Numerical Simulation on the Leading Edge of Coronal Mass Ejection in the Near-Sun Region

The coronal mass ejections (CMEs) observed by white-light coronagraphs, such as the Large Angle and Spectrometric Coronagraph (LASCO) C2/C3, commonly exhibit the three-part structure, with the bright leading edge as the outermost part. In this work, we extend previous work on the leading edge by performing a large-scale 3D magnetohydrodynamic numerical simulation on the evolution of an eruptive magnetic flux rope (MFR) in a near-Sun region based on a radially stretched calculation grid in spherical coordination and the incorporation of solar wind. In the early stage, the new simulation almost repeats the previous results, i.e., the expanding eruptive MFR and associated CME bubble interact with the ambient magnetic field, which leads to the appearance of the helical current ribbon/boundary (HCB) wrapping around the MFR. The HCB can be interpreted as a possible mechanism of the CME leading edge. Later, the CME bubble propagates self-consistently to a larger region beyond a few solar radii from the solar center, similar to the early stage of evolution. The continuous growth and propagation of the CME bubbles leading to the HCB can be traced across the entire near-Sun region. Furthermore, we can observe the HCB in the white-light synthetic images as a bright front feature in the large field of view of LASCO C2 and C3.


Introduction
Solar coronal mass ejections (CMEs) are spectacular phenomena originating from the corona due to the eruption of magnetic structures triggered or maintained by magnetohydrodynamic (MHD) instabilities and magnetic reconnections (Lin et al. 2005;Priest 2014).The CMEs generally eject a mass in the range of 10 11 -10 13 kg to interplanetary space, with radial propagation velocity in the plane of the sky (orthogonal to the Sun-Earth line) ranging from 20 km s −1 to >2000 km s −1 (Yashiro et al. 2004;Chen 2011).The angular width of the CMEs is typically between 20°and 120° (Webb & Howard 2012), with some narrow CMEs with a width of <20°, and a tiny fraction of wide CMEs with a width of >120°.The morphological structure of the CME shows large variability in the white-light coronagraph, which observes that the brightness of the K-corona originates from Thomson scattering of free electrons.Among diverse morphologies, the three-part CME and "loop" CME stand out (Vourlidas et al. 2013).The three-part CME includes a bright front/leading edge, a bright core, and a dark cavity between them (Illing & Hundhausen 1986).The "loop-like" CME has only one bright loop-like structure, similar to the bright front in the three-part CME.Observations of white-light coronagraphs in the past two decades show that more than 30% of CMEs have three parts (Webb & Howard 2012).Song et al. (2017Song et al. ( , 2019b) ) demonstrated that extra three-part CMEs can be observed in the early stage of the CME evolution in the extreme-ultraviolet (EUV) passbands.
Researchers had early recognized that the CMEs result from the eruption of the magnetic flux rope (MFR; Webb & Howard 2012), where helical magnetic field lines wrap around their central axial.MFRs may exist prior to the eruption (Patsourakos et al. 2013), such as filaments/prominences and hot channels, or form during the CME eruption (Song et al. 2014) due to ongoing magnetic reconnection.The eruptive structure of a CME always contains an MFR, which results in a large-scale ejection of magnetized coronal plasma.The MFR models and numerical MHD simulations for the CMEs had self-consistently reproduced many observed features of CMEs, including a three-part structure (Wu et al. 2001;Wood & Howard 2009;Mei et al. 2020a), the current sheet and magnetic reconnection (Lin & Forbes 2000;Mei et al. 2012Mei et al. , 2017;;Reeves et al. 2019;Ye et al. 2019;Jiang et al. 2022), the global coronal EUV disturbances (Chen et al. 2002;Delannée et al. 2008;Downs et al. 2012;Xie et al. 2019;Mei et al. 2020b), large-scale evolution in the near-Sun region or interplanetary space (Roussev et al. 2012;Jin et al. 2017;Török et al. 2018;Yang et al. 2021), and so on.The three-part CMEs are observed in the white-light coronograph to consist of a bright core, a dark void, and a bright circular front/leading edge (Illing & Hundhausen 1985).Because of the complexity of the CME appearance, it is still a challenge to precisely describe how the erupting MFRs evolve into three parts observed (Manchester et al. 2017).The cores of CMEs used to be interpreted as eruptive filaments according to the corresponding Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.photospheric magnetogram and multiband observations (Gopalswamy et al. 2006).However, recent works challenge the filament-core connections for some events (Howard et al. 2017;Song et al. 2017Song et al. , 2019aSong et al. , 2022Song et al. , 2023;;Wang et al. 2022).They found that more than half of the three-part CMEs are not associated with filaments, and speculated that the geometrical projection of the eruptive MFR may also produce CME cores.
For the CME front/leading edge, its physical nature is also challenging to explore, although it has been studied extensively (Chen 2011).The CME front had been interpreted as a fastmode MHD shock (Nakagawa et al. 1975;Vourlidas et al. 2003).However, the researcher realized the difference between the fast shock and the CME front because the legs of the CME front do not expand laterally (Howard et al. 1982;Sime et al. 1984) and some CMEs are too slow to drive a fast shock.Furthermore, the fast shock can be observed as faint features ahead of the CME in white-light coronagraphs (Gopalswamy et al. 2009;Vourlidas et al. 2013;Lee et al. 2014), and some researchers have used the stand-off distance between the fast shock and the CME front to develop technology to deduce coronal magnetic field parameters (Gopalswamy & Yashiro 2011;Lee et al. 2017;Ying et al. 2022).It is widely known that the CME front is the density enhancement of ambient plasma caused by the eruptive magnetic structures (Forbes 2000;Ciaravella et al. 2005;Cheng et al. 2012).The tricky part is how the magnetic structure evolution leads to the density increase of the CME front (Chen 2017).It has been interpreted as a 3D bubble-like MFR (or hollow croissant) expanding and broadening in the corona (Mouschovias & Poland 1978;Gibson & Low 1998;Chen et al. 2000;Thernisien et al. 2006), a coronal plasma pileup region ahead of the MFR Forbes (2000), nonwave components of the global EUV disturbances (Chen 2009;Cheng et al. 2011), and a helical current boundary (HCB) between the MFR and the ambient magnetic field (Mei et al. 2020a).The hollow croissant model requires an inner coronal plasma distribution that peaks near the shell of the croissant, such that the plasma density enhancement of the CME front comes from integrating the inner plasma of the MFR along the line of sight.Chen (2009) has found in a limb event that the EUV wave front is nearly cospatial with the CME front, and the following EUV dimming is also cospatial with the CME cavity.They thus suggest that the CME front is generated by the successive stretching of magnetic field lines as the eruptive MFR continues to push overlying field lines along its path.Concretely, the successive interaction between the MFR and the overlying field lines can lead to the appearance of an expanding HCB between them (Török et al. 2004;Mei et al. 2018).Mei et al. (2020a) have seen a fast shock front followed by a bright HCB in the Solar Dynamics Observatory (SDO)/Atmospheric Imaging Assembly (AIA) synthetic images of their high-resolution 3D MHD simulation, and furthermore suggest that the HCB is a possible mechanism for the nonwave component of the CME front and EUV disturbances.Haw et al. (2018) have proposed a reverse current model for the formation of the dark cavity and the front.The reverse current is highly similar to the HCB (see Figure 2 in the following), as both come from the interaction of the rising MFR with the ambient magnetic field.Due to the absence of kink processes, the reverse current does not have a helicity geometrical feature.
In this work, we perform a large-scale 3D MHD simulation of CMEs with the Titov & Démoulin (1999) model (TD99 hereafter; also in Lin et al. 2002).Utilizing the radially stretched calculation grid, we can simulate the early-stage eruptive processes of the MFR and sequence the evolution of the corresponding CME in the near-Sun region.To compare the simulation results with realistic observations from SDO/AIA images (Lemen et al. 2012;Pesnell et al. 2012), Solar and Heliospheric Observatory/the Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995;Domingo et al. 1995), and ground-based K-Coronagraph (K-cor; K-Cor Team 2013) located at Mauna Loa Solar Observatory (MLSO), we employ forward modeling (Van Doorsselaere et al. 2016) to obtain synthetic EUV and white-light images.Section 2 describes the TD99 model in the spherical coordinate system for the initial magnetic structure, and the background isothermal solar wind extending from the solar surface to the entire near-Sun region.Section 3 presents our main results, and the last section summarizes our work.

Setup of Simulation
The initial magnetic structure in our simulation comes from the TD99 model for solar eruptive events (Török et al. 2004;Török & Kliem 2005;Mei et al. 2017Mei et al. , 2020a)).The TD99 model has three parts, as shown in Figure 1.The first part models the filament/prominence by an MFR with minor radius a, major radius R, and total toroidal current I.The second one is to confine the MFR by a dipole of strength q, lying on the MFR symmetry axis below the bottom of the photosphere r = R e , at a distance d.Here, the photosphere has a thickness of r p and R e = 6.95 × 10 10 cm is the solar radius.The third, another dipole, is buried below the photosphere bottom at a distance d d , and has been adopted to introduce an additional magnetic field near the MFR to control the twist profile of the MFR (Mei et al. 2020b).However, this dipole has also introduced an unnecessary background field already given by the second part.Instead, this work uses a circular solenoid overlapping on the MFR to introduce an additional toroidal magnetic field inside the MFR, essentially leaving the outside field unchanged.The magnetic field from the solenoid is given by Here, ( . These formulae are written in Cartesian coordination system --   .The MFR is lying with the  - plane, and its center is located at = d  below the origin point of --   .Parameters q so and s are to control the magnetic field of the solenoid.The formulae for the first and second parts of the modified TD99 model can refer to Mei et al. (2018Mei et al. ( , 2020b)), but also in --   .For the spherical coordinate system of the computational domain R-θ-f, the solar center locates at its origin point, and the solar equator is lying with θ = π/2.R-θ-f are related to a Cartesian coordination system x-y-z by sin cos sin sin cos .3 The relationship between x-y-z and --    can be expressed by a series of transformations: Here, (x td , y td , z td ) is the coordinates of the toroidal MFR center in x-y-z; α x , α y , and α z are angles rotating along x-, y-, and zaxes, respectively.In order to obtain formulae of the magnetic field in --   , we first rewrite the coordinates in r-θ-f with the coordinates in --   , according to Equations (3) and (4); then utilize formulae of the modified TD99 model to get magnetic field (B B B , , Later, by Cartesian to spherical coordination system transformation, we get magnetic field formulae in r-θ-f.Notice here that we ignore the curvature of the solar surface because the scale of the TD99 model is smaller when compared with the solar disk, so we can treat the spherical surface as a plane and insert the TD99 model into our calculation domain directly by a series of transformations.The magnetic structure of the TD99 model located in the lower atmosphere is shown in the upper left panel of Figure 1, in which the colorful curves are magnetic field lines.The upper right panel of Figure 1 plots the magnetic field components along the y-axis.
Our initial atmosphere also comes from analytical formulae, including a gravitationally stratified high-density layer representing the photosphere, and an extended corona in the near-Sun region from the standard Parker solution for solar wind.The atmosphere below r = R e + r p = 1.0086R e is isothermal with constant temperature T p = 5000 K and in static equilibrium under pressure and gravity forces.The pressure distribution is given by where p is the pressure, 1 is abundance of helium, m p = 1.67 × 10 −24 g is the mass of a proton, and g e = 2.74 × 10 4 cm s −1 is the gravity constant on the solar surface.This high-density layer below r = R e + r p is the purpose of realizing the line-tied environment on the photosphere, which fixes the footpoints of the coronal magnetic structures because of the high plasma β environment and the frozen condition.The dynamical evolution of the coronal magnetic structure cannot reach the bottom of the simulation box due to the high-density layer.We are able to keep all the physical quantities unchanged, bypassing the possible complicated behavior when implementing the line-tied condition without this layer.In addition, we adopt open boundary conditions for all the other five boundaries, where all physical variables on the ghost cell are derived from extrapolations of neighboring internal cells.
In this simulation, we assume that the atmosphere above r = R e + r p and the extended atmosphere, i.e., solar wind, in the near-Sun region are also isothermal with temperature T c = 1 × 10 6 K.Because of neglecting the effect of the initial magnetic structure from the TD99 model on the velocity field distribution of solar wind and not including a large-scale initial background magnetic field, such as a coronal helmet streamer, the plasma velocity field of the solar wind has only radial component v r , independent of θ and f.The solar wind is governed by the differential equation for v r to r 2 is essential since r = r c = 5.78R e and v r = c s = 1.165 × 10 7 cm s −1 is a sonic point.If the radial velocity v r reaches the sound speed c s , then r must equal r c or else dv/dr becomes infinite.This equation can be integrated to give a transcendental equation as where C is the constant of integration; only a physically reasonable solution passes through the sonic point, which requires C equal to −3.The radial plasma speed is given by Inserting the velocity gradient from Equation (9), and integrating Equation (12) from r p to r yields The dashed line in the lower right panel of Figure 1 gives the corresponding distribution of plasma thermal pressure.
The governing resistive MHD equations are numerically solved by a shock-capturing finite volume method, which consists of a stretched calculation grid in spherical coordination, a robust total variation diminishing (TVD) Lax-Friedrichs spatial discretization, a three-order slope limiter (Čada & Torrilhon 2009), and three-step Runge-Kutta time-marching scheme.All algorithm components are well coded in the Message Passing Interface (MPI)-parallelized adaptive mesh refinement code (Keppens et al. 2012;Porth et al. 2014;Xia et al. 2018), where various schemes are implemented to solve hyperbolic partial differential equations.For convenience, we normalize all physical quantities to R e , c s , 1.66 hr, 2.34 × 10 −14 g cm −3 , 3.18 Pa, and 6.32 G for length, velocity, time, density, temperature, pressure, and magnetic field, respectively.The relevant parameters for the initial improved TD99 model are set to R = 0.17, a = 0.017, d = 0.065, q so = − 6, s = 20, (x td , y td , z td ) = (0, 1, 0), α x = 0, α y , and α z = π/2.The remaining parameters related to the first and second parts of the improved TD99 model are set to q = 2.2 × 10 −3 , α = 0.9997, and T f = 2 × 10 5 for the magnetic strength, the contribution of the thermal pressure on the internal equilibrium of the MFR, and the plasma temperature inside the MFR.The computational domain is a 3D sphere of size 1 r 45, 0.05 π θ 0.95 π and 0 f 2 π, resolved by 320 × 160 × 160 grid points.Regions close to the polar axis have been excluded from our domain to avoid singularities in the computational mesh along the polar axis.The stretched grid has been used to cover the near-Sun region while maintaining sufficient resolution in the lower atmosphere region and in the MFR vicinity.The grid has been stretched along the r-axis, θaxis, and f-axis with the same scaling factor 1.02.The scaling factor is the ratio of edges of two neighboring cells along the raxis, θ-axis, or f-axis.Along the θ-axis and f-axis direction, the calculation grid has been stretched symmetrically, with the center at θ = π/2 and f = π, where the initial MFR is located.The resultant cell near the initial MFR is close to the cube with an edge length of about 5.0 × 10 −3 so that there are 12 cells along the r-axis to resolve the MFR.

Main Results
Since the initial MFR deviates from equilibrium, a net upward force is acting on the MFR, leading to an eruption immediately after the start of the simulation.Figure 2 gives the evolution of the magnetic structure in the early stage of simulation, similar to the results in Mei et al. (2020a).The colored curves in the upper panels are the magnetic field lines.The current isosurfaces in the lower panels show the 3D geometrical features of the current sheet (CS) and the MFR.With the MFR takeoff, the overlying magnetic field has been stretched, leading to the appearance of a 3D CS under the MFR (Forbes & Priest 1995;Lin & Forbes 2000;Lin et al. 2002).Then, magnetic reconnection takes place inside the CS, and the newly formed field lines are continuously attached to the MFR.The following refers to the sum of MFR and new appended magnetic field lines as the CME bubble, whose boundary is marked in Figure 2. Due to the twist feature of the MFR, it undergoes a kink process as it moves upwards.The kinking MFR leads to the rotation movement of the CME bubble, which, as the proxy of the MFR, interacts with ambient magnetic field lines and leads to the formation of the HCB, wrapping around the bubble (Mei et al. 2018).In front of the CME bubble and the HCB, there is a fast shock (FS) with a dome shape in the outermost part, as shown by the current distribution on the cuts x = 0 and z = 0 in Figure 2. Ahead of the FS, the isothermal solar wind is unperturbed and maintains an initial pressure and velocity distribution given by the analytical Parker solution.
Based on the simulation datum, we performed forward modeling to obtain SDO/AIA images by the forward modeling code (FoMo; Van Doorsselaere et al. 2016), which calculates the EUV emission from the corona plasma with the optically thin approximation.Figure 3 shows the synthetic images for AIA 193 Å, 171 Å, and 131 Å in the image plane (also known as the plane of the sky) at the early stage of evolution with time t = 0.05 and 0.3, respectively.The frame of reference of the observation ¢ x -¢ y -¢ z connects the Cartesian coordination x-y-z by two angles  and -.The line of sight (LOS) is along the ¢ z -axis, and the image planes lay on the panel ¢ = z 0.  and - are angles between the LOS and the z-axis and between the LOS and the y-axis, respectively.Similar to Mei et al. (2020a), we can see a bright MFR at t = 0.05, which later disappears at t = 0.3, and an S-shape structure appears in the original place of the MFR, which is related to CS and the magnetic reconnection (Aulanier et al. 2010;Roussev et al. 2012;Jiang et al. 2022).The MFR becomes visible immediately after the start of the simulation due to its internal untwisting process, which intrigues magnetic reconnection and internal plasma heating.Due to the continuous expansion of the MFR, the heated MFR gradually darkens again and disappears in the synthetic AIA images.We can also see a bright FS propagating across the solar disk in the AIA bands.The FS front in the higher region, 0.2R e away from the solar surface, can only be seen in the AIA multiband images in the early stage of evolution.Even in the enhanced images with log scale, the FS front cannot be identified in the higher region clearly after t = 0.1, as shown by the second row of Figure 3.
As for the HCB, we are not able to identify the HCB following the FS front in the synthetic AIA images in this work.Because the FS and HCB are highly close to each other at the early stages of the simulation, the spatial resolution of the simulation is critical to resolving the HCB from the bright FS front.Since a large fraction of computational resources have been devoted to capturing the large-scale evolution of the CME in the near-solar region, there is not sufficient resolution to distinguish the perturbation of the density and temperature distributions by the HCB from the perturbation due to the nearby FS front at the early stage.In our previous work (Mei et al. 2020a), a relatively higher resolution has been used, and an HCB has been observed following the FS in the AIA synthetic images.The HCB, MFR, and the relatively dim region between them are similar to the typical three parts of the CME observed.The presence of the HCB suggests that it could be the physical mechanism of the CME front.Furthermore, we suggest it may also be responsible for the nonwave component of the EUV disturbance, according to the bimodal interpretation of the EUV disturbance (Shibata & Magara 2011;Warmuth 2015).
However, in Mei et al. (2020a) the evolution of the CME has stayed within the 2R e away from the solar center.We still need to confirm that the HCB can also be observed in LASCO C2 and C3 in the near-solar region.Can the HCB still be a plausible theoretical explanation for CME fronts in large spaces?Originally, the CME front was a concept based on extensive large field-of-view observations of white-light coronagraphs.In this work, we extend the calculation region of our previous work to cover the large near-Sun region to simulate the continuous expansion and propagation of the MFR.Benefitting from the utilization of the stretched calculation grid, the calculation resolution for this simulation can effectively catch the propagation of the CME in the near-Sun region.Figure 4 gives the large-scale evolution snapshots of magnetic structures in the region from 2 ∼ 40R e , which can cover the field of view of LASCO C2 and C3.Please note that the colors of curves are only used to highlight that they are different curves; the curves with the same color in different panels are not the same at different times.Following the early stage evolution in Figure 2, the MFR continues to erupt outward and be stretched in the near-Sun region.Similar to the early evolution of the CME bubble (Mei et al. 2012(Mei et al. , 2017;;Janvier et   magnetic reconnection inside the CS following the MFR results in the continuous growth of the CME bubble in the large space, by attaching new magnetic field lines to the outermost part of the bubble.
Figure 5 gives the corresponding current distribution on the cut z = 0, which makes it easier to identify different features than to find in a bundle of magnetic field lines shown in Figure 4.When the magnetic structure is continuously expanding in the near-solar region, the current density decreases dramatically.We have adjusted the color bar for each snapshot of Figure 5 to highlight the dynamical evolution of the current features.The MFR, the FS, and the HCB show continuous growth and propagation in larger space and can still be identified in current distribution even when they propagate or exceed 30R e away from the solar center.The FS nearly degenerates into sound waves in the near-Sun region and thus propagates isotropically outward as the magnetic pressure decreases faster than the solar wind thermal pressure.
The propagation of the HCB in the near-Sun region depends on the continuous expansion of the CME bubble, which further depends on the expansion of the MFR.Because the MFR legs are anchored in the solar disk, the expansion of the CME bubble is significantly confined on both flanks, as is the HCB.The expansion angle of the HCB is about 90°in our simulation, as shown in Figure 5.In the right front of the MFR (i.e., along the yaxis), the MFR has not been confined; thus, the HCB can propagate far in this direction, closely following the FS.Moreover, the HCB leads to a considerably stronger perturbation on the large-scale background magnetic field than the FS leads.
Figure 6 shows the distributions of current, plasma density, and thermal pressure along two lines (L1 and L2), marked in Figure 5, one along the y-axis and another one with an angle of 45°on the y-axis.The HCB can be easily identified in the distribution of current along the lines L1 and L2.The FS can only be identified in the current distribution along the line L2.Thus, the FS is more easily detected in the two flanks of the CME than in the right front.Along the L2, the disturbances for the HCB are considerably stronger than disturbances from the FS.The background magnetic field on the way of the propagating FS comes from the dipole buried under the photosphere, which is used to confine the MFR in the initial magnetic configuration.This background field decreases with the cube of distance away from the solar surface, so the FS along the y-axis is weaker than the other directions because it is farthest away from the solar disk.Nevertheless, a more realistic background field should be used in the future to study the propagation of the CME in a large space.For the HCB, it comes from the interaction of the CME bubble with the ambient background field, so that the strength of disturbance from the HCB on the current depends on the magnetic field of the CME bubble.Although the CME bubble undergoes expansion during its propagation, its magnetic strength is determined by its initial eruptive magnetic structure and decreases slowly with distance away from the solar surface than the dipole's field.This results in an HCB involving a stronger magnetic field than the FS involved.Thus, in the current distribution, the HCB is more prominent than the FS when propagating in the near-Sun region.Along the L2, we can also identify two peaks for the HCB and the FS on the curve for density distribution.This means that both HCB and FS contribute to the density enhancement in front of the CME bubble, with HCB contributing more than FS to the density enhancement.Along the L1, only the peak from the HCB can be identified, which means that only the HCB is responsible for the density enhancement in the right front of the CME bubble.To compare our numerical results with coronagraph observations of the CME, we also present synthetic, Thomsonscattered, white-light images by MLSO K-cor in Figure 7 and by LASCO C2 and C3 in Figures 8 and 9. Several groups have performed synthetic images for CMEs in the recent two decades (Chen et al. 2000;Manchester et al. 2008;Riley et al. 2008;Lugaz et al. 2009;Shen et al. 2014;Chen 2017;Manchester et al. 2017).Similar to the EUV synthetic images, we also performed forward modeling of the white-light observations with the FoMo code.The plasma density from our simulations is translated into total and polarized brightness.Then, it is integrated along the LOS to generate synthetic LASCO C2 and C3 observations by using the formula (from chapter 6 of Billings 1966;Lugaz et al. 2005) here, μ is the limb-darkening parameter, which is set to 0.5; A, B, C, and D are functions of the half-angle subtended by the Sun; θ is the angle between the plane of sky and the line connecting the Sun center and the plasma element in the LOS.
Here, we use polarized brightness polar  to obtain synthetic images for K-cor, and use the total brightness total  to obtain the synthetic white-line images for LASCO, because the total brightness observations are made far more frequently than the polarized brightness (Morgan 2015).
Corresponding to Figure 3, Figure 7 gives the early stage evolution of a three-part structure with different viewing angles at t = 0.1 and t = 0.2 in the white-line images observed by MLSO K-cor with field of view 1.05 ∼ 3R e away from the solar center.In this figure, we can clearly see that an FS propagates outward in the outermost part, followed by a threepart structure with a bright CME front followed by a relatively dark cavity containing a bright core associated with the eruptive MFR.The bright CME front corresponds to the HCB, shown in Figure 2. Different from the EUV bands shown by Figure 3, the HCB can be identified in the white-line images.Excepting the very beginning stage t = 0.1, we cannot distinguish the FS and the HCB for some viewing angles, as shown by the lower left panel of Figure 7.
Figure 8 shows the synthetic images with different viewing angles at t = 1, 2, and 3 for LASCO C2 with field of view 2 ∼ 7R e away from the solar center.In the left panels of this figure for t = 1, we can see an FS in the outermost part, whose front is more obvious in the two flanks than in the right front of the CME.After the FS, a three-part structure is seen.The CME front of this three-part structure corresponds to the HCB, shown in Figures 5 and 6.These features in the synthetic image agree nicely with observations of CME by white-light coronagraph (such as Vourlidas et al. 2013;Lee et al. 2017).The CME front is the dominant feature in the white line observations, and the FS is a relatively faint feature immediately ahead of the CME front.With the continuous expansion of the eruptive magnetic structure, the three-part features become more and more obscure in the middle panels at time t = 1.In order to enhance these features in a larger space 4 ∼ 30R e away from the solar center, the right panels of Figure 8 show the synthetic images for LASCO C3 with the log scale and also Figure 9.We can see the FS, the HCB, and the MFR in the right panels of Figure 8.In Figure 9, the FS and the HCB can still be identified in the whole field of view of the LASCO C3.In Figure 9, the MFR becomes indistinguishable, although this figure has a logarithmic enhancement, as the plasma density inside the MFR decreases with its continuous expansion.The FS and the HCB, however, can still be identified in the whole field of view of the LASCO C3.This strengthens the reliability of the suggestion that HCB is a possible mechanism for the CME fronts observed in the EUV bands, in the white-light coronagraphs of LASCO C2 and C3, and even of SECCHI COR1 and COR2 (Howard et al. 2008;Kaiser et al. 2008).

Conclusion and Discussion
We performed a large-scale 3D MHD numerical simulation to study the three-part structure of CMEs, with emphasis on the leading edge.The TD99 model has been used to set up the initial magnetic field, which consists of an MFR, a solenoid overlapping on the MFR, and a dipole under the photosphere.The analytical Parker solution has been used to set the density and velocity of isothermal solar wind in the near-Sun space.A spherical stretched calculation grid has been adopted to enable us to simulate the early evolution of the eruptive MFR and its propagation in the near-Sun region in the meantime.To comprehensively compare the numerical simulation result with the realistic multiband observations, we performed the forward modeling of the SDO/AIA, MLSO K-cor, and LASCO C2 and C3 by the FoMo code.In the early stage of simulation, the evolution of CME almost repeats our forgoing work (Mei et al. 2020a).Immediately after the launch, the MFR erupts upward, and the CME bubble grows continuously.When the CME bubble lifts upward, it interacts with the ambient background field, leading to the appearance of the HCB between them.The interaction between the CME bubble and the ambient background magnetic field leads to the appearance of the HCB and the plasma pileup around the HCB in the meantime.Ahead of the HCB, an FS with a 3D dome shape appears and propagates outward.In the synthetic SDO/AIA images, we can observe the FS front as the outermost bright feature and the three-part structure, composed of the bright FS, the bright MFR, and the relatively dim region between them.However, we were not able to distinguish the HCB from the FS front in the SDO/AIA synthetic images in this work because we did not have sufficient computational grid points to resolve the adjacent HCB and HCS.In the synthetic MLSO K-cor images, we can easily distinguish the FS and the HCB, both of which can serve as the leading front of the three-part structure.
As the simulation progresses, the CME leaves the region less than R e away from the photosphere and propagates into the larger near-Sun region, which has not been done in our previous simulations.With the stretched computational grid, we are able to model the early eruption of the MFR with its footprint frozen into the photosphere and the subsequent propagation of the CME in the near-Sun region in a selfconsistent way.Similar to the evolution in the early stage, the FS continues to propagate in a larger space, and the CME bubble continues to grow and propagate in a larger space.The HCB, as a result of the interaction between the bubble and the ambient background field, also expands and propagates into outer space.The three-part features are seen in the synthetic observations of LASCO C2 and C3.This confirms our previous suggestion that HCB is the mechanism responsible for the CME leading edge/front observed by SDO, AIA, and LASCO C2 and C3.Moreover, our simulations still support that the CME front is a plasma pileup before the eruptive structure and that the CME front corresponds to the nonwave component of the coronal disturbance during the eruption.Moreover, our conclusion is more concrete, namely that the plasma pileup is due to the interaction between the CME bubble and the ambient magnetic field.

Figure 1 .
Figure1.Upper left: initial magnetic structure (colored curves) with footpoints rooted on the solar surface (gray shading), the translucent isosurface illustrates the surface of the magnetic flux rope (MFR).Lower left: stretched grid on cut z = 0 in a spherical coordination system, whose relationship with the Cartesian system is given by Equation (3).Upper right: distribution of three components of the magnetic field line along the y-axis.Lower right: pressure and radial velocity distributions of solar wind in the near-Sun region along the y-axis.The units in all plots in this work are dimensionless with conversions as given in the main text.
a small value near r = r e and becomes supersonic for r > r c , as shown by the solid line in the lower right panel of Figure1.Here,  is the Lambert  Function, and i is an imaginary unit.The plasma thermal pressure is given by 

Figure 2 .
Figure 2. Evolution snapshots of magnetic configuration during the early stage of numerical simulation, t = 0.2, similar to Mei et al. (2020a).Upper left: cut x = 0 shows the current distribution; colored curves are magnetic field lines showing an eruptive flux rope.Upper right: cut y = 0 shows the distribution of the z-component current, in which the HCB is similar to the reverse current layer given by Haw et al. (2018).Lower left and lower right: gray isosurfaces of the current show the 3D geometrical shapes of the CS and the twisted flux rope.The spherical cut r = 1.0086 is the surface of the photosphere; the texts marked on the figure are a fast shock (FS), the HCB, and the MFR.

Figure 3 .
Figure 3. Synthetic AIA 193 Å (first and second rows), 171 Å (third row), and 131 Å (fourth row) images in the plane of the sky with viewing angles =  41  and =  14  at t = 0.05 (left) and 0.3 (right).Texts marked on the figure are FS, MFR, and S-shape sigmoid.The second row with the log scale aims to enhance the features during the eruption.

Figure 4 .
Figure 4. Large-scale evolution snapshots of magnetic structure (colorful curves) at different times (t = 0.5, 1, 2, and 4), showing the continuously expanding CME.Note that curves with the same color at different times are not identical.Colors are used only to distinguish different curves.

Figure 5 .
Figure 5. Large-scale evolution snapshots of current distribution on cut z = 0 at different times (t = 0.5, 1, 2, and 4 minutes), which demonstrate the expanding HCS, MFR, and FS in the near-Sun region.

Figure 6 .
Figure 6.The distributions of current, density, and thermal pressure along line L1 (left) and line L2 (right), marked in Figure 5. L1 is along the y-axis, and L2 has an angle of 45°on the y-axis.

Figure 8 .
Figure 8. Synthetic images in plane of sky with viewing angles =  60  and =  23  (upper) and =  1  and =  3  (lower) at times t = 0.5 (left) and t = 1 (middle) for LASCO C2 observation with FOV 2 ∼ 7R e .To enhance the features in the middle panels, synthetic images with log scale are given in the right panels.

Figure 9 .
Figure 9. Synthetic images with log scale in plane of sky with viewing angles =  60  and =  23  (upper) and =  1  and =  3  (lower) at different times t = 2, 4, and 6 for the LASCO C3 with FOV 4 ∼ 30R e .