D^0.049 for the combined data sets. It is important to mention that the observed variability of the solar wind speed with D can be expected to depend, amongst other factors, on the particular initial speed of individual plasma bubbles, and on their interaction with other structures in interplanetary space.

Density. Assuming conservation of mass and a constant solar wind speed as predicted by the Parker’s solar wind model, the proton number density is expected to fall off as n_p ∝ D⁻². Bougeret et al. (1984) derived the exponent for the solar wind electron number density, as n_e ∝ D^−2.10 based on Helios 1 and Helios 2 observations. More recently, Venzmer & Bothmer (2018) reported a radial dependence for the solar wind proton number density of n_p ∝ D^−2.01 by performing an analysis of the full Helios database. Furthermore, Hellinger et al. (2011) and Perrone et al. (2019a) reported radial dependencies of n_p ∝ D^−1.8 and n_p ∝ D^{−1.96 ± 0.07}, respectively, based on Helios observations of solar wind high-speed streams (v ≥ 600 km s⁻¹). In the slow solar wind (v < 400 km s⁻¹), Hellinger et al. (2013) reported a radial dependence of n_p ∝ D^−1.9 based on Helios observations. Under the assumption of quasi-neutrality and by treating the solar wind as composed by protons and electrons only (neglecting the presence of α particles, which have typical abundances between 1 and 5%; see e.g., Kasper et al. 2012), the number density of protons and electrons are identical, namely, n_p = n_e, meaning that the above dependencies hold for both particle populations.

Magnetic field magnitude. For an adiabatic solar wind expansion and assuming conservation of mass and magnetic flux, the three components (radial, latitudinal, and longitudinal) of the interplanetary magnetic field on the ecliptic plane are expected to fall off as B_r ∝ D⁻², B_θ ∝ D⁰, and B_ϕ ∝ D⁻¹, respectively. Therefore, the total magnetic field magnitude can be expected to decay with a fall off index between a_B = 0 and a_B = −2. Observationally, the radial dependence of the interplanetary magnetic field magnitude in the inner heliosphere was first characterised using early measurements carried out by the Helios missions. Mariani & Neubauer (1990) provides a summary of these early findings. In particular, Helios 1 data were best fitted by a radial dependence of B ∝ D^{−1.56 ± 0.09}, while Helios 2 data were best fitted via the relation B ∝ D^{−1.84 ± 0.08}. More recently, Venzmer & Bothmer (2018) reported a radial dependence of the total magnetic field as B ∝ D^−1.55 by performing an analysis of the full Helios database. Previous analyses also reported a dependence of the scaling of the interplanetary magnetic field magnitude with the solar wind speed, such as Mariani & Neubauer (1990), who fitted Helios 2 data with B ∝ D^{−1.86 ± 0.05} for the fast solar wind (v > 550 km s⁻¹) and B ∝ D^{−1.64 ± 0.24} for the slow solar wind (v < 450 km s⁻¹). More recently, Hellinger et al. (2011) and Perrone et al. (2019a) obtained radial dependencies in fast solar wind streams of B ∝ D^−1.6 and B ∝ D^{−1.59 ± 0.06}, respectively. For the slow solar wind, Hellinger et al. (2013) obtained a radial dependence of B ∝ D^−1.6.

Temperature. Assuming an ideal gas in adiabatic expansion, the solar wind temperature (considering all populations in thermal equilibrium) is expected to fall off as T ∝ D^−4/3. However, the description of the actual solar wind temperature is much more complex, as the various species composing the solar wind plasma (i.e. protons and electrons, but also α particles and ionised heavy elements) exhibit distribution functions often characterised by different bulk temperatures and deviating from Maxwellian or isotropic conditions (see e.g., Marsch 2006, 2012; Verscharen et al. 2019, and references therein). Therefore, the temperature properties of the various particle populations have to be considered independently. Furthermore, the existence of a speed–proton temperature relation in the solar wind has been also recognised (e.g., Lopez & Freeman 1986; Lopez 1987; Perrone et al. 2019b), requiring us to treat the radial dependence of the solar wind temperature in slow and fast solar wind streams on an individual basis. Limiting our attention to proton populations, Schwenn et al. (1981) obtained the following radial dependencies: T_p ∝ D^−1.21 for the slow solar wind and T_p ∝ D^−0.69 for the fast solar wind, which assumes the existence of an additional heating mechanism to explain the non-adiabatic expansion of fast solar wind streams in interplanetary space (e.g., Totten et al. 1995). More recently, for the fast solar wind, Hellinger et al. (2011) reported a dependence of T_p ∝ D^−0.74, while Perrone et al. (2019a) reported a dependence of T_p ∝ D^{−0.9 ± 0.1}. Hellinger et al. (2013) reported a dependence of T_p ∝ D^−0.58 in the slow solar wind. An average proton temperature exponent of T_p ∝ D^−0.79 was also reported by Venzmer & Bothmer (2018). A detailed investigation of the physical origin of the temperature–speed relation in the solar wind, including a description of its relation to heating mechanisms and adiabatic expansion, can be found in Démoulin (2009).

Proton pressure and proton β. Based on the above expected theoretical scalings for the solar wind speed, density, magnetic field, and temperature, we can estimate the scaling of relevant derived quantities such as the proton thermal pressure, magnetic pressure, and proton β. In particular, P_p, th ∝ n_pT_p ∝ D^{a_np + a_Tp} (i.e. a_Pp, th = a_np + a_Tp ∼ −10/3), P_mag ∝ B² ∝ D^2a_B (i.e. a_Pmag = 2a_B ∼ −4 ↔ 0), and hence β_p = P_p, th/P_mag ∝ D^{a_np + a_Tp − 2a_B} (i.e. a_βp = a_Pp, th − a_Pmag ∼ −10/3 ↔ 2/3). As a reference, Perrone et al. (2019a) reported a dependence of the proton thermal pressure P_p, th ∝ D^{−2.9 ± 0.1}, and a dependence of the magnetic pressure P_mag ∝ D^{−3.2 ± 0.1} in high-speed solar wind streams. The resulting radial dependence of the proton β, defined as the ratio between the proton thermal and magnetic pressures, namely, β_p = P_p, th/P_mag, was β_p ∝ D^{0.4 ± 0.1}.

3.2. Modelling results and comparison with expectations

To characterise the radial dependence of the solar wind parameters in EUHFORIA, we analysed the modelled time series at the virtual spacecraft distributed along the Sun–Earth line (Fig. 2). In order to make sure that similar solar wind conditions were considered at each spacecraft (i.e. evolution with heliocentric distance) and because according to EUHFORIA simulations, the CME propagated through a slow solar wind background along the Sun–Earth line (Fig. 3), we identified periods of slow solar wind using the criteria v ≤ 400 km s⁻¹ (similar to Hellinger et al. 2013). For each of these time periods at different virtual spacecraft, we calculated the maximum, minimum, mean, and median values of the solar wind parameters observed at each location and we then performed a linear fit of the mean conditions in log–log space. By re-writing Eq. (1) in log–log space for a reference heliocentric distance D₀ = 1 au, we obtain a linear fitting relation equal to:

(2)

where log X(D) is the logarithm of the given solar wind parameter X considered, log(D) is the logarithm of the generic heliocentric distance in au, a_x is the power law exponent for the solar wind parameter X (same as in Eq. (1)), and log(X₀) is the logarithm of the typical value of the particular solar wind parameter X at 1 au. The quality of the fitting is evaluated in terms of the Pearson correlation coefficient (r_P), and of the Spearman’s rank correlation coefficient (r_S). The results from this procedure are shown in Figs. 5 and 6. Table 1 reports the radial dependencies (a_x parameter in Eqs. (1) and (2)) of the mean slow solar wind parameters in EUHFORIA, derived from the fitting of the individual values at different heliocentric distances using Eq. (1) in log–log space. To assess the consistency between the mean- and median-based values (with the former typically being more sensitive to outliers than the latter), in the rightmost column in Table 1, we also provide the dependencies estimated from the median values; for both, we indicate the uncertainties derived from the linear fitting based on 95% confidence intervals. The two methods give similar results, particularly for the magnetic field strength and for the proton density, speed, temperature, and β. Such a result is also suggested by the close mean and median values reported in the small panels (for 0.2 and 1.0 au) in Figs. 5 and 6. Because of the larger number of observational studies investigating mean values rather than median values, in the following, we focus on mean values only. A comparison with mean-based observational values (when available) is provided in Table 1.

Fig. 5. Ambient solar wind from EUHFORIA: Radial dependence of the mean solar wind parameters extracted at virtual spacecraft located along the Sun–Earth line (red dots): (a) proton number density ⟨np⟩sw; (b) speed ⟨v⟩sw; (c) magnetic field ⟨B⟩sw; (d) proton temperature ⟨Tp⟩sw. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Observation-based relations from Hellinger et al. (2013) are indicated as dashed green lines. Observed mean values from Wind in the 24 h prior to the CME arrival are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

Fig. 6. Ambient solar wind from EUHFORIA: Radial dependence of the mean solar wind parameters extracted at virtual spacecraft located along the Sun–Earth line (red dots): (e) proton thermal pressure ⟨Pp, th⟩sw; (f) magnetic pressure ⟨Pmag⟩sw; (g) total pressure ⟨Ptot⟩sw; (h) proton β ⟨βp⟩sw. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Mean observed values from Wind in the 24 h prior to the CME arrival are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

Overall, the fall-off (i.e. the slope) of the mean of each solar wind parameter with heliocentric distance is consistent with observational values (see summary of the exponents a_x for the various parameters and solar wind components in Table 1) and most parameters considered also exhibit a strong correlation (≥0.9) with the heliocentric distance D in log–log scale. The two parameters that exhibit the weakest dependencies on the heliocentric distance are: the solar wind speed (a_v = 0.04, correlations between 0.81 and 0.92) and proton β (a_βp = 0.52, correlations between 0.20 and 0.39), which is consistent with theoretical and observational arguments. Regarding v, it is important to note that a modelled non-zero dependence with heliocentric distance might, in turn, affect the modelled decay of all the other parameters, whose radial devolution is strictly dependent on the radial evolution of the speed.

Since our characterisation of the solar wind radial evolution has the ultimate scope of assessing how the solar wind modelling in EUHFORIA affects the ultimate radial evolution of the CME, we focus here on the discussion of the observational–modelling discrepancies in light of their potential influence on the CME propagation. First of all, we note that the drag force is primarily dependent on the CME geometry, that is, its cross-section and width, as well as on its density and speed relative to the ambient solar wind (e.g., Vršnak et al. 2010). The solar wind modelled in EUHFORIA shows that ⟨n_p⟩_sw (⟨v⟩_sw) is slightly overestimated (underestimated) compared to the mean values measured by Wind at 1 au, suggesting that the drag acted by the solar wind on the CME is probably slightly overestimated in our simulation than it was in reality. Secondly, the CME expansion is primarily controlled by the relative pressure of the CME and the surrounding solar wind (contributing to the momentum equation primarily via the thermal and magnetic pressure gradients; see Eq. (2) in Paper I, and Démoulin & Dasso 2009a; Gulisano et al. 2010); namely, the most relevant solar wind parameters are: ⟨P_p, th⟩_sw, ⟨P_mag⟩_sw, and ⟨P_tot⟩_sw. We find that ⟨P_p, th⟩_sw is slightly overestimated in EUHFORIA, while ⟨P_mag⟩_sw is slightly underestimated (as a consequence of the slight underestimation of ⟨B⟩_sw). However, since ⟨β_p⟩_sw is higher than 1 at all heliocentric distances, that is, the total pressure in the solar wind is primarily determined by the thermal pressure rather than by the magnetic pressure, we do not expect this modelling discrepancy to significantly affect the expansion of the CME. Most importantly, we see from Fig. 6g that ⟨P_tot⟩_sw well matches Wind observations at L1, and therefore the CME expansion in EUHFORIA is most probably not (or only weakly) affected by any discrepancy with in situ observations.

Overall, we can thus conclude that the EUHFORIA modelling of the slow solar wind in the range between 0.2 and 1.9 au reproduces the observational expectations sufficiently well to ensure a realistic modelling of the CME radial evolution for the aspects concerning the CME–solar wind interactions that result in drag and expansion.

4. Results: Radial evolution of the CME

4.1. Identification of the sheath and ejecta boundaries in EUHFORIA

Characterising the radial evolution of the CME properties in EUHFORIA simulations requires us to first distinguish among the various CME substructures in EUHFORIA time series, that is, the CME-driven shock, the CME sheath, and the magnetic ejecta. This passes through the determination of the shock time (t_shock), and of the start (t_start) and end time (t_end) of the magnetic ejecta from the EUHFORIA time series extracted at different heliocentric distances. To identify the arrival time of the CME-driven shock at various radial distances, we apply the following threshold conditions to EUHFORIA time series:

(3)

where t_i is a generic time in the time series, v(t), n_p(t), B(t) are the speed, proton number density and magnetic field magnitude time series obtained from the simulation of the CME with EUHFORIA, and v_sw(t), n_p, sw(t), B_sw(t) are the speed, proton number density, and magnetic field magnitude time series of the ambient solar wind as determined by performing one simulation of the ambient solar wind alone. We note that in our simulations, time series are characterised by a cadence of Δt_out = t_i − t_i − 1 = 10 min. The arrival time of the CME shock t_shock at each spacecraft is marked based on the earliest time t_i in the EUHFORIA time series at which Eq. (3) is satisfied. This set of conditions is an adapted version of the conditions used to detect interplanetary shocks from in situ solar wind measurements employed by the Heliospheric Shock Database, and it was chosen due to its reliability in detecting the arrival of CME fronts in EUHFORIA in situ time series at various heliocentric distances, as verified by their visual inspection (see also Fig. A.2).

In the inner heliosphere, several proxies have been used to identify magnetic ejecta from in situ solar wind plasma and magnetic field time series, including proton temperature (Lopez & Freeman 1986; Lopez 1987; Richardson & Cane 1995), the proton and plasma β (Richardson & Cane 1995; Lepping et al. 2005), and the characteristics of the magnetic field (e.g., Burlaga et al. 1981; Gulisano et al. 2012). For completeness, we note that the identification of magnetic ejecta in the outer heliosphere (beyond ∼5 au) is even more complex, both because of the blurring of some of their characteristics through interaction with the ambient solar wind and because of the limited instrumentation precluding the observations of some of the key parameters on board of past and current missions (Richardson et al. 2006). As a result, the most reliable characteristics for tracing magnetic ejecta in the outer heliosphere are those less affected by the interaction with solar wind streams, e.g., the elemental abundance ratios and the relative charge state abundance for given elements (e.g., Goldstein et al. 1998; Paularena et al. 2001).

At 1 au, a common proton β threshold used to automatically identify magnetic ejecta from in situ solar wind measurements is β_p, obs ≤ 0.3 (e.g., Lepping et al. 2005). To be as consistent as possible with this observational approach, in this work, we identify the start time t_start and end time t_end of CME or magnetic ejecta based on an similar analysis of the proton β EUHFORIA time series. Because of the fundamental differences between the actual solar wind characteristics and its single-fluid description performed by EUHFORIA, we tested a wide range of thresholds, spanning between β_p ≤ 0.1 and β_p ≤ 1.0. We find that a β_p ≤ 0.5 threshold provides the best compromise to allow a robust identification of the CME boundaries over a wide range of latitudes, namely, between 0.2 au and 1.9 au (see Appendix A for the comparison of different thresholds). In order to remove possible spurious results before and after the CME passage, we limit the application of the β_p threshold condition to times following the detection of the CME shock. We define the start time of the CME/magnetic ejecta t_start as the first time when β_p(t_i)≤0.5, with t_i ≥ t_shock. Furthermore, the passage of the CME rear edge is marked by the first time at which β_p passes from β_p ≤ 0.5 to β_p > 0.5, with t_end ≥ t_start. We further impose that the β_p > 0.5 condition is met for a period of at least 1 h in order to make sure to identify the same feature at various heliocentric distances (the results of this approach are shown in Fig. A.2). We note that the use of a fixed β_p threshold condition to identify the magnetic ejecta at different heliocentric distances is also justified by the almost constant solar wind β_p with respect to the heliocentric distance (Fig. 6h), which provides a stable reference value for distinguishing between the magnetic ejecta and the solar wind, regardless of the heliocentric distance.

From the identification of the time of the shock and of the start and end times of the ejecta, we identify the periods associated with the sheath and magnetic ejecta at each virtual spacecraft. Figure A.2 shows the results obtained applying the shock identification conditions (Eq. (3)) and the chosen plasma β_p threshold to the EUHFORIA time series at various radial distances.

4.2. Radial evolution of mean CME values

Based on the determination of the sheath and CME boundaries discussed above, we calculate the mean values of the parameters characterising the CME structure at various heliocentric distances and we compare their radial evolution with results reported in previous studies. As discussed below, we limit our attention to mean values because of the comparatively greater number of observational studies investigating mean values rather than median values (despite the latter being more robust and resistant to outliers; see Janvier et al. 2019). In particular, we focus on the determination of the CME mean magnetic field ⟨B⟩_CME, mean proton number density ⟨n_p⟩_CME, mean speed ⟨v⟩_CME, and mean proton temperature ⟨T_p⟩_CME. Figures 7 and 8 show the radial dependence of the mean CME properties in a similar manner as done in Figs. 5 and 6 for the ambient solar wind. Table 2 reports the heliocentric distance dependencies (a_x parameter in Eqs. (1) and (2)) derived from the fitting of the mean individual values at different heliocentric distances using Eq. (1) in log–log space. To assess the consistency between mean and median values, we also include the dependencies estimated from the median-based values in Table 2. For both, we indicate the uncertainties derived from the linear fitting based on 95% confidence intervals. In this case, the two methods give a broader range of different results than in the solar wind case, as also suggested by the different mean and median values reported in the small panels (for 0.2 and 1.0 au) in Figs. 7 and 8. Because of the larger number of observational studies investigating mean values rather than median values, in the following we focus on mean values only. A comparison with mean-based observational values (when available) is also provided in Table 2.

Fig. 7. Radial dependence of the mean CME parameters in EUHFORIA, extracted at virtual spacecraft located along the Sun–Earth line (red dots): (a) speed ⟨v⟩CME; (b) proton number density ⟨np⟩CME; (c) magnetic field ⟨B⟩CME; (d) proton temperature ⟨Tp⟩CME. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Observation-based relations are indicated as dashed green lines or green shaded areas. Observation-based values at selected heliocentric distances are indicated as green dots. Mean values from Wind are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

Fig. 8. Radial dependence of the mean CME parameters in EUHFORIA, extracted at virtual spacecraft located along the Sun–Earth line (red dots): (e) thermal pressure ⟨Pth⟩CME; (f) magnetic pressure ⟨Pmag⟩CME; (g) total pressure ⟨Ptot⟩CME; (h) proton β, ⟨βp⟩CME. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Observation-based relations are indicated in green. Mean values from Wind are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

Speed. The CME mean speed in EUHFORIA simulations is found to be mildly decreasing with heliocentric distance, with a slope of a_v = −0.08. This result is consistent with values reported by Liu et al. (2005), namely, −0.002 ± 0.02, and by Salman et al. (2020), namely, −0.305 ± 0.2. At 1 au, the CME mean speed from EUHFORIA (354 km s⁻¹) is lower than observed at Wind (488 km s⁻¹) by ∼135 km s⁻¹, and the ratio of modelled and observed values is equal to 0.73. Such underestimation of the CME impact speed at 1 au can be explained by the slower-than-observed (higher-than-observed) speed (density) in the modelled solar wind (as discussed in Sect. 3), which leads to a higher drag in our simulation than occurred in reality. Another possible concause to such a discrepancy is a slight underestimation in the CME initial speed based on coronal images.

Density. The CME mean proton density in EUHFORIA simulations decreases with heliocentric distance. The fall-off is well-described by a power law with exponent a_np = −2.38, which is comparable to the value of −2.32 ± 0.07 reported by Liu et al. (2005) based on a statistical study. At 1 au, the CME mean proton density modelled with EUHFORIA (2.37 cm⁻³) is similar to the one observed by Wind (1.90 cm⁻³), with the ratio of modelled and observed values equal to 1.25.

Magnetic field magnitude. The CME mean magnetic field magnitude in EUHFORIA simulations is found to decrease with heliocentric distance, with fall-off of a_B = −1.90. This value is well within the error bars of the values reported by Gulisano et al. (2010), namely, −1.89 ± 0.10 for perturbed CMEs, −1.85 ± 0.11 for non-perturbed CMEs, and −1.85 ± 0.07 for the full set. On the other hand, our value is higher than those reported in earlier studies by Liu et al. (2005), Wang et al. (2005), Leitner et al. (2007), who recovered exponents between −1.30 ± 0.09 (Leitner et al. 2007) and −1.52 (Wang et al. 2005). As pointed out by Gulisano et al. (2010), the discrepancy among different observational studies is most probably the result of different selection criteria used to identify the CME events, and of the different ranges of heliocentric distances considered in various studies. More recently, Winslow et al. (2015) reported an exponent of −1.95 ± 0.19 between Mercury and 1 au, while Janvier et al. (2019) and Salman et al. (2020) did not perform a fitting of the CME mean magnetic field at different heliocentric distances. However, as visible in panel c of Fig. 7, their values at selected heliocentric distances appear overall consistent with our modeling results and with previous observational studies. At 1 au, the mean CME magnetic field magnitude from EUHFORIA (10.0 nT) is underestimated with respect to the value observed at Wind (16.5 nT), and the ratio of modelled and observed values is ∼0.61. In this respect, we note that the CME internal magnetic field has traditionally been the most difficult parameter to reproduce in numerical simulations, because of the severe observational limitations affecting our knowledge of the global magnetic structure of CMEs and ICMEs (e.g., Owens 2016; Wood et al. 2017). The results shown in Fig. 4 are in line with this general trend and additionally represent a significant improvement compared to other modelling attempts for the same CME event (e.g., Shen et al. 2014; Singh et al. 2020). We refer to Paper I for a more detailed discussion on the observational limitations and uncertainties affecting the reproduction of the CME internal magnetic field.

Temperature. The CME mean temperature in EUHFORIA simulations decreases with heliocentric distance with exponent a_Tp = −1.19, which is significantly faster than reported by Liu et al. (2005), i.e. −0.32 ± 0.06. At 1 au, the CME mean proton temperature from EUHFORIA (8.70 × 10⁴ K) is higher than observed at Wind (5.44 × 10⁴ K), and the ratio of modelled and observed values is ∼1.60.

Pressures and proton β. The CME mean proton thermal pressure in EUHFORIA simulations decreases with heliocentric distance with a slope of a_Pp = −4.05, while CME mean magnetic pressure decreases with heliocentric distance with a slope of a_Pmag = −3.96. As a result, the mean proton β shows a mild increase with heliocentric distance, with a slope of a_βp = 0.11 (r_P = 0.62,   r_S = 0.74). We note that this behaviour might be a result of the particular threshold condition imposed to identify the magnetic ejecta in EUHFORIA time series, as also suggested by the histogram distributions in Fig. 8h, which show the median shifting towards lower values between 0.2 and 1.0 au, while the mean value remains approximately constant. Because of the lack of statistical studies investigating the radial evolution of the pressure terms and of the proton and plasma β in the case of real CMEs, we cannot directly compare the results from our simulations with observational values at other distances than 1 au. We note that at 1 au, the mean pressure terms inside the magnetic ejecta in EUHFORIA simulations are underestimated compared to the observed values at Wind. In particular, the ratio between modelled and observed values are: 0.92 for the mean proton thermal pressure, 0.34 for the mean magnetic pressure, and 0.36 for the mean total pressure. The resulting mean β_p ratio is 2.63.

4.3. Radial evolution of sheath and CME sizes

Similarly to Gulisano et al. (2010), we fit the temporal speed profile v_CME(t) in the magnetic ejecta in both EUHFORIA and Wind time series using a least square fit with a linear function of time, that is,

(4)

where t is the time since the start of the magnetic ejecta, and v_fit is the slope of the fitted linear function. The linear fit is used to define the following velocities: the velocity of the CME front, v_start = v_CME, fit(t_start), the velocity of the CME back, v_end = v_CME, fit(t_end), and the velocity of the CME centre, v_c = v_CME, fit(t_c), with t_c = (t_start + t_end)/2. As can be seen from Fig. 10, v_start, v_end, and v_c are generally close to the non-fitted velocities, v_CME(t_start), v_CME(t_end), and v_CME(t_c), although the exact differences depend on the particular heliocentric distance considered. We further define the full expansion velocity of the magnetic ejecta as Δv_CME ≃ v_CME, fit(t_start)−v_CME, fit(t_end) and the magnetic ejecta duration as Δt_CME = t_end − t_start. We also compute an estimate of the CME size as S_CME = v_c ⋅ Δt_CME (we note that this is likely a slight overestimation of the CME size due to CME ageing, further discussed in Sect. 5).

In the sheath region, we calculate the mean sheath velocity ⟨v⟩_sh and the duration of the sheath as Δt_sh = t_start − t_shock. These two quantities are used to compute an estimate of the sheath size, as S_sh = ⟨v⟩_sh ⋅ Δt_sh. Figure 9 shows the duration and size of the sheath and of the magnetic ejecta, as a function of the heliocentric distance D. Table 2 reports the radial dependencies (a_x parameter in Eqs. (1) and (2)) derived from the fitting of the individual values at different heliocentric distances using Eq. (1) in log–log space. A comparison with theoretical and observational values (when available) is provided.

Fig. 9. Durations and radial sizes of the CME and its driven sheath as a function of D, based on EUHFORIA simulations: (a) sheath duration; (b) sheath size; (c) ejecta duration determined starting from the condition βp ≤ 0.5; (d) ejecta size determined starting from the condition βp ≤ 0.5; (e) ejecta to sheath duration ratio; (f) ejecta to sheath size ratio. Panels a, c, e, and f: observational values at selected heliocentric distances are indicated as green dots, based on Masías-Meza et al. (2016) and Janvier et al. (2019). Panel b: the sheath size expected from Eq. (8) in Lee et al. (2017) is indicated as a magenta dashed line. Panel d: the range of values recovered from observation-based relations are indicated as a green shaded area. The values recovered from Al-Haddad et al. (2019) based on numerical simulations are indicated as dashed and dotted-dashed magenta lines. Observation-based values in the solar corona reconstructed in Paper I are shown as magenta diamonds. The position of the EUHFORIA inner boundary is indicated as a dashed grey line. Values observed at Wind are shown as blue diamonds.

CME duration and size. The linear fitting of the CME duration and size in log–log scale from EUHFORIA simulation provide slopes of a_ΔtCME = 0.77 and a_SCME = 0.69. For comparison, fitting the values reported by Janvier et al. (2019) at selected planetary locations, we recover a slope for the evolution of the CME duration with heliocentric distance equal to 1.08; slopes for the evolution of the CME size with heliocentric distance reported in previous studies range between 0.61 (Wang et al. 2005) and 0.92 (Liu et al. 2005). Distinguishing between perturbed and non-perturbed CMEs, Gulisano et al. (2010) reported a size slope of 0.89 ± 0.15 for non-perturbed magnetic ejecta, and a slope of 0.45 ± 0.16 for perturbed magnetic ejecta, with a value of 0.78 ± 0.12 for the full set of magnetic ejecta (perturbed and non-perturbed) considered (see Eq. (3) therein). In this regard, EUHFORIA results fall between the two categories, that is, the lower limit of non-perturbed ones, and the upper limit of perturbed ones. The growth of the CME size in time in our simulation appears overall consistent with typical observations.

However, we note that in the EUHFORIA simulation, the magnetic ejecta results significantly more extended both in time and in the radial direction than typically reported from in situ observations and from self-consistent numerical simulations of flux-rope CMEs in the corona and heliosphere (Al-Haddad et al. 2019). This can be seen by comparing the intercept of the linear fitting S_CME, 0 with that reported by previous studies. In EUHFORIA, we obtain S_CME, 0 = 0.82 au and a CME duration of Δt_CME, 0 = 4.00 days. For comparison, the values reported in literature range between 0.16 au (Gulisano et al. 2010, perturbed CMEs) and 0.32 au (Gulisano et al. 2010, non-perturbed CMEs), that is, the CME size in EUHFORIA is between 2.56 and 5.13 times larger than typically observed. Janvier et al. (2019) reported a typical CME duration of 0.8 days at 1 au, meaning that the duration of the CME in EUHFORIA is about five times longer than typically observed. The overestimation in the CME size in EUHFORIA is also apparent when we compare the modelled values with in situ observations at Wind (1 au) for the specific CME considered: in fact, the CME duration in EUHFORIA is 2.1 times longer than observed, while the CME size is about 1.5 times larger.

Sheath duration and size. The linear fitting of the sheath duration and size in log–log scale from EUHFORIA simulation provide slopes of a_Δtsh = 2.03 and a_Ssh = 1.88. For comparison, fitting the values provided by Janvier et al. (2019) at selected planetary locations, we recover a slope of 1.77 for the growth of the CME-driven sheath duration. We also calculate the upper limit of the expected sheath size as predicted by Eq. (8) in Lee et al. (2017), with θ_CME = 0°, γ = 1.5, R_CME = D tan(ω_CME/2) as in Paper I, and the Alvfén Mach number M_A calculated similarly to Scolini et al. (2020a). This relation predicted a slope of 0.99. The sheath size in EUHFORIA simulations therefore grows faster than expected, although values near 1 au appear consistent with theoretical expectations. On the other hand, the growth in the duration Δt_sh (a_Δtsh = 2.03) matches the observational trend well, making us confident of the performance of the model in reproducing this particular aspect of CME evolution (as the duration of a structure from in situ time series provides a more direct comparison to observations than its inferred size).

In the EUHFORIA simulation, the sheath also results significantly more extended in the radial direction than observed in situ. This can be seen by comparing the intercept of the linear fitting S_sh, 0 with that reported by previous studies. In EUHFORIA, we obtain S_sh, 0 = 0.27 au. The sheath duration is Δt_sh, 0 = 0.88 days. For comparison, Janvier et al. (2019) reported a typical sheath duration of 0.5 days at 1 au, that is, the duration of the sheath size in EUHFORIA is about 1.75 times larger than typically observed. This result is also confirmed for the specific CME considered, with the sheath duration in EUHFORIA that is about 1.9 times longer than observed at Wind, while the sheath size is about 1.4 times larger.

Duration and size ratio of CME substructures. We find that the ejecta to sheath duration and size ratios both decrease with heliocentric distance (with fall offs of a_RΔt = −1.25 and a_RS = −1.19), reflecting the different rates of duration and size increase in the sheath and magnetic ejecta, with the sheath growing faster than the ejecta. This result is consistent with those reported by Janvier et al. (2019) based on actual observations of interplanetary CMEs between Mercury and 1 au. The nature of the size increase is interpreted in different ways for the sheath and the magnetic ejecta. While both structures expand in the solar wind while propagating away from the Sun, the faster increase in the sheath size is interpreted as the result of the additional solar wind piling up via a ‘snow plow’ effect. The ultimate thickness of the sheath therefore depends on the amount of plasma and magnetic field that is accumulated at the front of the CME into the sheath and on the amount of it that is able to escape towards the CME sides, in addition to the aforementioned intrinsic expansion (Siscoe & Odstrcil 2008). Additional magnetic reconnection phenomena leading to erosion or flux injection effects (Démoulin et al. 2016) are also contributing to a reduction or increase in the size of both the sheath and the magnetic ejecta, although to a lesser extent. Overall, it seems more plausible for the increase of the sheath duration (compared with that of the magnetic ejecta) to be due to a pile-up of material combined with sheath expansion, rather than to magnetic reconnection phenomena.

4.4. Radial evolution of CME expansion

One key parameter providing insights into the radial evolution of interplanetary CMEs is the non-dimensional CME expansion rate, defined by Démoulin et al. (2008) as:

(5)

We compute ζ_CME at different heliocentric distances in EUHFORIA and compare it with the value estimated from Wind observations at 1 au. This dimensionless parameter takes into consideration that faster (higher v_c) and longer (higher Δt_CME) CMEs have higher expansion speeds (higher Δv_CME), while the normalisation by the heliocentric distance considered (D) allows us to compare observations at different distances from the Sun. From observations, this parameter is found to range between 0.5 and 1.5 for most non-perturbed CMEs (with a median value around 0.8; see Démoulin et al. 2008; Démoulin 2010; Gulisano et al. 2010) and between −1 and 2 for perturbed CMEs (Gulisano et al. 2010). In EUHFORIA, we find that ζ_CME varies between 0.63 and 1.09 depending on the heliocentric distance considered (Fig. 10d). From Wind observations, we find ζ_CME = 0.82 by fitting the observed speed profile with a linear fit throughout the whole magnetic ejecta period as indicated in Fig. 1. Expected, observed, and modelled values are overall consistent among each other. The implications of the recovered expansion rates for the global evolution of the CME under study are further discussed in Sect. 5.

Fig. 10. Non-dimensional CME expansion rate evolution based on EUHFORIA simulations. Left column: derivation of the non-dimensional CME expansion rate ζ from the linear fitting of the CME speed profile in EUHFORIA simulations at D = 0.2 au (a), D = 1.0 au (b), and D = 1.8 au (c). Right column: non-dimensional CME expansion rate ζCME as a function of D (d), vc, CME (e), and Δvx, CME (f). Values observed at Wind are shown as blue diamonds. The green lines represent the fit for perturbed CMEs (dashed line), non-perturbed CME (thin continuous line) and for both sets of CMEs (thick continuous line) as reported by Gulisano et al. (2010).

Correlations. Theoretically, ζ_CME is expected to be independent from Δt_CME, v_c, and D in the case of non-perturbed CMEs. In EUHFORIA simulations, we observe this is not the case (as reported in Fig. 10). In fact, the linear fitting of ζ_CME as a function of the heliocentric distance provide correlation coefficients of r_P = −0.78 and r_S = −0.94, indicating the two are strongly (anti-)correlated. For comparison, Gulisano et al. (2010) reported a very weak correlation for non-perturbed magnetic ejecta, and a modest correlation (r_P = 0.49) for perturbed ones. The fitting of ζ_CME as a function of v_c provides r_P = 0.94 and r_S = 0.93, indicating a strong correlation of the two parameters. On the other hand, Gulisano et al. (2010) reported a weak correlation (r_P ≤ 0.3) regardless of the magnetic ejecta type considered. Finally, fitting ζ_CME as a function of Δv_CME we report correlation coefficients of r_P = 0.98 and r_S = 0.94, meaning that in EUHFORIA, the two parameters are very strongly correlated. For comparison, Gulisano et al. (2010) reported a strong correlation coefficient (r_P = 0.79) in the case of perturbed magnetic ejecta only. This analysis suggests that the propagation behaviour of the CME in EUHFORIA simulations resembles that of perturbed magnetic ejecta. This is also confirmed by the left column panels in Fig. 10 (panels a–c), where we observe that in EUHFORIA, the CME speed profile is characterised by an irregular speed decrease that is significantly deviating from a monotonic decrease, as observed in the case of perturbed CMEs. This behaviour is observed at all the heliocentric distances considered, with more perturbed conditions closer to the Sun, where the fitting of the CME speed time profile v_CME(t) with a straight line gives a minimum Pearson correlation coefficient of r_P = −0.86. The physical origin of this perturbation in EUHFORIA simulations is investigated in the following paragraphs and Sect. 5.

Comparison of expansion and translation speeds. To further explore the nature of the CME, we compute at all heliocentric distances the relative importance of its expansion speed, calculated as v_exp = Δv_CME/2, and of its translation speed v_trans ≃ v_c. Figure 11 summarises the results. We observe that closer to the Sun, the ratio of the expansion and translation speeds decreases faster than farther away. In particular, close to the model inner boundary (i.e. at D = 0.2 au), the expansion speed accounts for 67% of the translation speed, dropping by almost 50% between 0.2 and 0.7 au, where it reaches a value of 35%. Between 0.7 au and D = 1.9 au the decrease is more modest, reaching 20% at the outer distance considered. At 1 au the ratio drops to 28% (100 km s⁻¹ vs 354 km s⁻¹). For comparison, the 3D reconstruction of the CME kinematics performed in Paper I estimated the expansion speed being 66% of the translation speed (503 km s⁻¹ vs 763 km s⁻¹) at a height of about 0.07 au (∼14.9 solar radii), which is similar to the value extracted from EUHFORIA simulations at 0.2 au. At 1 au, Wind data indicate the expansion speed accounted for ∼22% of the translation speed at 1 au (110 km s⁻¹ vs. 488 km s⁻¹), that is, a fraction of only 8 percentage points lower than estimated with EUHFORIA. These numbers are also consistent with the statistical observational picture, with previous studies reporting that the expansion speed is typically ∼57% of the translation speed of a CME in the corona (below 0.1 au; Dal Lago et al. 2003; Schwenn et al. 2005; Gopalswamy et al. 2009b). At 1 au, the typical fraction drops below 20% (Lugaz et al. 2020). Simulations and observations therefore agree in supporting a picture where the contribution of the expansion speed reduces as the CME approaches a more relaxed state towards the equilibrium with the surrounding solar wind, as further discussed in Sect. 4.5.

Fig. 11. Ratio between the expansion speed vexp and the translation speed vtrans of the CME, as a function of the heliocentric distances. The reconstructed value in the corona (at 14.9 solar radii, corresponding to 0.07 au) from Paper I is indicated as a magenta diamond. The observed value at Wind is indicated as a blue diamond. Reference values measured in the corona (< 0.1 au, from Gopalswamy et al. 2009a) and at 1 au (from Lugaz et al. 2020) are indicated as green and orange dashed lines, respectively. The location of the EUHFORIA heliospheric inner boundary is indicated by the dashed vertical grey line.

4.5. Radial evolution of CME magnetic field profile

Recent works by Masías-Meza et al. (2016), Janvier et al. (2019), and Démoulin et al. (2020) highlighted the close relation between the asymmetry in the magnetic field time profile within a magnetic ejecta, and the spatial asymmetries of the magnetic structure possibly related to its level of relaxation. In this work, we investigate this relation in EUHFORIA simulations and we quantify the asymmetry in terms of two parameters: the time shift of the magnetic field peak compared to the centre of the magnetic ejecta (Δt_max), and the asymmetry parameter (Δt_asym), corresponding to the first moment of the magnetic field within the magnetic ejecta, introduced by Janvier et al. (2019).

Firstly, we define the time shift of the peak of the magnetic field with respect to the centre of the magnetic ejecta as:

(6)

In order to account for the progressive expansion of the CME structure and to compare the results at different heliocentric distances, we further normalise Δt_max with respect to the magnetic ejecta duration as:

(7)

Secondly, we compute the asymmetry parameter in the same way as Janvier et al. (2019) and Lanabere et al. (2020), namely:

(8)

which we compute from discrete EUHFORIA time series as:

(9)

In order to account for the progressive expansion of the CME structure during propagation and to compare the results at different heliocentric distances, we further normalise Δt_asym with respect to the magnetic ejecta duration as:

(10)

We note that Δt_asym, norm is a more robust indicator than Δt_max, norm due to the fact that it considers the whole magnetic field profile (through the integral over time), while Δt_max, norm only considers the single point associated with the peak of the magnetic field profile.

Figure 12 summarises the main characteristics of the CME magnetic field profile at different distances. To better compare between the time profiles at different heliocentric distances, in the top panel of Fig. 12 we also plot the magnetic field profile normalised over the range of magnetic field values observed within the CME at different heliocentric distances, calculated as:

(11)

Fig. 12. CME magnetic field profile evolution as a function of D, based on EUHFORIA simulations. Top panel: CME magnetic field profile in EUHFORIA at various heliocentric distances, expressed in terms of the normalised magnetic field Bnorm as a function of the normalised time tnorm. The normalised magnetic field profile at Wind is indicated by the dashed-dotted blue line. Middle panel: normalised time shift between the magnetic field peak and the centre of the magnetic ejecta ΔtBpeak, norm, as a function of the heliocentric distances. The observed value at Wind is indicated as a blue diamond. Bottom panel: normalised asymmetry parameter Δtasym, norm for the magnetic field profile, as a function of the heliocentric distances. The green shaded area indicates the range of asymmetry of the mean CME profiles at different heliocentric distances, based on the results by Janvier et al. (2019). The observed value at Wind is indicated as a blue diamond.

The temporal axis shows the time since the start of the magnetic ejecta, normalised over its duration at different heliocentric distances:

(12)

In the middle panel of Fig. 12, we plot the asymmetry in the magnetic field profile using as proxy the normalised shift of the magnetic field peak compared to the centre of the magnetic ejecta at different heliocentric distances. In the bottom panel, we plot the normalised asymmetry parameter in function of the heliocentric distance.

Results from both metrics show that the CME magnetic field profile is more asymmetric closer to the Sun than at higher heliocentric distances. Moreover, EUHFORIA simulations also show that the asymmetry indicated by the time shift of the magnetic field peak at D = 0.2 au is initially lower, it increases up to D = 0.3 au and then it is progressively reduced. On the other hand, the magnitude of the asymmetry parameter follows a monotonically decreasing trend with heliocentric distance. Beyond D ≃ 0.7 au, both metrics indicate that the magnetic field asymmetry changes less rapidly than closer to the Sun. Comparing EUHFORIA results at 1 au with observed values at Wind, we report a ratio of modelled (−0.36) and observed (−0.44) values equal to ∼0.82 for the normalised time shift (corresponding to an underestimation of the asymmetry in EUHFORIA compared to observed values). The ratio of modelled (−0.12) and observed (−0.10) values is ∼1.20 for the normalised asymmetry parameter (corresponding to an overestimation of the asymmetry in EUHFORIA compared to observations). We note that the two metrics considered here provide two (complementary) ways of looking at the magnetic field time profile asymmetry of CMEs, and that the overestimation or underestimation provided should not be considered as particularly informative as it mostly relates to the specific definition of the parameter or metric considered. Rather, it is preferable to focus on the overall trend with heliocentric distance, which in both cases tends towards lower asymmetry values at high distances from the Sun. A decrease in the asymmetry parameter with heliocentric distance has been previously observed in situ and interpreted as due to the relaxation of a CME magnetic structure tending towards a state of equilibrium with the surrounding solar wind (Janvier et al. 2019). While a monotonic trend in the asymmetry parameter is typically observed, the non-monotonic behaviour of the time shift of the magnetic field peak as a function of the heliocentric distance recovered from EUHFORIA simulations is intriguing and we further discuss its nature and origin in Sect. 5.

4.6. Radial evolution of CME wake

Previous studies have assessed that perturbed conditions in the solar wind after the passage of a CME often have a long recovery period that can sometimes exceed the duration of the CME disturbance itself (Liu et al. 2014; Temmer et al. 2017). Observations at 1 au and at inner heliocentric distances have highlighted that such a phenomenon, referred to as ‘solar wind pre-conditioning’, can last up to five days after the end of the CME observed in situ, having important implications for CME propagation models and space weather forecasting algorithms, as the propagation of a CME precursor event might significantly affect the propagation of a following CME even when the two are launched several days apart. However, the actual duration of this pre-conditioning in EUHFORIA numerical simulations has never been assessed prior to this study.

In this work, we quantify the duration of the solar wind CME-induced perturbation after the passage of the magnetic ejecta, which we refer to as ‘CME wake’, based on a 5% threshold condition, namely, by applying the following threshold condition to EUHFORIA time series at different heliocentric distances:

(13)

where t_i is a generic time in the EUHFORIA time series, X and X_sw are time profiles of a generic MHD variable obtained from simulations of the CME and ambient solar wind with EUHFORIA, and |X(t_i)−X_sw(t_i)|_max corresponds to the maximum (peak) variation recorded in background-subtracted time series of that same variable. At each spacecraft, the end of the CME wake relative to a given variable (t_X, wake) is marked based on the first time t_i after the end of the magnetic ejecta (t_end) at which Eq. (13) is not satisfied anymore in EUHFORIA speed, proton number density, magnetic field magnitude, and proton temperature time series. The duration of the CME wake after the end of the magnetic ejecta is further computed as Δt_X, wake = t_X, wake − t_end. Figure A.3 shows the result of this analysis at selected heliocentric distances. We note that the search for periods of perturbed solar wind speed and magnetic field conditions after the passage of a magnetic ejecta as indicator of the CME wake is consistent with the typical wake conditions reported by Temmer et al. (2017) and Janvier et al. (2019). On the other hand, to the best of our knowledge, no previous study has ever quantified the effect of CME pre-conditioning to the solar wind with respect to density and temperature.

Based on Fig. A.3, we note that all the variables except the temperature present a similar duration of the perturbation after the end of the CME passage. The temperature, on the other hand, appears to be restored to pre-CME values already within the magnetic ejecta. At a first consideration, this might indicate that taking β_p as only parameter to distinguish the CME duration in EUHFORIA time series might not be sufficient, and it may lead to a misinterpretation of the location of the CME rear edge that might also explain the longer ejecta duration and radial size reported in Fig. 9. If, in fact, the CME wake as determined from the temperature profile were to be considered as marking the real end of the magnetic ejecta, the estimated ejecta size would be reduced by a factor of ∼2 compared to current estimated in Fig. 9, resulting in values more similar to observations at Wind. While it should be noted that the identification of proxies to identify the exact duration of the CME wake is non-trivial and it most probably requires further verification via an in-depth investigation of the 3D magnetic structure of the CME, there are two main reasons that make us consider the temperature proxy as not representative of the actual CME wake duration: (1) in EUHFORIA, the choice of both the solar wind and CME initial temperatures at 0.1 au (i.e. the model inner boundary) relies on severe assumptions that are likely to significantly affect the modelling of the plasma temperature throughout the whole modelling domain; (2) the derivation of the temperatures of the various particle populations in the solar wind based on in situ measurements are subject to higher uncertainties than the other plasma and magnetic field parameters; as such, temperature estimations are not considered very reliable even when it comes to solar wind in situ measurements. While leaving future investigations on this matter for future studies, here we point out that because of the time consistency between low β_p signatures and other typical signatures indicating the passage of a magnetic ejecta (e.g., high speed, intense magnetic field, low densities) in the in situ time series, and of the lack of any evidence supporting a more reliable modelling of the solar wind temperature compared to other parameters, we are inclined to believe that the β_p condition do provide a reliable identification of the CME rear edge after all.

Figure 13 shows the duration of the CME wake and its ratio with respect to the duration of the magnetic ejecta, as a function of the heliocentric distance, for the different variables considered (except for the temperature). We observe that the CME wake at all heliocentric distances and for all the variables considered never exceeds the duration of one day after t_end (maximum of Δt_wake, np = 0.85 reached at 1.7 au). This duration is significantly lower than reported by previous studies (typically from two to five days, see e.g., Temmer et al. 2017), and it might be due to the specific characteristics of the CME considered. However, other possible explanations for the short duration of CME pre-conditioning detected in EUHFORIA simulations might be the non-ideal MHD nature of turbulence developing in the CME flank or wake region in the solar corona and interplanetary space as well as its short scales (e.g., Fan et al. 2018), which require higher spatial and temporal resolutions than the ones used in our simulation to be resolved. The analysis at different heliocentric distances also shows that the duration of the solar wind preconditioning grows as a function of D up to about 1.7 au, when it reaches a peak before starting to decrease. Modelling results also show that the duration of the CME wake remains significantly smaller than that of the ejecta, with their ratio never exceeding the value of 20% of the duration of the magnetic ejecta (maximum of Δt_wake, np/Δt_CME = 16% reached at 1.3 au). Their ratio remains almost constant between 0.3 and 1.3 au, meaning at these range of distances the two structures grow with a similar behaviour. After 1.3 au, simulations show the ratio decreasing because of the stable duration of the wake with respect to a progressive growth of the CME radial size. Within 0.3 au, we observe a much shorter duration of the CME wake, most probably due to a combination of poor detection performances of our algorithm, and modelling artifacts due to the vicinity to the model inner boundary.

Fig. 13. Duration of CME wake as a function of D, based on EUHFORIA simulations. Left: duration of the CME wake since the end of the magnetic ejecta for the magnetic field, density, and temperature, as a function of the heliocentric distance D (indicated by the solid dots). Right: ratio between the duration of the CME wake and of the magnetic ejecta as a function of the heliocentric distance, D, for the same quantities (indicated by the solid lines).

5. Discussion and physical interpretation

5.1. Evolution of the CME expansion rate

Figure 10d, showing higher-than-typical CME expansion rates close to the Sun, indicates that the magnetic ejecta undergoes a phase of rapid expansion in innermost region of the heliosphere (within 0.4 au from the Sun) that cannot be explained in terms of the sole balancing of the decreasing pressure in the surrounding solar wind. In particular, Démoulin & Dasso (2009a) showed that for a CME isotropic expansion controlled by the equilibrium with the surrounding solar wind, the expansion rate ζ_CME should be independent on the heliocentric distance and on the size of the ejecta. For such an expansion and assuming conservation of the magnetic flux, the total pressure of the ambient solar wind scales as ∝D^−a_Ptot, where the exponent a_Ptot relates to the CME expansion rate as a_Ptot ≈ −4ζ_CME (Démoulin & Dasso 2009a). Under these conditions, the mean magnetic field inside the magnetic ejecta should fall off as ∝D^a_Ptot/2, while the ejecta size should scale proportionally to ∝D^−a_Ptot/4. From Fig. 6g, we recover that a_Ptot ∼ −2.78 and, therefore, theoretical arguments would anticipate the ejecta size growing as S_CME ∝ D^0.70. Overall, this trend is well reproduced in our simulations, from which we find the CME size growing as S_CME ∝ D^0.69 (Fig. 9d), on average, throughout the whole heliospheric domain. On the other hand, the modelled mean magnetic field inside the ejecta falls off slightly faster than expected, that is, ⟨B⟩_CME ∝ D^−1.90 (from Fig. 7c), while the expected trend is ⟨B⟩_CME ∝ D^−1.39. When fitting heliocentric distances where strong and weak ζ_CME dependencies are observed separately, we recover a_Ptot ∼ −3.34, S_CME ∝ D^0.78, and ⟨B⟩_CME ∝ D^−1.82 between 0.2 and 0.4 au; and a_Ptot ∼ −2.50, S_CME ∝ D^0.61, and ⟨B⟩_CME ∝ D^−1.91 from 0.5 to 1.9 au. Within 0.4 au, theoretical arguments would expect the ejecta size to grow as S_CME ∝ D^0.83, while beyond 0.5 au the expected trend is S_CME ∝ D^0.63, i.e. both are very close to the modelled behaviour. In both cases, the mean magnetic field inside the ejecta decays faster than expected, as theoretical expectations would suggest that ⟨B⟩_CME ∝ D^−1.67 and ⟨B⟩_CME ∝ D^−1.25, respectively. We note, however, that such discrepancies for the scaling of the mean magnetic field are within observational uncertainties, that is, they are within the observed ranges reported in previous studies (e.g., Liu et al. 2005; Wang et al. 2005; Leitner et al. 2007; Gulisano et al. 2010; Winslow et al. 2015). Possible contributions to the rapid magnetic field decay with heliocentric distance (particularly between 0.5 and 1.9 au, where we have ruled out a significant contribution of magnetic erosion and over-expansion) include: a non-isotropic expansion of the CME, possibly related to a non-isotropic solar wind pressure, and numerical dissipation. Given our focus on the CME propagation along Sun-to-Earth radial direction only, we leave the investigation of the aforementioned phenomena for future studies.

Quantitatively, the ratio of the expansion and translation speeds of the CME appears consistent with observation-based estimates in the solar corona (in Paper I) and at 1 au (in this work) carried out for the specific event considered, and it is also in line with results obtained in previous works (e.g., Gopalswamy et al. 2009a; Lugaz et al. 2020).

By filling the observational gaps between 0.1 au and 1 au for the specific event considered, EUHFORIA simulations allow us to identify two regimes of CME expansion: a first phase between 0.1 and 0.4 au (i.e. around the orbit of Mercury), characterised by a rapid expansion of the ejecta structure in response to its continuous tendency to establish a pressure balance with the surrounding, expanding solar wind (controlled primarily by the interplay between the magnetic pressure inside the ejecta, and the thermal pressure in the surrounding wind; see Paper I for a full discussion on this subject). In a second phase of expansion, between 0.4 au and 1.9 au, the magnetic ejecta expands more gradually in response to the slowly-decaying pressure in the surrounding medium (Démoulin & Dasso 2009a). Overall, the radial dependence of ζ_CME in Fig. 10d appears therefore primarily linked to a non-exponential decay of the solar wind pressure (i.e. the radial dependence of a_Ptot), rather than to sources of expansion internal to the CME (consistent with what theorised by Démoulin & Dasso 2009a). This also implies that erosion or flux injection phenomena do not alter the expansion of the CME size in a significant way. As the efficiency of these phenomena in triggering magnetic reconnection at the solar wind–CME boundary primarily depends on the relative orientation of the magnetic fields therein, we note that this condition might be related to the magnetic configuration of the specific event considered. Further studies are therefore needed to generalise the picture to other magnetic configurations.

Additionally, by comparing the CME size extracted from EUHFORIA time series at 0.2 au with the reconstructed CME size in the solar corona at 0.07 and 0.1 au reported in Paper I, we observe that the size growth between 0.1 au and 0.2 au is consistent with a growth behaviour as S ∝ D^ζ_CME. We conclude that the overestimation in the CME radial size at 1 au is primarily the result of an overestimation in the CME radial size already in the range 0.7−0.1 au, and can therefore be traced back to the limited flexibility of the spheromak model in reproducing the plethora of evolutionary deformations observed to affect CME geometries in the corona and heliosphere. In particular, as a spherical structure the spheromak model in EUHFORIA relies on the assumption that during insertion in the heliospheric domain (i.e. at 0.1 au) the width of the CME is the same in all directions, (i.e. the structure is characterised by a circular cross section in all directions), therefore completely neglecting pancaking effects occurring before 0.1 au (e.g., Savani et al. 2011; Isavnin 2016). As such, the modelled expansion of the CME between 0.1 au and 1.9 au is entirely physical and is not undermined by any modelling artifact related to the insertion of the CME structure at 0.1 au. Although, in principle, the exact identification the ejecta boundaries is a delicate task both in terms of its observations and modelling, the fact that the initial CME radial size at insertion in the simulation domain is in line with further in situ estimates at virtual spacecraft suggests this most likely did not affect our estimation of the ejecta size in EUHFORIA simulations in a significant way.

Overall, the dependence of the CME expansion rate on the heliocentric distance suggests that this was a perturbed CME event. However, this appears to be limited to distances ≤0.4 au, and relates to a rapid expansion of the CME structure close to the model inner boundary. In our simulations, this is not related to the presence of any high speed solar wind stream compressing the magnetic ejecta from the back, although a high speed stream was present in reality as visible from Wind observations. This is a known limitation of the solar wind modelling in EUHFORIA that has been recently addressed by a number of publications and that is still currently under investigation (Asvestari et al. 2019; Hinterreiter et al. 2019; Samara et al. 2021).

5.2. Evolution of the magnetic field profile

The magnetic field inside CMEs is one of the most important factors affecting both the large-scale evolution of CME structures in interplanetary space, and their potential as space weather drivers (e.g., Gosling et al. 1991; Kilpua et al. 2017). Understanding its radial evolution and the factors affecting its in situ observations is therefore extremely important to better assess the potential space weather impact of a CME at a target location. At a given spacecraft, the asymmetry in the observed magnetic field profile within magnetic ejecta can be interpreted as the sum of two contributions: the expansion of the CME as it crosses the spacecraft (the so-called ‘ageing effect’; Démoulin et al. 2008, 2018), and the non-circular cross section of the magnetic ejecta itself (Démoulin & Dasso 2009b). Previous studies agree in their consideration of the latter as typically dominant at 1 au. However, the ageing effect has been found to be the main source of magnetic asymmetry for a significant minority of magnetic ejecta at 1 au, making the consideration and correction for its effects worthwhile, in particular, in the case of large events (Démoulin et al. 2020). Furthermore, we note that the effect of ageing at different heliocentric distances than 1 au, particularly those closer to the Sun, cannot be discarded a priori when interpreting the results obtained from EUHFORIA simulations.

The ageing effect is naturally linked to the ζ_CME parameter, which provides an estimate for the expansion rate of a magnetic ejecta regardless of its size, speed, or distance from the Sun. A number of previous studies estimated that for ζ_CME ∈ (0.6, 1.0), that is, the typical values at 1 au, the effect of ageing alone could not explain the asymmetry observed, which would typically require a much higher expansion rates (e.g., Démoulin et al. 2008, 2018). However, it should be stressed that more recent estimates considering magnetic ejecta with expansion rates ζ_CME ∈ (0.8, 1.2) reported the ageing effect as the main source of the observed magnetic asymmetry for more than one out of every four events (Démoulin et al. 2020).

From EUHFORIA simulations, we report values of ζ_CME ranging between 1.09 and 0.63 at the different heliocentric distances considered (Sect. 4.5). Such numbers are comparable with typical observational values at 1 au, which suggests the asymmetries in the CME magnetic field profile are primarily due to actual cross-section asymmetries, although ageing may still provide a significant contribution leading to a mix of the two effects (Démoulin et al. 2020). For the particular CME considered, and based on our simulations, we consider ageing to possibly be most relevant for heliocentric distances closer than 0.5 au, where ζ_CME was higher than 0.8, while at distances larger than 0.5 au the lower expansion rates (ζ_CME < 0.8) suggest cross-sectional asymmetries as the dominant source of magnetic asymmetry. EUHFORIA simulations therefore confirm previous estimates obtained from CME observations between the orbit of Mercury and 1 au (Janvier et al. 2019). Future investigations on the contribution of CME ageing to in situ observations within 0.5 au, including a comparison of simulation results with observations carried out by PSP in the range 0.1−0.3 au, are needed to corroborate EUHFORIA results at such small heliocentric distances.

The discussion above suggests that the asymmetry observed in situ is the manifestation of an actual asymmetry in the ejecta cross section. Overall, we find that the asymmetry is less pronounced at higher heliocentric distances than it is closer to the Sun. Therefore, we conclude that the results displayed in Fig. 12, which are qualitatively consistent with those presented in Fig. 3 in Janvier et al. (2019), should be interpreted as evidence of the progressive relaxation of the ejecta magnetic structure during propagation in interplanetary space. The fact that the time shift of the magnetic field peak does not follow the expected monotonic trend close to the Sun (within 0.7 au) might be due to a modelling artifact due to the fact that at the insertion into the heliospheric domain (at D = 0.1 au), the spheromak magnetic structure is defined as symmetric between its front and its back and it also lacks a pre-existing shock and sheath ahead. Only at D > 0.1 au (i.e. after insertion in the heliospheric domain), its cross-section starts becoming deformed as a consequence of the interaction with the surrounding solar wind, primarily in response to the formation of a sheath which develops conditions at the front and back of the ejecta that are different from one another. Results in Fig. 12b suggest that this phase of rapid sheath and asymmetry formation is fully complete between 0.3 and 0.4 au, and that it is only at higher heliocentric distances that the spheromak magnetic structure starts actually evolving towards a more relaxed configuration and thus exhibiting a trend consistent with observations. The asymmetry parameter appears to be a more stable metric that is not very sensitive to such a modelling artifact and it is found to better describe the evolutionary relaxation of the ejecta at the various heliocentric distance (Janvier et al. 2019).

6. Conclusions

In this work, as a case study, we investigated the interplanetary propagation of a CME. We modelled its radial evolution analysing the evolution of its mean properties, magnetic field profile, and pre-conditioning of the ambient solar wind after its passage. Aiming to bridge the gap between observational and modelling studies, we focused on these specific properties as they are the most often analysed in statistical studies of interplanetary CMEs. The CME under study was a fast halo CME directed towards the Earth observed on 12 July 2012 and previously analysed by the authors in Paper I. The double goal of this work was to validate numerical models of CME propagation, particularly the EUHFORIA model (Pomoell & Poedts 2018), which describes the CME internal magnetic field as a linear force-free spheromak configuration (Verbeke et al. 2019). We also aimed to perform a comprehensive analysis of the radial evolution of interplanetary CMEs, filling in observational gaps and providing a new context for observational studies of CME propagation in the inner heliosphere. The main results are as follows:

1. We find that the radial dependences of the mean solar wind and CME properties modelled in EUHFORIA are both consistent with observational and theoretical expectations based on statistical sets of solar wind and CME data.

2. The investigation of the CME expansion rate at different heliocentric distances indicated that the CME under study underwent a phase of rapid expansion up to ∼0.4 au, whereas farther out, the expansion rate was moderate and almost independent from the heliocentric distance. At all distances, the CME expansion was consistent with a trend controlled by the pressure balance with the surrounding solar wind (Démoulin & Dasso 2009a). The ratio between the expansion and translation speeds extracted close to the model inner boundary and at 1 au was also consistent with observational values in the corona and at Earth’s orbit.

3. The early rapid expansion alone was not sufficient to explain the overestimation in the CME radial size in EUHFORIA simulations, suggesting that the overestimation of the CME radial size by the spheromak model is most probably caused by an intrinsic limitation of the specific CME model to account for deformations of the CME structure such as pancaking (Savani et al. 2011; Isavnin 2016) or other possible asymmetric shapes (Démoulin & Dasso 2009b).

4. The analysis of the CME expansion rate extracted from EUHFORIA simulations also suggested that this was a perturbed CME event. However, the perturbation appears to be limited to distances ≤0.4 au and linked to a rapid expansion of the CME structure close to the model inner boundary. It is important to remark that the EUHFORIA simulations exhibited no evidence of a solar wind high speed stream compressing the magnetic ejecta from the back, although a high speed stream was present in Wind observations. The poor reproduction of solar wind high speed streams in EUHFORIA is a known limitation that was recently investigated in a number of studies (Asvestari et al. 2019; Hinterreiter et al. 2019; Samara et al. 2021).

5. The asymmetry in the magnetic field profile indicated a progressive relaxation of the magnetic ejecta structure during propagation. The analysis of the time shift of the magnetic peak with respect to the centre of the magnetic ejecta allowed to estimate the effects of the insertion through the inner boundary to last up to 0.4 au, that is, slightly longer than estimated by Scolini et al. (2020b) for simulations employing the cone CME model.

6. Moreover, the expansion rate ζ_CME ≤ 1.1 suggested that CME ageing is not likely to provide a significant contribution to the asymmetry in the magnetic field profile at any heliocentric distance sampled; this may be the case beyond 0.5 au in particular, where the expansion rate is found to be lower than 0.8 (Démoulin et al. 2020).

7. We report that the modelled duration of the CME wake is underestimated compared to typical observations (Temmer et al. 2017) and that this may be partially due to limitations of the ideal MHD and large-scale approach used in treating the formation of the CME wake. We argue that a better description of waves and turbulence might be required to realistically reproduce this feature in numerical simulations.

We conclude that EUHFORIA combined with the spheromak model are collectively capable of providing a physically consistent description of the radial evolution of the solar wind and CMEs throughout the inner heliosphere, at least in the proximity of the ecliptic plane and inside the CME and in its nearby environment. However, our study also highlights some intrinsic limitations of the spheromak model in the reproduction of the CME radial size, which will have to be improved in a refined version of EUHFORIA in order to achieve more accurate descriptions of the interplanetary propagation of CMEs. In this respect, it is also important to mention that modelling results depend on the initial configuration and the exact model used for the CME initiation, as well as for the ambient solar wind (e.g., Al-Haddad et al. 2019).

A future comparison with in situ observations of CMEs in the range 0.1−0.3 au carried out by PSP is needed in order to ultimately validate modelling results close to the corona–heliosphere boundary, that is, in the proximity of the Alfvén surface. As PSP orbits will access the LASCO/C3 coronagraph field of view (within 30 solar radii) in the near future, we will have the unprecedented opportunity to actually measure the CME characteristics as close as 15−30 solar radii and to directly compare them with results obtained from EUHFORIA simulations of the newly observed CME events. Results from ESA’s Solar Orbiter (Müller et al. 2020) mission will also help extending current observational data sets to heliocentric distances as close as than 0.28 au and at heliocentric latitudes as high as 25°. This calls for subsequent studies aimed at specifically assessing the performances of numerical models outside of the ecliptic plane.

Acknowledgments

C.S. was supported by the Research Foundation – Flanders (FWO, strategic base PhD fellowship No. 1S42817N). S.D. acknowledges partial support from the Argentinian grants UBACyT (UBA) and PIP-CONICET-11220130100439CO. S.D. is member of the Carrera del Investigador Científico, CONICET. L.R. and A.N.Z. thank the European Space Agency (ESA) and the Belgian Federal Science Policy Office (BELSPO) for their support in the framework of the PRODEX Programme. This project also received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 870405 (EUHFORIA 2.0; https://euhforia.com/euhforia-2-0/). These results were also obtained in the framework of the projects C14/19/089 (C1 project Internal Funds KU Leuven), G.0D07.19N (FWO-Vlaanderen), and C 90347 (ESA Prodex). Wind data can be accessed via the OMNIWeb Plus data portal (https://omniweb.gsfc.nasa.gov/ftpbrowser/wind_min_merge.html). This paper uses data from the Heliospheric Shock Database (http://ipshocks.fi), generated and maintained at the University of Helsinki. EUHFORIA is developed as a joint effort between the University of Helsinki and KU Leuven. The full validation of solar wind and CME modelling is being performed within the BRAIN-be CCSOM project (Constraining CMEs and Shocks by Observations and Modeling throughout the inner heliosphere; http://www.sidc.be/ccsom/) and BRAIN-be SWiM project (Solar WInd Modeling with EUHFORIA for the new heliospheric missions). The simulations were carried out at the VSC – Flemish Supercomputer Centre, funded by the Hercules foundation and the Flemish Government – Department EWI. C.S. thanks Jens Pomoell for useful discussions.

References

Appendix A: Radial evolution of CME time profile

This appendix provides a summary of the CME time series extracted at selected virtual spacecraft and the further analysis performed to identify the shock, magnetic ejecta boundaries, and CME wake. Figure A.1 shows the performance of different β_p thresholds in the identification of the CME magnetic ejecta boundaries in EUHFORIA time series at various heliocentric distances. Figure A.2 shows the identification of the sheath and CME magnetic ejecta boundaries in EUHFORIA time series (in red) at various heliocentric distances based on Eq. (3) and on the β_p = 0.5 threshold. Figure A.3 shows the identification of the start and end of the CME pre-conditioning of the background solar wind (CME wake) based on the 5% threshold in EUHFORIA time series at various heliocentric distances.

Fig. A.1. Identification of the CME boundaries in EUHFORIA time series at various heliocentric distances. At each heliocentric distance, the proton, βp, magnetic field intensity, B, speed, v, and proton number density, np, are provided. In all panels, the arrival time of the CME-driven shock is marked by the dashed blue line. Top panel: the boundaries of the magnetic ejecta identified by imposing different βp thresholds are marked by coloured vertical lines, while the ultimate boundaries of the magnetic ejecta obtained by applying the threshold condition βp ≤ 0.5 are marked as continuous blue lines in the second, third, and fourth panels.

Fig. A.2. Identification of the sheath and CME boundaries in EUHFORIA time series (in red) at various heliocentric distances. At each heliocentric distance, the magnetic field intensity, B, proton, βp, speed, v, and proton number density, np, are provided. The arrival time of the CME-driven shock is marked by the dashed blue line. The identification of the magnetic ejecta boundaries (marked as continuous blue lines, same as in Fig. A.1) has been obtained by applying the threshold condition βp ≤ 0.5 (indicated as blue shaded regions). The duration of the CME sheath, based on the identification of tshock and tstart, is marked in yellow. In the proton β panels, the βp = 0.5 threshold is indicated as an horizontal dashed black line.

Fig. A.3. Identification of the start and end of the CME perturbation to the background solar wind based on the 5% threshold (marked by the coloured horizontal rectangles in each panel) in EUHFORIA time series at various heliocentric distances. The CME wake for each variable is marked by the coloured vertical lines. The period of the sheath and CME passage (same as in Fig. A.2) are marked as yellow and blue shaded areas, respectively.

All Tables

All Figures

Fig. 1. 1 min averaged solar wind magnetic field and plasma parameters as detected by the Wind spacecraft at L1. From top to bottom: bulk speed (v), proton number density (np), magnetic field magnitude (B), magnetic field components in GSE coordinates (Bx, By, Bz), proton temperature (Tp), magnetic field elevation (θB), and proton β (βp). Bottom panel: 1 h Dst index. The duration of the sheath is indicated by the yellow area, while the duration of the magnetic ejecta is bounded by the blue area. The dashed blue line marks the start of the perturbation in the speed profile due to an approaching fast solar wind stream at the back of the CME. The temperature panel shows the comparison of the observed proton temperature, Tp, (in black) with the expected temperature, Texp, calculated using the speed-temperature relation by Lopez & Freeman (1986; in red). In the βp panel, the value βp = 1 is indicated by the dashed grey line.

In the text

Fig. 2. Location of the virtual spacecraft placed in the EUHFORIA heliospheric simulation domain, as seen from two different views. Blue dots: virtual spacecraft along the Sun–Earth line. Red dots: virtual spacecraft near the location of the Earth. Yellow dot: location of the Sun. Green dot: location of the Earth. We focus on the virtual spacecraft in blue to study the radial evolution of the Earth-directed CME event considered.

In the text

	Fig. 3. Overview of EUHFORIA results in the heliographic equatorial plane. Snapshots of the radial speed, vr (a), scaled number density, n (D/1 au)2 (b), co-latitudinal magnetic field, Bclt = −Bθ (c), and scaled radial magnetic field, Br (D/1 au)2 (d), on 15 July 2012 at 23:53 UT which is the time when the CME was crossing through Earth.
In the text

Fig. 4. Comparison of 10 min cadence EUHFORIA predictions at Earth (red solid line) and 1 min averaged Wind proton data (black solid line). The dark red and light red shaded areas show the maximum variation of EUHFORIA predictions at virtual spacecraft separated by Δσ = 5° and Δσ = 10° in longitude and/or latitude from Earth. From top to bottom: speed (v), proton number density (np), magnetic field magnitude (B), magnetic field components in GSE coordinates (Bx, By, Bz), and proton thermal pressure (Pp, th).

In the text

Fig. 5. Ambient solar wind from EUHFORIA: Radial dependence of the mean solar wind parameters extracted at virtual spacecraft located along the Sun–Earth line (red dots): (a) proton number density ⟨np⟩sw; (b) speed ⟨v⟩sw; (c) magnetic field ⟨B⟩sw; (d) proton temperature ⟨Tp⟩sw. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Observation-based relations from Hellinger et al. (2013) are indicated as dashed green lines. Observed mean values from Wind in the 24 h prior to the CME arrival are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

In the text

Fig. 6. Ambient solar wind from EUHFORIA: Radial dependence of the mean solar wind parameters extracted at virtual spacecraft located along the Sun–Earth line (red dots): (e) proton thermal pressure ⟨Pp, th⟩sw; (f) magnetic pressure ⟨Pmag⟩sw; (g) total pressure ⟨Ptot⟩sw; (h) proton β ⟨βp⟩sw. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Mean observed values from Wind in the 24 h prior to the CME arrival are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

In the text

Fig. 7. Radial dependence of the mean CME parameters in EUHFORIA, extracted at virtual spacecraft located along the Sun–Earth line (red dots): (a) speed ⟨v⟩CME; (b) proton number density ⟨np⟩CME; (c) magnetic field ⟨B⟩CME; (d) proton temperature ⟨Tp⟩CME. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Observation-based relations are indicated as dashed green lines or green shaded areas. Observation-based values at selected heliocentric distances are indicated as green dots. Mean values from Wind are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

In the text

Fig. 8. Radial dependence of the mean CME parameters in EUHFORIA, extracted at virtual spacecraft located along the Sun–Earth line (red dots): (e) thermal pressure ⟨Pth⟩CME; (f) magnetic pressure ⟨Pmag⟩CME; (g) total pressure ⟨Ptot⟩CME; (h) proton β, ⟨βp⟩CME. The red shaded areas show the maximum variation of EUHFORIA predictions as a function of the heliocentric distance. The results from the fitting of the mean values are indicated as dashed red lines. Observation-based relations are indicated in green. Mean values from Wind are shown as blue diamonds. Histograms showing the frequency of occurrence of each parameter are provided for D = 0.2 au and D = 1.0 au radial distances. The mean and median are indicated by the red and black vertical dashed lines, respectively.

In the text

Fig. 9. Durations and radial sizes of the CME and its driven sheath as a function of D, based on EUHFORIA simulations: (a) sheath duration; (b) sheath size; (c) ejecta duration determined starting from the condition βp ≤ 0.5; (d) ejecta size determined starting from the condition βp ≤ 0.5; (e) ejecta to sheath duration ratio; (f) ejecta to sheath size ratio. Panels a, c, e, and f: observational values at selected heliocentric distances are indicated as green dots, based on Masías-Meza et al. (2016) and Janvier et al. (2019). Panel b: the sheath size expected from Eq. (8) in Lee et al. (2017) is indicated as a magenta dashed line. Panel d: the range of values recovered from observation-based relations are indicated as a green shaded area. The values recovered from Al-Haddad et al. (2019) based on numerical simulations are indicated as dashed and dotted-dashed magenta lines. Observation-based values in the solar corona reconstructed in Paper I are shown as magenta diamonds. The position of the EUHFORIA inner boundary is indicated as a dashed grey line. Values observed at Wind are shown as blue diamonds.

In the text

Fig. 10. Non-dimensional CME expansion rate evolution based on EUHFORIA simulations. Left column: derivation of the non-dimensional CME expansion rate ζ from the linear fitting of the CME speed profile in EUHFORIA simulations at D = 0.2 au (a), D = 1.0 au (b), and D = 1.8 au (c). Right column: non-dimensional CME expansion rate ζCME as a function of D (d), vc, CME (e), and Δvx, CME (f). Values observed at Wind are shown as blue diamonds. The green lines represent the fit for perturbed CMEs (dashed line), non-perturbed CME (thin continuous line) and for both sets of CMEs (thick continuous line) as reported by Gulisano et al. (2010).

In the text

Fig. 11. Ratio between the expansion speed vexp and the translation speed vtrans of the CME, as a function of the heliocentric distances. The reconstructed value in the corona (at 14.9 solar radii, corresponding to 0.07 au) from Paper I is indicated as a magenta diamond. The observed value at Wind is indicated as a blue diamond. Reference values measured in the corona (< 0.1 au, from Gopalswamy et al. 2009a) and at 1 au (from Lugaz et al. 2020) are indicated as green and orange dashed lines, respectively. The location of the EUHFORIA heliospheric inner boundary is indicated by the dashed vertical grey line.

In the text

Fig. 12. CME magnetic field profile evolution as a function of D, based on EUHFORIA simulations. Top panel: CME magnetic field profile in EUHFORIA at various heliocentric distances, expressed in terms of the normalised magnetic field Bnorm as a function of the normalised time tnorm. The normalised magnetic field profile at Wind is indicated by the dashed-dotted blue line. Middle panel: normalised time shift between the magnetic field peak and the centre of the magnetic ejecta ΔtBpeak, norm, as a function of the heliocentric distances. The observed value at Wind is indicated as a blue diamond. Bottom panel: normalised asymmetry parameter Δtasym, norm for the magnetic field profile, as a function of the heliocentric distances. The green shaded area indicates the range of asymmetry of the mean CME profiles at different heliocentric distances, based on the results by Janvier et al. (2019). The observed value at Wind is indicated as a blue diamond.

In the text

Fig. 13. Duration of CME wake as a function of D, based on EUHFORIA simulations. Left: duration of the CME wake since the end of the magnetic ejecta for the magnetic field, density, and temperature, as a function of the heliocentric distance D (indicated by the solid dots). Right: ratio between the duration of the CME wake and of the magnetic ejecta as a function of the heliocentric distance, D, for the same quantities (indicated by the solid lines).

In the text

Fig. A.1. Identification of the CME boundaries in EUHFORIA time series at various heliocentric distances. At each heliocentric distance, the proton, βp, magnetic field intensity, B, speed, v, and proton number density, np, are provided. In all panels, the arrival time of the CME-driven shock is marked by the dashed blue line. Top panel: the boundaries of the magnetic ejecta identified by imposing different βp thresholds are marked by coloured vertical lines, while the ultimate boundaries of the magnetic ejecta obtained by applying the threshold condition βp ≤ 0.5 are marked as continuous blue lines in the second, third, and fourth panels.

In the text

Fig. A.2. Identification of the sheath and CME boundaries in EUHFORIA time series (in red) at various heliocentric distances. At each heliocentric distance, the magnetic field intensity, B, proton, βp, speed, v, and proton number density, np, are provided. The arrival time of the CME-driven shock is marked by the dashed blue line. The identification of the magnetic ejecta boundaries (marked as continuous blue lines, same as in Fig. A.1) has been obtained by applying the threshold condition βp ≤ 0.5 (indicated as blue shaded regions). The duration of the CME sheath, based on the identification of tshock and tstart, is marked in yellow. In the proton β panels, the βp = 0.5 threshold is indicated as an horizontal dashed black line.

In the text

Fig. A.3. Identification of the start and end of the CME perturbation to the background solar wind based on the 5% threshold (marked by the coloured horizontal rectangles in each panel) in EUHFORIA time series at various heliocentric distances. The CME wake for each variable is marked by the coloured vertical lines. The period of the sheath and CME passage (same as in Fig. A.2) are marked as yellow and blue shaded areas, respectively.

In the text