TY - JOUR
AU1 - Mei, Z, X
AU2 - Keppens,, R
AU3 - Cai, Q, W
AU4 - Ye,, J
AU5 - Xie, X, Y
AU6 - Li,, Y
AB - ABSTRACT We present a 3D magnetohydrodynamic numerical experiment of an eruptive magnetic flux rope (MFR) and the various types of disturbances it creates, and employ forward modelling of extreme ultraviolet (EUV) observables to directly compare numerical results and observations. In the beginning, the MFR erupts and a fast shock appears as an expanding 3D dome. Under the MFR, a current sheet grows, in which magnetic field lines reconnect to form closed field lines, which become the outermost part of an expanding coronal mass ejection (CME) bubble. In our synthetic SDO/AIA images, we can observe the bright fast shock dome and the hot MFR in the early stages. Between the MFR and the fast shock, a dimming region appears. Later, the MFR expands so its brightness decays and it becomes difficult to identify the boundary of the CME bubble and distinguish it from the bright MFR in synthetic images. Our synthetic images for EUV disturbances observed at the limb support the bimodality interpretation for coronal disturbances. However, images for disturbances propagating on-disc do not support this interpretation because the morphology of the bright MFR does not lead to circular features in the EUV disturbances. At the flanks of the CME bubble, slow shocks, velocity vortices and shock echoes can also be recognized in the velocity distribution. The slow shocks at the flanks of the bubble are associated with a 3D velocity separatrix. These features are found in our high-resolution simulation, but may be hard to observe as shown in the synthetic images. MHD, shock waves, waves, Sun: chromosphere, Sun: corona, Sun: coronal mass ejections (CMEs) 1 INTRODUCTION Magnetohydrodynamic (MHD) waves, shocks and other disturbances during solar eruptive events provide us with valuable information on fundamental physics of eruptions in the solar atmosphere. In past decades, many authors reported various disturbances in extreme ultraviolet (EUV), H α (⁠|$6563\, \mathring{\rm A}$|⁠), H i (⁠|$10830 \, \mathring{\rm A}$|⁠), soft X-ray (SXR), white light and type II and IV radio bursts (Warmuth 2007; Liu & Ofman 2014). Large-scale travelling disturbances were first observed in the chromosphere in H α and were known as Moreton waves (Moreton 1960), which propagate from the eruptive source region at speeds of around 1000 km s−1. Moreton waves usually appear as arc-shaped bright fronts confined to an angular extent of about 90° and they travel distances up to 3 × 105 km (Warmuth et al. 2004a). EUV waves are usually observed as moving fronts of increased EUV emission that travel quasi-radially away from the source regions (Thompson et al. 1998; Thompson & Myers 2009; Liu et al. 2010; Long, DeLuca & Gallagher 2011; Warmuth 2011; Shen & Liu 2012; Xue et al. 2013; Nitta et al. 2014; Shen, Song & Liu 2018). These disturbances can propagate beyond one solar radius. Their speed ranges from 50–700 km s−1 with a typical value between 200 and 400 km s−1. Usually, fast EUV waves manifest a large amplitude at the wave front; slow EUV waves have smaller amplitudes. Furthermore, fast EUV waves slow down during their propagation, and their amplitudes decrease as well. When the amplitude becomes small, the non-linear wave/shock degenerates into a linear wave. The shape and angular extent of the EUV wave fronts exhibit a large variety from case to case (Thompson & Myers 2009). Flares and coronal mass ejections (CMEs) are typical eruptive processes producing various disturbances in the solar atmosphere. These eruptive phenomena are closely related and manifest an underlying impulsive reconfiguration of a non-potential magnetic structure. The Moreton wave is always associated with flares (primarily M- and X-class) and type II radio bursts (Smith & Harvey 1971; Warmuth 2010). Studying 60 strong large-scale EUV wave events observed by the STEREO twin spacecrafts, Muhr et al. (2014) found the following association rates of EUV waves to other eruptive phenomena: |$95{{\ \rm per\ cent}}$| associated with a CME, |$74{{\ \rm per\ cent}}$| with a flare and |$22{{\ \rm per\ cent}}$| with type II bursts. Nitta et al. (2013) studied the association of flares, CMEs and type II bursts by investigating 138 global EUV disturbances in AIA |$193 \, \mathring{\rm A}$| images during 2010 April–2013 January. They found that only |$65{{\ \rm per\ cent}}$| of disturbances were associated with traceable CMEs beyond five solar radii, |$54{{\ \rm per\ cent}}$| associated with a type II radio burst and |$85{{\ \rm per\ cent}}$| associated with ≥C-class flare. Long et al. (2017) performed a statistical study on coronal disturbances in AIA |$193 \, \mathring{\rm A}$|⁠, in which 164 of the 362 solar flare events (45 per cent) have associated global disturbances with no disturbances found for the remaining 198 (55 per cent). Three major categories of models have been proposed for the coronal disturbance (Liu & Ofman 2014; Warmuth 2015), namely MHD wave/shock models (Uchida 1968, 1970; Vršnak & Lulić 2000; Warmuth et al. 2004b; Vršnak & Cliver 2008; Wang, Shen & Lin 2009; Downs et al. 2011; Mei, Udo & Lin 2012a; Selwa, Poedts & DeVore 2012; Wang et al. 2015), pseudo-waves or reconfiguration of magnetic structure (Delannée & Aulanier 1999; Chen et al. 2002; Delannée et al. 2008), as well as hybrid models involving both scenarios (Chen, Ding & Fang 2005; Downs et al. 2011). The fast-mode wave/shock models of EUV disturbances (Thompson et al. 1999; Vršnak & Lulić 2000; Wang 2000; Wu et al. 2001; Ofman & Thompson 2002; Vršnak & Cliver 2008; Warmuth & Mann 2011) postulate that the coronal disturbance is triggered by the piston-driven shock invoked by the fast-moving and expanding CME bubble. Selwa et al. (2012), Selwa, Poedts & DeVore (2013) have shown that the fast wave/shock front can also be seen in rotating active regions in their 3D numerical simulations. In addition, slow-mode shocks and velocity vortices at the flanks of CME bubbles (Wang et al. 2009; Mei et al. 2012a), slow-mode solitons (Wills-Davey, DeForest & Stenflo 2007) and magnetoacoustic surface gravity waves (Ballai, Forgács-Dajka & Douglas 2011) are also possible interpretations of EUV disturbances. The pseudo-wave model suggests that coronal waves are not true waves but signatures of the magnetic structure reconfiguration, such as successive stretching of magnetic field lines (Chen et al. 2002), a current shell surrounding the eruptive magnetic flux rope (MFR) (Delannée et al. 2008; Schrijver et al. 2011) and a reconnecting front between the laterally expanding eruptive magnetic field and the ambient field (Attrill et al. 2007, 2009). A long-term debate exists between the wave and the non-wave interpretations. In recent years, a converged view from hybrid models has been confirmed in recent observations showing bimodality characteristics. High-cadence EUV imaging observations with STEREO/EUVI (Kaiser et al. 2008) and SDO/AIA (Lemen et al. 2012) showed that EUV waves have both wave and non-wave structures (Chen et al. 2002; Zhukov & Auchère 2004). Many coronal EUV disturbances clearly show a fast wave-like bright front and a trailing intensity enhancement that is associated with an eruptive CME (Patsourakos et al. 2009; Ma et al. 2011; Guo, Ding & Chen 2015; Fulara et al. 2019). Dependent on the characteristics of the eruptive magnetic structure and the surrounding corona, the fast-mode wave/shocks or non-wave magnetic structure reconfiguration could dominate the observational signatures (Warmuth 2015). A fast eruption will generate a large-amplitude fast-mode shock. Conversely, a slow or gradual eruption may only launch the non-wave disturbances associated with reconfiguration of eruptive structure. This unified view for coronal disturbances has been supported by numerical simulations. Chen et al. (2002, 2005) identified two distinct disturbances in their 2D MHD numerical simulations and proposed hybrid models that incorporate both wave and non-wave components. This model captures the bimodality nature where two fronts of coronal waves consist of a leading fast-mode wave/shock and a slower trailing disturbance caused by the expanding CME bubble. Subsequent 3D global MHD simulations also confirmed the bimodality of EUV disturbances (Cohen et al. 2009; Downs et al. 2011, 2012; Selwa et al. 2013). Downs et al. (2012) found that the fast-mode wave is responsible for the EUV wave front, and the inner front or the CME flank is mainly a consequence of the expanding CME bubble. To study various types of disturbances, we perform a 3D realistic high-resolution MHD numerical experiment of an eruptive MFR based on the Titov & Démoulin (1999) model (TD99 hereafter), with gravity stratification in the atmosphere and thermal conduction, which are important ingredients for obtaining more realistic coronal variations. We employ forward modelling of the EUV observables to compare our numerical results with realistic EUV observations. In Section 2, our numerical experiment, the TD99 model for the initial magnetic structure, the background atmosphere and the boundary conditions are described. In Section 3, basic results of our numerical experiment are presented. In Section 4, the observational characteristics of coronal disturbances are given. In Section 5, we discuss the slow shock, echo and plasma flow structure in detail. Finally, our conclusions are given in the last section. 2 NUMERICAL METHOD AND MODEL DESCRIPTION The governing MHD equations for this numerical simulation are written as $$\begin{eqnarray} \partial _{t}\rho &+&\nabla \cdot (\rho \mathbf {v})=0, \end{eqnarray}$$ (1) $$\begin{eqnarray} &&{\partial _{t}e + \nabla \cdot \left[\left(e+p+\frac{1}{2} \left|\mathbf {B}\right|^{2}\right)- (\mathbf {v}\cdot \mathbf {B})\mathbf {B}\right]}\\ &&{\quad =\rho \mathbf {g} \cdot \mathbf {v}+\nabla \cdot (\kappa \cdot \nabla T),} \end{eqnarray}$$ (2) $$\begin{eqnarray} \partial _{t}(\rho \mathbf {v})+\nabla \cdot \left[\rho \mathbf {v} \mathbf {v}+\left(p+\frac{1}{2}\left|\mathbf {B} \right|^{2}\right)\mathbf {I}-\mathbf {B}\mathbf {B}\right]=\rho \mathbf {g}, \end{eqnarray}$$ (3) $$\begin{eqnarray} \partial _{t}\mathbf {B}&-&\nabla \times (\mathbf {v}\times \mathbf {B})=0. \end{eqnarray}$$ (4) Here, ρ, ρv, B and e = p/(γ − 1) + (1/2)ρv2 + |B|2/2 are gas density, gas momentum, magnetic field, and total energy density, respectively. Note that we include gravity and anisotropic thermal conduction, both relevant for the coronal environment that we simulate. The gravity ρg is along the z-direction, and |${\bf g} =\mathrm{g}_{\odot } \hat{z} /(1+z/\mathrm{R}_{\odot })^{2}$| in our set-up. Here, g⊙ = −2.742 × 10−4 cm s−1 is the downwards gravity acceleration at z = zp where zp = 0.4 (in our units) sets the height of the photospheric surface. The heat conduction is purely field aligned and temperature dependent, hence |$\kappa =\kappa _{\parallel }~T^{2.5}~\widehat{e}_B\widehat{e}_B$|⁠. The unit vector |$\widehat{e}_B=\vec{B}/|B|$| and κ∥ = 10−6 erg s−1 cm−1 K−3.5. These equations are complemented by the equation of state for an ideal gas. The adiabatic index γ = 5/3. These equations are solved by the MPI-parallelized adaptive mesh refinement code (MPI-AMRVAC) (Keppens et al. 2012; Porth et al. 2014; Xia et al. 2018). Various numerical schemes are implemented in this code to advance generic hyperbolic partial differential equations. We adopt a high-order accurate shock-capturing finite-volume spatial discretization, using a Harten–Lax–van Leer approximate Riemann solver (Harten 1983), a third-order limited reconstruction (Čada & Torrilhon 2009) and a three-step Runge–Kutta time-marching method. The initial magnetic configuration is based on the TD99 model, used by many authors to construct initial magnetic structures for solar eruptions (Roussev et al. 2003; Török, Kliem & Titov 2004; Török & Kliem 2005; Mei et al. 2017, 2018). As shown in Fig. 1, the whole magnetic structure consists of three parts. The first is an MFR, with major radius R, minor radius a and total toroidal current I, uniformly distributed over its circular cross-section. The second one is the background field resulting from a pair of magnetic sources ±q separated by a distance L, lying on the symmetry axis of the MFR, buried at z = d under the photospheric surface z = zp. The third ingredient is created by a dipole located at z = dd on the z-axis with strength qd and axis direction along the y-axis, unlike the line current in the original TD99 model. The vector magnetic potential of this ingredient is $$\begin{eqnarray} \vec{A_d}=\frac{q_{d}}{(x^2+y^2+(z+d_d)^2)^{1.5}} \left({\begin{array}{*{10}c}z+d_d \\ 0 \\ -x \end{array}}\right) . \end{eqnarray}$$ (5) The purpose of this component is to control the twist of magnetic field lines inside the MFR. However, the side effect of the line current in the original TD99 model is that it causes the background magnetic field to decay with height linearly, not with the square of the height. According to torus instability (Kliem & Török 2006), this favours the upward motion of the eruptive MFR. Instead of the line current, the dipole as the third component does not have this side effect. Figure 1. Open in new tabDownload slide Initial magnetic structure (coloured curves: golden for magnetic field lines inside the MFR and blue for the surrounding); the distribution of the normal magnetic field (red–green shading) on the photosphere z = zp and the distribution of density, which is gravitationally stratified (green shading), on the plane y = −2. Figure 1. Open in new tabDownload slide Initial magnetic structure (coloured curves: golden for magnetic field lines inside the MFR and blue for the surrounding); the distribution of the normal magnetic field (red–green shading) on the photosphere z = zp and the distribution of density, which is gravitationally stratified (green shading), on the plane y = −2. The MFR equilibrium can be decomposed into internal and external force equilibria under the thin MFR assumption a ≪ R (Isenberg, Forbes & Demoulin 1993; Lin et al. 1998; Lin, van Ballegooijen & Forbes 2002; Mei & Lin 2008). The external equilibrium is realized by a balance between an outward self-inducted force due to the MFR curvature and a confining force from the magnetic sources. Vanishing of the net force on the MFR gives a relation between the total toroidal current I inside the MFR and the critical strength ±qc of the magnetic sources: $$\begin{eqnarray} I=\frac{8\pi q_\mathrm{c} L R (R^2+L^2)^{-3/2}}{\ln (8R/a)-5/4}. \end{eqnarray}$$ (6) Considering the fact that the main purpose of this work is to study eruption-related disturbances in the lower layers, we rather start our simulation at a non-equilibrium state of the system with q = 0.8qc, such that the eruption immediately commences at the beginning of our simulation. For the internal equilibrium of the MFR, the inward force from the poloidal magnetic field towards the toroidal axis of the MFR is balanced by an outward pressure (Mei et al. 2018), which includes thermal pressure of the internal plasma and magnetic pressure of the toroidal field. The internal distribution of the plasma pressure is $$\begin{eqnarray} p = p_{m}+(1-\alpha)\frac{I^2}{4\pi ^2 a^2}\left(1-\frac{r_{a}^2}{a^2}\right), ~r_{a}\le a. \end{eqnarray}$$ (7) Here, $$\begin{eqnarray} r_{a}=\sqrt{x^2+(r_{\perp }-R)^2}, \end{eqnarray}$$ (8) $$\begin{eqnarray} r_{\perp }=\sqrt{y^2+(z-d)^2}, \end{eqnarray}$$ (9) and α is a parameter that controls the contribution from the thermal and magnetic pressure on the internal equilibrium of the MFR. The constant pm is set to the initial pressure of the background atmosphere at z = R − d, the highest point of the magnetic axis of the MFR. The toroidal field is written as $$\begin{eqnarray} B_\mathrm{t}&=& \frac{I_0}{2\pi }\sqrt{\frac{2 \alpha I^2}{a^2 I_0^2}\left(1-\frac{r_{a}^2}{a^2}\right)+\frac{1}{R^2}},~r_{a}\le a, \\ &=& \frac{I_0}{2\pi r_{\perp }},~r_{a}\gt a. \end{eqnarray}$$ (10) Here, I0 is the strength of the line current from the original TD99 model. In this experiment, we used the dipole located at (0, 0, dd) instead of the line current at (0, 0, d) to control the twist inside the MFR. However, we can still use this formula to calculate the toroidal field Bt. We only need to control the parameters dd and qd so that the magnetic field of this dipole near the MFR axis is approximately the same as the magnetic field of the alternative line current near the axis. That is, two conditions should be satisfied. First, the magnetic field in the vicinity of the MFR generated by qd should run along the toroidal direction of the MFR approximately. Second, the magnetic strength at the surface of the MFR given by qd equals that given by the line current I0. More detailed formulae for the poloidal field of the TD99 model are found in TD99 and extended with internal pressure variations in Mei et al. (2018). The computational domain is a 3D uniform box of size −10 ≤ x ≤ 10, −10 ≤ y ≤ 10 and 0 ≤ z ≤ 20 in a Cartesian coordinate system, and 320 × 320 × 320 grid points are employed. The relevant parameters for the TD99 model are set to R = 1.2, a = 0.2, d = −0.1, dd = −1.2, L = 0.32 and qc = 38.0. For convenience, all physical quantities are normalized to 5 × 109 cm, 1.165 × 107 cm s−1, 4.293 × 102 s, 2.342 × 10−14 g cm−3, 3.175 Pa and 6.317 G for length, velocity, time, density, pressure and magnetic field, respectively. In addition, we set α = 0.997 and the uniform plasma temperature inside the MFR Tin = 0.04 to control the total mass inside the MFR, which affects the acceleration of the MFR during the eruption. Thus, the maximum density inside the MFR is 35 (in our units), and the total mass inside the MFR is about 1.62 × 1016 g. We use a two-layered gravity stratification, with z ≤ zp and z > zp representing the photosphere and the extended corona, respectively. The dimensionless value of the initial temperature is Tp = 0.006 below the photosphere and Tc = 1 above the photosphere so that we obtain a layer with high density and a sharp increase in pressure from z = zp to the bottom of the simulation box. The initial pressure distribution in our atmosphere is then given by $$\begin{eqnarray} p&=&p_c~ \text{exp}\left[\frac{1}{\lambda _c}\frac{z_p-z}{(1+z/\mathrm{r}_\odot)(1+z_p/\mathrm{r}_\odot)}\right],~~z\lt =z_p \\ &=&p_c~\text{exp}\left[\frac{1}{\lambda _p}\frac{z}{1+z/\mathrm{r}_\odot }\right],~~z\gt z_p \end{eqnarray}$$ (11) and the corresponding density distribution is $$\begin{eqnarray} \rho &=& p/T_p,~z\lt =z_p \\ &=&p/T_c,~z\gt z_p \end{eqnarray}$$ (12) where $$\begin{eqnarray} \lambda _c=\frac{k_\mathrm{B} T_c}{m_p \mathrm{g}_\odot } \end{eqnarray}$$ (13) and $$\begin{eqnarray} \lambda _p=\frac{k_\mathrm{B} T_p}{m_p \mathrm{g}_\odot } \end{eqnarray}$$ (14) are the scale heights of the corona and the rigid atmosphere under our photosphere. The dimensionless value of plasma pressure of corona pc at z = zp has been set to 1. We used the Boltzmann constant kB = 1.38 × 10−16 erg k−1, the proton mass mp = 1.67 × 10−24 g, the solar radius r⊙ = 6.95 × 1010 cm and the gravity constant on the solar surface g⊙ = 2.74 × 104 cm s−1. In the artificial dense layer below, the plasma β is much larger than unity. This allows us to keep all plasma parameters in the ghost zone underneath this layer fixed, since any change in the magnetic field and atmosphere above the photosphere cannot impact our bottom boundary. In this way, we realize a line-tied environment in the bottom region that includes a high-density layer such that the foot-points of magnetic field lines are anchored into the photosphere. We notice here that the chromosphere and the transition region have not been included in our setting, because they would require us to resolve large temporal and spatial variations in physical quantities and so need unaffordable computational costs for us. Furthermore, according to our previous practices (Wang et al. 2009, 2015; Mei et al. 2012a; Xie et al. 2019), the two-layer atmosphere is actually enough for our physical process, the large-scale EUV disturbance and its corresponding observational characteristics in SDO/AIA images. For all other five boundaries, we can use open boundary conditions, in which ghost cell values of all physical variables are deduced via extrapolation of internal cells. 3 BASIC RESULTS OF THE NUMERICAL EXPERIMENT The MFR starts to rise immediately after the launch of our experiment. After 3.5 min acceleration of 1.57 km s−2, the MFR moves upward with an almost constant speed of 336 km s−1. The consequent evolution of the magnetic structure (coloured curves) and the reconnecting current sheet (CS, illustrated by showing a pink isosurface of the plasma temperature) are given in Fig. 2. With the MFR take-off, the overlying magnetic field lines become stretched, and a 3D CS forms under the MFR (cf. Forbes & Priest 1995; Lin & Forbes 2000; Mei et al. 2012a, 2017). Magnetic reconnection then takes place inside the CS and heats the plasma in the CS up to 5 MK. Ahead of the MFR, the plasma following the fast shock (FS) front with dome shape (Selwa et al. 2012, 2013) has also been heated compressionally up to 4 MK. Figure 2. Open in new tabDownload slide Evolution snapshots of the eruption process at t = 1, 2, 3 and 4 respectively. The dimensionless units of time and length are 4.293 × 102 s and 5 × 109 cm. Red lines show the boundary of the CME bubble; golden curves are magnetic field lines of the MFR; blue curves show the ambient background magnetic field. The red–green shading is the distribution of the normal magnetic field on the photosphere z = zp, as in Fig. 1. The pink temperature isosurface reaches T = 3 MK and marks the reconnecting current sheet and the dome front of the fast shock (FS). Figure 2. Open in new tabDownload slide Evolution snapshots of the eruption process at t = 1, 2, 3 and 4 respectively. The dimensionless units of time and length are 4.293 × 102 s and 5 × 109 cm. Red lines show the boundary of the CME bubble; golden curves are magnetic field lines of the MFR; blue curves show the ambient background magnetic field. The red–green shading is the distribution of the normal magnetic field on the photosphere z = zp, as in Fig. 1. The pink temperature isosurface reaches T = 3 MK and marks the reconnecting current sheet and the dome front of the fast shock (FS). During the eruption, the upward reconnection outflow and the newly formed magnetic field lines continue to become the outermost part of the CME bubble, connecting to the upper edge of the 3D CS below the MFR. In the following, we refer to the outermost magnetic field line of the CME bubble as its ‘boundary’, which separates the magnetic system inside the bubble from the outside. The CME bubble is larger and envelops the MFR, which fills with twisting magnetic field lines and has a toroidal current along its axis. The toroidal current governs the twist of the magnetic field lines inside the MFR and the MFR boundary is a surface where the internal toroidal current disappears. As shown in Fig. 2, the MFR is embedded in the centre of the CME bubble. Significant expansion of the bubble can be seen during the eruption, which is mainly due to the continuous process of newly formed circular magnetic field lines as a result of magnetic reconnection. In addition, the MFR also experiences continuous expansion, as shown by the golden curves in Fig. 2 (cf. Isenberg et al. 1993; Wang et al. 2009, 2015; Mei et al. 2017). In order to demonstrate the various types of disturbances during the eruption, Fig. 3 gives the distributions of temperature T, density ρ, velocity divergence |$\triangledown \cdot \vec{v}$| and velocity curl |$\triangledown \times \vec{v}$| on cut y = 0 at t = 3. For the distributions of velocity divergence and curl, we follow Forbes (1990) and Wang et al. (2009) to identify locations of the FS and the slow-mode shock (SS) respectively. The divergence of plasma velocity reflects how much plasma is compressed due to the propagating fast shock. The curl of plasma velocity gives more information about the SS, in which disturbances in the tangential velocity are expected. For the SS, the normal velocity across it in the switch-off limit can at most change by a factor γ = 5/3 in the limit of zero β (Forbes 1990). However, the tangential velocity is capable of undergoing an enormous change. Therefore, the velocity vorticity |$\triangledown \times \vec{v}$| reveals more information about the slow shock. Figure 3. Open in new tabDownload slide Distributions of temperature T, density ρ, velocity divergence |$\triangledown \cdot \vec{v}$| and velocity curl |$\left|\triangledown \times \vec{v}\right|$| on cut y = 0 at time t = 3. The fast shock (FS), the slow shock (SS) and the echo are marked on the panels. Figure 3. Open in new tabDownload slide Distributions of temperature T, density ρ, velocity divergence |$\triangledown \cdot \vec{v}$| and velocity curl |$\left|\triangledown \times \vec{v}\right|$| on cut y = 0 at time t = 3. The fast shock (FS), the slow shock (SS) and the echo are marked on the panels. As a dominant feature in Fig. 3, the FS propagates outward and appears as a dome overlying the CME bubble. It expands sidewards and reaches the photosphere, which leads to the appearance of disturbances in the lower atmosphere. Ahead of the FS front, the plasma is undisturbed and keeps the gravity stratification that we started from: an equilibrium with constant temperature T = 1. After the FS front, the plasma has higher density and higher pressure than the unshocked ambient plasma, which is the typical characteristic of an FS. Because the plasma following the FS moves outward continuously and the plasma ahead of the FS front cannot enter the dome, the outward propagation of the dome yields an adiabatic expansion of the volume of our eruptive source region. Thus, a dimming area with lower density and temperature than the ambient atmosphere can be recognized (Wang et al. 2009, 2015). In addition, we can also see the SS at the flanks of the CME bubble and the shock echoes in the bottom of the corona in Fig. 3. These details have already been shown and are well discussed in the previous 2D MHD simulations of Wang et al. (2009, 2015), Mei et al. (2012a) and Xie et al. (2019). Wang et al. (2009) noticed that the SS and the velocity vortex excited by the upward-moving MFR may be responsible for the formation of EUV waves. Wang et al. (2015) and Xie et al. (2019) further proposed that there exist various disturbances, such as a secondary wave or a pile-up, which may also be the source of EUV waves. In the following sections, we revisit these disturbances by translating our 3D numerical result to realistic observations. We also link the formation of the SS and the echoes to details that we find in the 3D plasma flows. 4 OBSERVATIONAL CHARACTERISTICS OF DISTURBANCES In order to make a direct comparison between our numerical results and realistic observations, we perform forward modelling of SDO/AIA observables by fomo (the forward modelling code, Van Doorsselaere et al. 2016), developed to calculate EUV emission from optically thin typically coronal plasma. The plasma density and temperature from our experiment are translated to EUV emissivity using data (Dere et al. 1997; Landi et al. 2013; Van Doorsselaere, Antolin & Karampelas 2018). Then, emissivity is integrated along the line-of-sight (LOS) to generate synthetic SDO/AIA imaging observations. The coordinate system x–y–z in our numerical experiment and x′–y′–z′ in the rotated frame of reference of the observation are connected by two rotations. In particular, a first angle |$\mathcal {L}$| is between the LOS and the z-axis of our calculation box, and a second |$-\mathcal {B}$| is between the LOS and the y-axis of the calculation box, and they are connected as $$\begin{eqnarray} {\begin{pmatrix}x^{\prime } \\ y^{\prime } \\ z^{\prime } \end{pmatrix}}= {\begin{pmatrix}\cos\mathcal {B} &\quad 0 &\quad -\sin\mathcal {B}\\ 0 &\quad 1 &\quad 0\\ \sin\mathcal {B} &\quad 0 &\quad \cos\mathcal {B} \end{pmatrix}}\! {\begin{pmatrix}\cos\mathcal {L} &\quad -\sin\mathcal {L} &\quad 0\\ \sin\mathcal {L} &\quad \cos\mathcal {L} &\quad 0\\ 0 &\quad 0 &\quad 1 \end{pmatrix}}\! {\begin{pmatrix}x \\ y \\ z \end{pmatrix}}. \\ \end{eqnarray}$$ (15) Here x′–y′ are the coordinates in the image plane (also known as the plane-of-the-sky) and z′ is the direction along the LOS. The synthetic images of SDO/AIA |$193 \, \mathring{\rm A}$| in the image plane at different times during the eruption, viewed with |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=45^\circ$|⁠, are shown in Fig. 4 to demonstrate the EUV wave observed at the limb of the solar disc. Similarly, viewed with |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=86^\circ$| we obtain Fig. 5 to show the EUV wave observed at the centre of the solar disc. Synthetic images of SDO/AIA at other wavelengths |$171\, \mathring{\rm A}$| and |$131\, \mathring{\rm A}$| with |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=86^\circ$| are given in Fig. 6 for comparison. In all panels, both the propagation of the FS dome and the expansion of the bright MFR can be seen. Unlike Fig. 4, the FS front in Fig. 6 appears as a dark dome instead of a bright one, because of the shape of the contribution function for its passbands |$171\, \mathring{\rm A}$| and |$131\, \mathring{\rm A}$| (Lemen et al. 2012). In the beginning, plasma inside the MFR has been heated to a temperature of T = 4 MK, so that the initial dark MFR appears bright in the synthetic AIA |$193\, \mathring{\rm A}$| and |$131\, \mathring{\rm A}$| images (see panels at t = 0 and 1 in Figs 4 and 5). The magnetic field inside the MFR undergoes a significant untwisting and magnetic reconnection operates inside the MFR because of the highly twisted internal magnetic field lines (Mei et al. 2018). Part of the magnetic energy gets converted into plasma thermal energy inside the MFR, and so the EUV emission inside the MFR is enhanced. According to Fig. 2, the MFR is different from the CME bubble. The MFR with a clear toroidal current along its axis is only the central part of the CME bubble. However, we cannot distinguish the bubble from the bright MFR and identify the true boundary of our CME bubble in synthetic images. As the eruption process continues, the heated MFR experiences a significant expansion and also becomes stretched. The increase of total MFR volume leads to a dramatic decrease of its internal temperature and so the MFR cannot be distinguished easily from the ambient region in the panels of Figs 4–6 after t = 2. Figure 4. Open in new tabDownload slide Synthetic AIA |$193 \, \mathring{\rm A}$| snapshots in the image plane (also known as the plane-of-the-sky) at different times during the eruption. The viewing angles are |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=86^\circ$|⁠. The magnetic flux rope (MFR), the FS and the echo are marked on the panels. Figure 4. Open in new tabDownload slide Synthetic AIA |$193 \, \mathring{\rm A}$| snapshots in the image plane (also known as the plane-of-the-sky) at different times during the eruption. The viewing angles are |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=86^\circ$|⁠. The magnetic flux rope (MFR), the FS and the echo are marked on the panels. Figure 5. Open in new tabDownload slide The same as Fig. 4, but with |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=45^\circ$|⁠. Figure 5. Open in new tabDownload slide The same as Fig. 4, but with |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=45^\circ$|⁠. Figure 6. Open in new tabDownload slide Synthetic images of AIA |$171 \, \mathring{\rm A}$| (left-hand column) and |$131 \, \mathring{\rm A}$| (right-hand column) with |$\mathcal {L}=-45^\circ$|⁠, |$\mathcal {B}=86^\circ$|⁠. Figure 6. Open in new tabDownload slide Synthetic images of AIA |$171 \, \mathring{\rm A}$| (left-hand column) and |$131 \, \mathring{\rm A}$| (right-hand column) with |$\mathcal {L}=-45^\circ$|⁠, |$\mathcal {B}=86^\circ$|⁠. Ahead of the MFR, the FS front can be seen clearly as a bright 3D dome in the AIA images, as already shown in Fig. 3. Following the FS front, the temperature and density increase significantly so the EUV emission is enhanced. In the early stage of eruption, the heated MFR can be identified as following the FS front in Figs 4 and 6. The FS and the bright MFR together constitute the typical bimodality interpretation of the EUV disturbance, i.e. the FS is a true wave component of our EUV disturbance and the MFR is the non-wave component. This scenario has been suggested by Chen et al. (2002) and Zhukov & Auchère (2004) and is supported by lots of STEREO/EUVI and SDO/AIA observations (Patsourakos et al. 2009; Ma et al. 2011; Guo et al. 2015; Fulara et al. 2019) and 3D numerical simulations (Cohen et al. 2009; Downs et al. 2011, 2012). In our numerical experiment, the CME leading edge in the synthetic observation of EUV waves corresponds to the outside edge of the MFR in Fig. 4. However, as indicated by Fig. 3, the CME bubble boundary and the MFR are close to each other, and we cannot distinguish the bubble boundary from the outside edge of the MFR. In realistic eruptive events, actual morphologies of the MFR and the CME bubble may be more diverse, and both of them may contribute to the detailed appearance of the leading edge of the CME bubble. The panels in Fig. 7 show the propagation of the EUV disturbances in AIA synthetic images for a limb event. In the upper-left-hand panel, the AIA |$193\, \mathring{\rm A}$| image with angles |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=90^\circ$| is given. Along line L1 as marked on this panel, the changes of EUV emission versus time are plotted in the upper-right-hand panel. According to equation (15), the dimensionless coordinates in the upper-left-hand panel of Fig. 7 are equal to dimensionless units of length 5 × 104 km in our experiment. The propagation speeds of the FS and the MFR are almost the same, about 1.65 (i.e. 192 km s−1). Because of the expansion and stretching of the MFR, the temperature of the bright MFR decreases quickly so its EUV emission decreases and becomes invisible in EUV channels. For comparison, the panels in the lower row of Fig. 7 show fig. 1 of Ma et al. (2011), which presents a typical EUV disturbance with bimodality structure. This event was observed at 05:40:08 ut on 2010 June 13 and occurred on the limb of the solar disc. Figure 7. Open in new tabDownload slide Upper left: synthetic images of AIA |$193\, \mathring{\rm A}$| with angles |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=90^\circ$| at time t = 1. Upper right: changes of EUV emission with time along line L1 marked in the upper-left-hand panel. The horizontal axis is the distance from the left endpoint of the line L1. Lower row: morphology of the CME and the FS front in a realistic eruptive event that occurred at 05:40:08 ut on 2010 June 13 (Ma et al. 2011), which was observed on the limb of the solar disc by SDO/AIA |$193\, \mathring{\rm A}$|⁠. The original image performed low-pass filtering (lower-left-hand panel) and high-pass filtering (lower-right-hand panel) to highlight both structures. Figure 7. Open in new tabDownload slide Upper left: synthetic images of AIA |$193\, \mathring{\rm A}$| with angles |$\mathcal {L}=-45^\circ$| and |$\mathcal {B}=90^\circ$| at time t = 1. Upper right: changes of EUV emission with time along line L1 marked in the upper-left-hand panel. The horizontal axis is the distance from the left endpoint of the line L1. Lower row: morphology of the CME and the FS front in a realistic eruptive event that occurred at 05:40:08 ut on 2010 June 13 (Ma et al. 2011), which was observed on the limb of the solar disc by SDO/AIA |$193\, \mathring{\rm A}$|⁠. The original image performed low-pass filtering (lower-left-hand panel) and high-pass filtering (lower-right-hand panel) to highlight both structures. The maximum distance away from the centre of the eruptive source region where the bright MFR can be distinguished depends on several factors, such as its geometrical characteristics and strength of the magnetic structure, the plasma contrast between inside and outside the MFR, and the viewing angles. In our experiment, the parameters of the initial magnetic structure and atmosphere are close to a realistic coronal environment. The resultant disappearance time of the MFR in our images is about t = 9 min. The corresponding maximum distances that the leading edge of the bright MFR can reach are about 5.5 × 104 km in the solar disc and 1.75 × 105 km along the direction perpendicular to the solar disc. If the angle |$\mathcal {L}=0^\circ$|⁠, the maximum distance in the horizontal direction can be 8 × 104 km. This means that the leading edge of the MFR in our experiment can only be observed less than |$0.2\, \mathrm{R}_\odot$| away from its source region. In realistic observations, the EUV disturbances have been observed to cover distances 350–850 Mm (⁠|$0.5\!-\!1.3 \, \mathrm{R}_\odot$|⁠), with an average value of about 500 Mm (⁠|$0.7 \, \mathrm{R}_\odot$|⁠) (Thompson & Myers 2009; Patsourakos & Vourlidas 2012; Warmuth 2015). Nevertheless, we can speculate that a scaling law generalizes our numerical MHD result, which is in essence free of dimensions. If the characteristic length of the initial MFR is thus enlarged by a factor of three, the bright MFR would be observable |$0.6\, \mathrm{R}_\odot$| away from the eruptive source region. Unlike Fig. 4, Fig. 5 shows synthetic images of AIA |$193\, \mathring{\rm A}$| in the centre of the solar disc. In this almost-top view, the bright MFR has a highly anisotropic geometrical shape so that its morphology is not like the non-wave component of EUV disturbances on the solar disc, which usually has a dispersive and quasi-circular geometrical shape. Instead of the bimodality structure in Fig. 4, only the FS can be seen to propagate outward in Fig. 5. Fig. 8 gives the propagation of EUV disturbances along different lines marked on its top-left-hand panel. The kinematic behaviours along L1 and L2 are significantly different, which means that the global kinematics of the disturbances will depend on the angle between the axis of the erupting MFR and the direction used to estimate the kinematics. The disturbances along L1 and L2 propagate outward with an average speed of about 1.9, but with weak acceleration and deceleration respectively. Figure 8. Open in new tabDownload slide Upper row: synthetic images of AIA |$193\, \mathring{\rm A}$| observed from the top (⁠|$\mathcal {L}=-45^\circ$|⁠, |$\mathcal {B}=0^\circ$|⁠) for t = 0.88 and 3.96. Lower row: changes of the EUV emission with time during the eruption along L1 and L2 as marked on the upper-left-hand panel. The horizontal axis is the distance from the left endpoint of the line L1 and L2. Figure 8. Open in new tabDownload slide Upper row: synthetic images of AIA |$193\, \mathring{\rm A}$| observed from the top (⁠|$\mathcal {L}=-45^\circ$|⁠, |$\mathcal {B}=0^\circ$|⁠) for t = 0.88 and 3.96. Lower row: changes of the EUV emission with time during the eruption along L1 and L2 as marked on the upper-left-hand panel. The horizontal axis is the distance from the left endpoint of the line L1 and L2. The FS can be seen to propagate outward with a velocity of 1.9 along L1 and L2. Following the FS, a dimming region expands outward continuously. When the distance of the FS away from the centre of the eruptive source region exceeds about |$0.5\, \mathrm{R}_\odot$|⁠, the atmosphere near the centre of the source region begins to recover from its lower-temperature and low-density state, and EUV emission near the centre of the eruptive source becomes stronger than the dimming region, as shown in Fig. 5. In addition, it is difficult to identify the SS and the echoes in these synthetic AIA images of Fig. 5. In Fig. 6, the echoes can be seen, which means that the echoes are more likely to be observed on the limb of the solar disc rather than on the solar disc. Thus, for the EUV disturbances observed on the solar disc, our experiment does not support the non-wave features. It is more difficult to identify features consistent with a bimodal interpretation for these disturbances when observing the disturbance on-disc versus on the limb. 5 VELOCITY SEPARATRIX, SLOW SHOCKS AND ECHOES The slow shocks (SS) and the echoes have not (yet) been observed, except for a few cases (Liu et al. 2012; Liu & Ofman 2014), presumably because realistic imaging observations cannot provide high enough temporal (several seconds) and spatial resolution (less than 100 km) for the 2D or even 3D distribution of velocity components (Fisher & Welsch 2008; McKenzie 2013; Tian et al. 2013; Tremblay et al. 2018), which are necessary to confirm these discontinuities’ physical nature, for example, by checking the Rankine–Hugoniot relations (Shiota et al. 2005; Priest 2014). However, echoes and SS are very common phenomena in numerical experiments (Wang et al. 2009, 2015; Mei et al. 2012a; Xie et al. 2019). In this section, we further discuss SS and echoes by investigating the associated plasma velocity field. Fig. 9 gives disturbances of |$\triangledown \times \vec{v}$|⁠, density ρ and temperature T along a line z = 1.2 on the cut x = 0 at different times. We can see the echo, the SS and other disturbances after the SS rather clearly. After a short-term acceleration, the SS propagates outward at a speed of about 105.0 km s−1 in the early stage. Later, its speed continues to decrease slowly, and eventually it spreads outward with a uniform speed of about 58.3 km s−1. Figure 9. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (a), plasma density ρ (b) and temperature T (c) along the line z = 1.2 on the cut x = 0 at different times during the eruption. Figure 9. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (a), plasma density ρ (b) and temperature T (c) along the line z = 1.2 on the cut x = 0 at different times during the eruption. Compared to the FS, an important feature of the SS is that it causes significant changes in the velocity field, instead of the density and temperature distributions, as shown by Fig. 9. Thus, it is hard to identify in realistic EUV images, which depend on the plasma density or temperature distributions. We present in Figs 10 and 11 the clear relationship between the plasma velocity field and the SS, suggestive for future high-time-resolution observations and reliable velocity-field observations. The arrows in Fig. 10 show the velocity distribution at t = 2. Above the CME bubble, the plasma follows the FS front and moves outward, which leads to an enhancement of the density and the temperature in the region after the FS and a dimming region around the CME bubble. On the flanks of the CME bubble, the velocity field shows a much more complex pattern. Velocity vortices appear in the flanks of the CME bubble; these are marked on the right-hand panel of Fig. 10. Part of the plasma moves away from the CME bubble and follows the FS front while another part of the plasma moves toward the 3D reconnecting CS and the CME bubble. Figure 10. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (coloured shading) and velocity (3D yellow arrows) on the cut y = 0 at t = 2. The red curve shows the boundary of the CME bubble. The right inset zooms into the chosen region (black box) to show more details around the vicinity of the SS. The pink curve on this inset marks the velocity separatrix. Figure 10. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (coloured shading) and velocity (3D yellow arrows) on the cut y = 0 at t = 2. The red curve shows the boundary of the CME bubble. The right inset zooms into the chosen region (black box) to show more details around the vicinity of the SS. The pink curve on this inset marks the velocity separatrix. Figure 11. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (coloured shading) and velocity (3D yellow arrows) on the layer z = 1.2 at t = 2. The red curve shows the boundary of the CME bubble, and the pink curve marks the velocity separatrix. Figure 11. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (coloured shading) and velocity (3D yellow arrows) on the layer z = 1.2 at t = 2. The red curve shows the boundary of the CME bubble, and the pink curve marks the velocity separatrix. Between the FS dome and the CME bubble, there inevitably exists a 3D velocity separatrix in the lower atmosphere (below about z < 4.5, i.e. at the height of the vortex centre), as shown by the pink curve in Fig. 10. This curve originates from the vortex centre and then extends downward to the bottom of the corona. It separates the outward plasma flow from the inward plasma flow. Near the velocity separatrix, the SS is marked at the most distorted place on the distribution of |$\triangledown \times \vec{v}$|⁠. The velocity separatrix does not overlap with the SS. Considering that both the FS and the reconnecting CS are very common phenomena during solar eruptions (Wang et al. 2009, 2015; Mei et al. 2012b; Xie et al. 2019), this velocity separatrix must also exist in many eruptive events, although it has not been reported in detail by researchers. Behind the SS, there exist other disturbances in the velocity distribution, as shown in Figs 9 and 10. The formation of these disturbances relates to the CME bubble expansion and the reconnecting CS, because of the feedback between the plasma flows, the CME bubble and the reconnection CS. On the one hand, the plasma behind the SS moves toward the CS. Part of the plasma becomes the reconnection inflow and the other enters into the outermost newly formed shell of the CME bubble. On the other hand, the upward-moving CME bubble affects the reconnection rate of the CS and the inward plasma flow. In addition, the turbulent region (marked by a green box in the left-hand panel of Fig. 10) under the lower edge of the CME bubble can disturb the velocity distribution in the ambient atmosphere and the reconnection outflow inside the 3D reconnecting CS may also lead to disturbances outside the CS (Mei et al. 2012b; Takahashi, Qiu & Shibata 2017; Ye et al. 2019). In our 3D experiment, Fig. 11 demonstrates the velocity field on another cut at z = 1.2. Together with Fig. 10, they give us a more complete understanding of the velocity-field structure than previous 2D MHD experiments (Chen et al. 2002; Wang et al. 2009, 2015; Mei et al. 2012a). The pink curve in Fig. 11 marks the front of the velocity separatrix on the cut z = 1.2, which appears to have a quasi-circular shape and locates between the FS and SS fronts. The front of the velocity separatrix propagates outward at a speed of about 93.2 km s−1 in the beginning and later decreases to a speed of about 58.3 km s−1. In addition, the inward-moving plasma leads to a slow increase of the density and the temperature in the eruptive source region behind the velocity separatrix, so this region can recover from the dimming appearance in the AIA images, as shown in Fig. 5. The last feature in Fig. 9 is the echoes following the FS. In the early stage, we cannot see the echoes of the fast shock although the FS front reaches the bottom of the corona and is reflected by the photosphere. Fig. 12 shows the velocity distribution near the FS front at t = 1.6 and t = 4. At t = 1.6, the SS front, the velocity separatrix and the FS front are closely arranged from left to right, as indicated by the upper panel of Fig. 12. We cannot see a similar velocity distribution of echoes, as already shown in fig. 10 of Xie et al. (2019). The inward plasma toward the CS and the CME bubble and the outward plasma following the FS dominate the velocity distribution after the FS front. The corresponding velocity distributions of the echoes from the bottom are depressed in the early stage. In the later stage of our simulation, the SS front and the velocity separatrix gradually separate from the FS. The inward and outward plasma flows no longer dominate the whole region after the FS any more, and then the velocity field due to the echoes can exist after the FS as shown in the lower panel of Fig. 12. The velocity separatrix breaks into two branches when it is close to the bottom of our simulation box after t = 3. This velocity separatrix isolates this echo from other plasma flows. Figure 12. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (coloured shading) and velocity (3D yellow arrows) on the cut y = 0 at t = 1.6 (upper panel) and t = 4 (lower panel) respectively. The pink curves show the SS front, the velocity separatrix and the FS front respectively. Figure 12. Open in new tabDownload slide Distributions of |$|\triangledown \times \vec{v}|$| (coloured shading) and velocity (3D yellow arrows) on the cut y = 0 at t = 1.6 (upper panel) and t = 4 (lower panel) respectively. The pink curves show the SS front, the velocity separatrix and the FS front respectively. 6 CONCLUSIONS In this work, we performed a 3D MHD numerical experiment based on the TD99 model (Titov & Démoulin 1999) for a solar eruption and the EUV signatures that result from it. We used a current-carrying magnetic flux rope (MFR) that is the typical structure that can support prominences or filaments in the corona. A gravitationally stratified background atmosphere with a temperature of 106 K is used to model the corona, while a thin layer with a temperature of 6000 K at the bottom represents the photosphere. Our treatment thus realizes a line-tied environment for the erupting MFR at the photosphere. Following recent foregoing 2D numerical experiments (Wang et al. 2009, 2015; Mei et al. 2012a; Xie et al. 2019), we put emphasis on the various types of disturbances during the eruption, such as the fast-mode shock (FS), the MFR, the CME bubble and the slow-mode shocks (SS) at the flanks of this bubble. Our main conclusions are as follows: At the very beginning of the simulation, the MFR erupts upward because we deliberately started from a global non-equilibrium in our MFR. Above the MFR, a clear FS front with a 3D dome shape forms and expands outward. Under the MFR, the 3D CS grows continuously, in which magnetic reconnection takes place, relaxing the stretched field lines into closed ones that account for the growing flare loop system and the expanding CME bubble. A dimming region with lower density and temperature can be seen after the FS front. At the flanks of the CME bubble, the SS and velocity vortices appear in the lower atmosphere. The FS echoes due to reflection on the photosphere cannot be seen at the early stages of the eruption, but do appear at later stages when its corresponding velocity distribution is no longer suppressed by overlying plasma flows. We employed forward modelling of EUV observations by fomo (Van Doorsselaere et al. 2016) to make a direct comparison between the numerical results and realistic EUV observations. Due to the untwisting process inside the MFR, magnetic reconnection operates inside the MFR such that it is heated significantly and a hot MFR can be observed at the beginning of our simulation in synthetic SDO/AIA images. Above the MFR, the FS covering the whole eruptive magnetic structure forms and begins to propagate outward. About 9 min later, the bright MFR expands outward, its brightness decays continuously and disappears in the dark cavity/dimming region surrounding it. When EUV disturbances are observed on the limb, the bright MFR and the FS can be seen clearly, in favour of the bimodality interpretation of EUV disturbances. However, images for disturbances propagating on the solar disc do not support this, because the morphology of the bright MFR has no typical circular dispersive feature in the EUV disturbances. In addition, the echoes can be seen in synthetic images on the limb in the later stages of our numerical experiment. In the lower atmosphere, a 3D velocity separatrix and vortices appear between the FS front and the CME bubble. Outside this separatrix, the plasma moves toward the expanding front of the FS. Inside the separatrix, the plasma moves into the CME bubble or the reconnecting CS below the MFR. The inward and outward velocity distributions, separated by the velocity separatrix, also lead to the formation of the SS. However, these structures are hard to detect in our synthetic SDO/AIA images, which agree with current realistic observations. Nevertheless, we suggest that the velocity separatrix and the SS should be ubiquitous during the eruption, just like the FS and the CME bubble. ACKNOWLEDGEMENTS The authors thank Jun Lin, Ilia I. Roussev and other colleagues for fruitful discussions. This work was supported by the Strategic Priority Research Program of CAS with grants XDA17040507, QYZDJ-SSWSLH012 and XDA15010900, the National Science Foundation of China (NSFC) under the grant numbers 11933009, U1631130, 11273055, 11303088, 11573064, 11403100, 11333007, and 11603070, the project of the Group for Innovation of Yunnan Province grant 2018HC023, the Yunnan Ten-Thousand Talents Plan-Yunling Scholar Project, as well as the Yunnan Science Foundation of China under grant 2019FB005, as well as a joint FWO-NSFC grant G0E9619N and received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 833251 PROMINENT ERC-ADG 2018), and from Internal Funds KU Leuven, project C14/19/089 TRACESpace. REFERENCES Attrill G. D. R. , Harra L. K. , van Driel-Gesztelyi L. , Démoulin P. , 2007 , ApJ , 656 , L101 10.1086/512854 Crossref Search ADS Crossref Attrill G. D. R. , Engell A. J. , Wills-Davey M. 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TI - 3D numerical experiment for EUV waves caused by flux rope eruption
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/staa555
DA - 2020-04-21
UR - https://www.deepdyve.com/lp/oxford-university-press/3d-numerical-experiment-for-euv-waves-caused-by-flux-rope-eruption-ZXGji5yRdF
SP - 4816
VL - 493
IS - 4
DP - DeepDyve
ER -