3.5 Acceleration and propagation

From above it is seen that, as the closed magnetic structure is initiated to rise, a current sheet is newly formed above the magnetic neutral line. The current sheet exerts a resisting force over the CME progenitor. If magnetic reconnection is then excited in the current sheet, the reconnection leads to the rapid energy release from magnetic energy to thermal, kinetic, and nonthermal energies, which is just the picture of the CSHKP model: the downward reconnection outflow collides with the flaring loops which are line-tied to the solar surface, whereas the upward reconnection outflow pushes the core field of the CME source region, which is then accelerated to move up. The up-moving core field, e.g., a flux rope, further stretches up the overlying magnetic field lines, leading to the large-scale magnetic restructuring along with the mass ejection. Such a resistive MHD paradigm is probably applicable to a majority of CME events, especially those energetic ones, as evidenced by the strong relationship between CMEs and flares. If magnetic reconnection is not excited in the newly formed current sheet, from a theoretical point of view, the CME progenitor, after being triggered, may also erupt via an ideal MHD process, e.g., the ideal MHD instabilities in the low plasma beta case or the dragging force from the solar wind, if the outward force can overtake the resisting force from the current sheet. Such an ideal MHD paradigm is particularly discussed in Section 4.1.

The CME propagation was mainly observed by imaging telescopes. However, in imaging observations, it is always a problem to distinguish mass motions from waves and apparent motions (for instance, the ribbon separation in solar flares is a kind of apparent motion). Regarding the CMEs, we describe the acceleration and propagation of their components separately, i.e., the bright core, the cavity, and the frontal loop. The rising of the bright core, i.e., the filament, is definitely a mass motion. The CME frontal loop, however, might be much more complicated (see discussions in Section 4.4).

3.5.1 Bright core

Kahler et al. (1988Jump To The Next Citation Point) traced four CME/filament eruption events, and found that the Hα filaments in all cases were moving upward slowly before the flare onset, indicating that the CMEs have already been triggered before the flare. Moreover, they found that two of the filaments showed strong acceleration during the impulsive phase of the flares, indicating that the main magnetic reconnection plays an important role in accelerating the filaments (by “main reconnection” here we distinguish it from the localized reconnection in the triggering process).

EIT 304 Å images can trace the erupting filament further out, and it was found that the filament rises with an almost constant velocity after the impulsive acceleration phase (e.g., Joshi and Srivastava, 2007).

The upward reconnection outflow would hit the bottom of the flux rope, forming SXR plasmoids just below the filament (Ohyama and Shibata, 2008Jump To The Next Citation Point). Therefore, it is not surprising that the SXR plasmoid, which traces the filament, would also show a strong acceleration during the flare impulsive phase, as revealed by Ohyama and Shibata (1997). It is also expected that the SXR plasmoid also becomes a part of the CME bright core. In this case, the plasmoid motion, moving along with the filament, is different from the upward reconnection outflow, with the latter being much faster.

The erupting filament, so as the bright CME core, may have inverse or normal polarities. While little attention was paid to the dynamics of the eruption of normal-polarity filaments, a lot of efforts have been made to investigate the dynamics of the filaments with the flux rope configuration. Note that, in the case of the flux rope configuration, it is believed that the filament is located near the bottom of the flux rope (Kuperus and Raadu, 1974). The modeling of the flux rope dynamics includes analytical formulations and MHD simulations.

Similar to Anzer (1978), the dynamics of the flux rope in the flux injection model of Chen (1996Jump To The Next Citation Point) is determined by Newton’s second law for simplicity, i.e., d2Z- M dt2 = FR , where M is the total mass per unit length of the flux rope, Z is the apex height of the flux rope axis, FR is the integrated force including the drag term related to the solar wind, and t is time. The solution reproduces the impulsive acceleration of CME bright core, which is followed by an almost uniform propagation as revealed by the right panel of Figure 19View Image.

With the assumption that the plasmoid (or flux rope) is accelerated solely by the momentum of the reconnection outflow, Shibata and Tanuma (2001Jump To The Next Citation Point) derived an analytical solution to the motion of the plasmoid, i.e.,

V = -----VA-exp(ωt)------, (2 ) p exp(ωt) − 1 + VA ∕V0
where Vp is the plasmoid velocity, VA is the Alfvén speed of the ambient corona, V0 is the initial velocity of the plasmoid before reconnection, ω = ρVA ∕(ρpLp), ρ is the density of the ambient corona, ρp the density of the plasmoid, Lp the vertical length of the plasmoid, and t the time. As seen in Figure 27View Image, the simple model, for the case of V0 = 0.01VA, can also reproduce the typical observational features, i.e., the CME core, or erupting prominence, is strongly accelerated in the impulsive phase of the flare, and then moves up with an almost constant velocity. Such a model is rather simple, where the restraining force from the coronal loops overlying the flux rope is neglected.
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Figure 27: Temporal variations of the plasmoid velocity (V plasmoid, dotted line), its height (solid line), and inflow velocity (Vinflow, dashed line) in the analytical model of Shibata and Tanuma (2001).

A more sophisticated approach is the circuit model by Martens and Kuin (1989), which is equivalent to the modeling of inertialess MHD equations. Their model can be regarded as the quantitative version of the CSHKP model. It turned out to well explain the impulsive acceleration of the flux rope, while fail to explain the later propagation phase. This model was improved by Lin and Forbes (2000Jump To The Next Citation Point), as shown in Figure 28View Image. Their analytical solution reproduced the three-phase feature for the CME core very well, i.e., a slow rise in the preflare phase, a strong acceleration in the flare impulsive phase, and the final eruption with a nearly constant velocity in the gradual phase.

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Figure 28: Diagram of the magnetic configuration including an erupting flux rope and a current sheet between the flux rope and the reconnected flare loop used for the setup of the analytical solution by Lin and Forbes (2000Jump To The Next Citation Point).

Numerical simulations of the full set of the MHD equations provide the CME dynamics that can be better compared with observations. The 2D simulations by Chen and Shibata (2000Jump To The Next Citation Point) showed that, as the flux rope loses its equilibrium due to the emerging flux (see Section 3.4.2), a current sheet forms near the null point below the flux rope. As anomalous resistivity sets in, magnetic reconnection proceeds in the current sheet. On one hand, it forms a cusp-shaped flare below the reconnection site as described by the standard flare model. On the other hand, with reconnection, the constraint of the closed field lines overlying the flux rope is removed, and the flux rope is accelerated, as depicted by Figures 17Watch/download Movie and 18Watch/download Movie. The time evolution of the flux rope height is compared with that of the reconnection rate in Figure 29View Image. It is seen that the flux rope is intensively accelerated near the peak of the reconnection rate, i.e., the flare impulsive phase, after which the flux rope rises with an almost invariable velocity, a typical feature found in the erupting filament observations (Kahler et al., 1988Jump To The Next Citation Point; Sterling and Moore, 2005).

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Figure 29: Temporal evolutions of the flux rope height (h) and the magnetic reconnection rate (R) in the numerical simulation of Chen and Shibata (2000Jump To The Next Citation Point), which shows that the strong acceleration of the flux rope is coincident with the peak reconnection rate.

The consistency of the above models with observations of many CMEs implies that magnetic reconnection is the key to linking the violent CME and flare eruptions as a unity. However, as we will see in Section 4.1, some factors other than reconnection may also contribute to the acceleration of CMEs, e.g., ideal MHD instabilities (Amari et al., 2004; Fan and Gibson, 2007), which is the reason why some events present continual acceleration before and after the impulsive phase (e.g., Kahler et al., 1988Jump To The Next Citation Point). They are these factors that might accelerate some CMEs without the involvement of magnetic reconnection, e.g., via ideal MHD instabilities such as those discussed in Section 3.4.2. In MHD simulations, numerical resistivity is generally unavoidable, therefore, it is still premature to see the ideal MHD models of CME eruptions. Instead, the corresponding dynamics of CMEs is often studied by solving the motion equation of the flux rope under the integrated force, especially the toroidal force. As demonstrated by Chen (1989Jump To The Next Citation Point, see Figure 19View Image) and Kliem and Török (2006), the flux rope can indeed be accelerated and then approach an almost constant velocity. It is pointed out here that one thing is missing in these works, i.e., the existence of a newly-formed current sheet below the flux rope, which resists the flux rope from further erupting.

It is noted in passing that the circular flux rope in the low corona may evolve into a pancake structure in the interplanetary space (Riley et al., 2003; Manchester IV et al., 2004).

3.5.2 Cavity

An expanding cavity often immediately follows the CME frontal loop, with the bright core being embedded in (Illing and Hundhausen, 1985). The white-light deficit means that the density in the cavity is relatively low. The density depleted cavity would also be weak in EUV wavelength, as illustrated by Chen et al. (2000Jump To The Next Citation Point) and Chen (2009aJump To The Next Citation Point). The mass outflow in the dimming region was measured by spectroscopic observations (Harrison and Lyons, 2000; Harra and Sterling, 2001Jump To The Next Citation Point), which was interpreted as the result of magnetic field line stretching pushed by the erupting flux rope (Chen et al., 2002Jump To The Next Citation Point, 2005cJump To The Next Citation Point).

The CME cavity, which is observed above the solar limb, should have disk counterpart. Theoretically, Chen and Fang (2005) proposed that whereas the CME frontal loop corresponds to the EIT wave front on the solar disk, the CME cavity corresponds to the extended dimmings, which follow the EIT wave fronts. With the gap of the fields of view between the SOHO/LASCO coronagraph and SOHO/EIT, Thompson et al. (2000) speculated that the angular span of EUV extended dimmings is roughly the same as the corresponding CME. With the overlapping fields of view of Mauna Loa Solar Observatory (MLSO)/MK4 coronagraph and SOHO/EIT, Chen (2009aJump To The Next Citation Point) verified that EIT wave front, which borders the extended dimmings, is the EUV counterpart of the CME frontal loop, while EUV extended dimmings are the disk counterpart of the CME cavity. Such a result is understandable since both EIT wave fronts and CME frontal loops are characterized by density enhancement, whereas both EUV extended dimmings and the CME cavity are characterized by density depletion.

It is noted that, even before CME eruption, a coronal cavity may be visible surrounding the torpid prominence (e.g., Vaiana et al., 1973; Hudson et al., 1999), as a part of the helmet streamer (Illing and Hundhausen, 1986Jump To The Next Citation Point). Consensus has not been achieved regarding the relationship between the pre-eruption streamer cavity and the CME cavity. While it was claimed that the streamer cavity simply swells and erupts to become the CME cavity (Illing and Hundhausen, 1986; Gibson et al., 2006bJump To The Next Citation Point), an alternative explanation is that the CME cavity results from successive stretching of the closed magnetic field lines from the inner field to the outside (Delannée and Aulanier, 1999Jump To The Next Citation Point; Chen et al., 2002Jump To The Next Citation Point, 2005cJump To The Next Citation Point). I tend to believe that the streamer cavity is a steady structure, which is believed to correspond to the filament channel (Engvold, 1989) and is, therefore, related to the filament formation. However, the CME cavity is a dynamic structure, with its footprints expanding laterally as more and more field lines are stretched up, i.e., the streamer cavity is only the initial part of the CME cavity, and more and more overlying field lines are involved into the cavity during the CME eruption.

Two issues were often ignored about the CME cavity. The kinetic energy of the cavity is often neglected (Vourlidas et al., 2000), probably because the plasma density of the CME cavity (or the extended dimmings) is small. However, the cavity volume is very large, where outflows with a high velocity are often induced (Harra and Sterling, 2001), therefore, to which extent it may contribute to the total kinetic energy should be clarified. On the other side, as the gas pressure in the cavity decreases during the CME eruption, the pressure gradient is expected to drive outflow from the transition region and even chromosphere (Jiang et al., 2003; Jin et al., 2009). This process, in addition to the interchange reconnection between the stretched magnetic field and small coronal loops (Attrill et al., 2008), refills the depleted corona, making the coronal dimming recover to normal in tens of hours.

3.5.3 Frontal loop

The kinematics of the frontal loop, sometime called leading edge, was extensively studied. Combining the Mark III (MK3) K coronameter at Mauna Loa Solar Observatory and the SMM space-borne coronagraph, St Cyr et al. (1999Jump To The Next Citation Point) found that among the 141 CME events, the frontal loop in 87% of the sample showed a positive acceleration, which is averaged at 0.264 km s–1. The strong acceleration happens generally below 2 R⊙, beyond which CMEs move with an almost constant velocity. With the unprecedented high-resolution and wide-view observations by SOHO/LASCO coronagraph, Zhang et al. (2001aJump To The Next Citation Point) investigated four CME events covered down to 1.1 R ⊙ and found that the CME frontal loop propagation can be described in a three-phase scenario, as illustrated by Figure 30View Image:

(1) The initiation phase, when the frontal loop rises slowly with a velocity of ∼ 80 km s–1 for tens of minutes before the main flare. During this phase, the CME is being triggered.

(2) The impulsive acceleration phase, when the frontal loop is accelerated rapidly. This phase lasts several to tens of minutes, which is coincident with the impulsive phase of the main flare.

(3) The propagation phase, when the frontal loop moves with an almost constant velocity.

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Figure 30: Temporal evolutions of the SXR flux of the flare (solid line) and the propagation velocity of the CME frontal loop (dotted line), showing a three-phase pattern (from Zhang et al., 2001aJump To The Next Citation Point).

Of course, the CME frontal loop also shows smaller acceleration or deceleration in the propagation phase. Statistical study by Zhang and Dere (2006) indicates that the main acceleration in the impulsive phase widely ranges from 2.8 × 10–3 to 4.464 km s–2, with an averaged value at 0.331 km s–2; the corresponding main acceleration duration widely ranges from 6 to 1200 min, with an averaged value at 180 min. On the other hand, the residual acceleration in the propagation phase ranges from –131 to 52 m s–2, with an averaged value at 0.9 m s–2, much smaller than the main acceleration. They also found a scaling law between the main acceleration (a) and its duration (T), i.e., a = 104 T−1 (m s− 2), where T is in unit of min.

As the CMEs propagate in the interplanetary space, it is found that fast CMEs decelerate and slow CMEs accelerate, so that the ICME velocity tends to approach the ambient solar wind speed. Gopalswamy et al. (2000) derived an empirical formula for the CME propagation from the coronagraph field of view to 1 AU:

a[m s−2] = 1.41 − 0.0035u [km s− 1], (3 )
where a is the mean acceleration in the interplanetary space and u is the CME propagation velocity measured in the coronagraph field of view. The asymptotic solution of the above formula, as a approaches 0 and u becomes constant, is u = 1.41/0.0035 = 403 km s–1.

In contrast to the bright core and the cavity, the CME frontal loop is actually much less understood. It is generally taken for granted that the CME frontal loop propagation is regarded as a mass motion, and in most modelings, the CME frontal loop was generally considered as the apex of the flux rope (see Forbes et al., 2006, for details). However, the propagation of the CME frontal loop might not be a mass motion (see Section 4.4 for a completely different explanation). For instance, as described in Forbes (2000Jump To The Next Citation Point), the CME frontal loop may be a plasma pileup region rather than the flux rope. Therefore, we do not explicate the modeling of the CME frontal loop here, leaving the topic to be discussed in Section 4.4.

3.5.4 Lateral expansion

The CME propagation features mentioned in the previous two subsections are focused on the radial direction. In this subsection, we briefly talk about the lateral expansion of CMEs.

Lacking the observations in the low corona, CME evolutions above 2R ⊙ gave an impression that CMEs generally have a fixed angular width. For instance, Figure 25 of Schwenn (2006) shows that the 60° cone angle and the general shape of the 2000 February 27 CME event are maintained during the whole 12 hour passage through the LASCO field of view. This maintained “self-similarity” is characteristic for most CMEs (Low, 2001), and is also the foundation for the cone model that describes the CME geometry (Howard et al., 1982Jump To The Next Citation Point; Fisher and Munro, 1984Jump To The Next Citation Point). However, low corona observations indicated that CMEs experience significant lateral expansion in the early stage. For example, St Cyr et al. (1999Jump To The Next Citation Point) found that the average width of CMEs in the lower field of view of the MK3 coronagraph is 12° smaller than that measured in SMM. It is just due to such an expansion, which is often asymmetric on the two sides, that a CME becomes much wider than the accompanying flare, and the flare is often near one footpoint of the CME, rather than being centered under the CME (Harrison, 1986). Such an expansion of the CME and its asymmetry to the solar flare can be understood as follows: as the CME frontal loop was suggested to be formed due to successive stretching of magnetic field lines (Chen, 2009aJump To The Next Citation Point), it is expected that, if the background magnetic field is asymmetric, a larger area of magnetic field is involved to form the CME on the side with weaker magnetic field, and we are ready to see the expanding CME covers a larger span on this weak field side as depicted by Figure 31View Image. Therefore, an asymmetric ambient magnetic configuration naturally results in a CME asymmetric to the underlying flare.

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Figure 31: Schematic sketch showing how a CME frontal loop expands to be asymmetric to the underlying flare in the early stage. The black solid lines correspond to the initial magnetic field, the black dashed lines to the deformed magnetic field, and the green dashed lines to the CME frontal loop formed by the successive stretching of magnetic field lines. The arrows indicate that the flux rope is erupting.

It is interesting to note that in most CME/flare modelings, only a central bipolar magnetic structure erupts as a CME. Even in the magnetic breakout model where a quadrupolar configuration is present (Antiochos et al., 1999), the ejected magnetic structure is still the bipolar field near the central neutral line. However, there often exist interconnecting loops above the source region of the CME eruption, and the erupting bipolar structure would interact with the overlying interconnecting loops. If the overlying loops have roughly the same magnetic orientation as the erupting bipolar structure (maybe with a small angle), the latter would pull up the overlying loops, leading to a global CME (Zhukov and Veselovsky, 2007) as illustrated in Figure 32View Image (Delannée and Aulanier, 1999Jump To The Next Citation Point; Delannée et al., 2007Jump To The Next Citation Point) and numerically modeled by Roussev et al. (2007). If the overlying or even lateral loops have roughly opposite magnetic orientation as the erupting bipolar structure, interchange reconnection could happen between them, leading to a CME with a re-organized morphology (Attrill et al., 2006Jump To The Next Citation Point; Gibson and Fan, 2008).

All these three situations, i.e., the asymmetric expansion of the source active region, its merging with the overlying interconnecting loops, and the reconnection with the ambient or overlying field, could happen in combination (Cohen et al., 2009Jump To The Next Citation Point). They may explain why many flares are located near one footpoint of the corresponding CMEs.

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Figure 32: Schematic sketch showing how a CME expands to a global scale as it pushes up the overlying interconnecting loops. The gray lines corresponds to small-scale magnetic field, black lines to the interconnecting lines (adapted from Delannée and Aulanier, 1999Jump To The Next Citation Point).

3.5.5 CME-associated shocks

Similar to the launch of a rocket, the eruption of a CME would also drive a piston-driven shock wave ahead of it. The shock, which can locally accelerate electrons to ∼ 10 keV and hence excite in situ plasma emissions, was observed as type II radio bursts (Wild and McCready, 1950). Such a piston-driven shock was identified in the interplanetary type II radio bursts at ∼ 290 kHz (Malitson et al., 1973), as well as the decametric coronal type II radio bursts at ∼ 2.5 MHz (Reiner et al., 2000Jump To The Next Citation Point). It is still on debate whether such a piston-driven shock may show up at metric wavelengths (Vršnak and Cliver, 2008Jump To The Next Citation Point).

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Figure 33: The LASCO/C2 white-light images showing a faint bow shock straddling over the 1999 April 2 CME. The solar disk is revealed by the EIT 195 Å images (adapted from Vourlidas et al., 2003Jump To The Next Citation Point).

The white-light imaging signature of this piston-driven shock should be the so-called “CME forerunner” found by Jackson and Hildner (1978), which was later inferred based on the streamer deflections (Sheeley Jr et al., 2000) and clearly detected by Vourlidas et al. (2003Jump To The Next Citation Point) and Eselevich and Eselevich (2008), as shown by Figure 33View Image. The spectroscopic properties of the CME-driven shock are well documented by Raymond et al. (2000), Mancuso et al. (2002), and Ciaravella et al. (2006Jump To The Next Citation Point). Such a piston-driven shock extends down to the solar surface, especially in the early stage. Generally speaking, the strength of such a bow shock would be the strongest at the top since the driver, the erupting flux rope, is moving upward. Gopalswamy et al. (2001) proposed that, however, in a non-uniform corona, the shock strength at one flank may also be strong enough to form a second type II radio burst, e.g., at metric wavelengths.

The footpoint of the shock sweeping the chromosphere was proposed to generate Hα Moreton waves (Chen et al., 2002Jump To The Next Citation Point, 2005cJump To The Next Citation Point), which were originally explained due to the flare-induced blast waves (Uchida, 1968Jump To The Next Citation Point). It is expected that the shock strength at the footpoint decays drastically as the piston, i.e., the erupting flux rope, moves higher and higher. This may explain why Moreton waves are visible for a relatively short distance.

According to the standard CME/flare model, it is expected to see another two fast-mode termination shocks associated with the CME/flare eruption, i.e., the upward reconnection outflow collides with the flux rope, forming a reverse fast-mode shock, and the downward reconnection outflow collides with the flaring loops, forming another fast-mode shock, as illustrated in Figure 34View Image. The termination shock below the flux rope apparently propagates up along with the erupting CME core. This shock may also excite type II radio bursts, and Magara et al. (2000) applied it to interpret the metric type II burst below the faster-moving decametric burst, which was found by Reiner et al. (2000). Similarly, the termination shock at the flare loop top might be observed as high-frequency slowly drifting radio structures found by Karlický et al. (2004), though they interpreted it as accelerated electrons trapped in the erupting plasmoids.

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Figure 34: Schematic sketch showing the possible shocks associated with CMEs in the standard model. Besides the piston-driven shocks, another two fast-mode termination shocks, which are formed as reconnection outflows (long blue arrows) collide with the flux rope and the flare loop, are indicated by red bars. Black solid lines correspond to the magnetic field, dashed lines to the slow-mode shock fronts.

Another open question is whether the pressure pulse of the flaring loop can generate a blast wave which produces metric type II bursts. According to the standard CME/flare model shown in Figure 34View Image, heat and nonthermal particles are transferred down from the reconnection site to the solar surface, the resulting high pressure drives chromospheric evaporation, with hot and dense plasma filling the flare loop. It was suggested that the high pressure would excite a blast shock wave, which accounts for the metric type II radio burst and Moreton waves (Uchida, 1974; Wagner and MacQueen, 1983; Vršnak and Lulić, 2000). Some authors have a different view (e.g., Cliver et al. 1999; Chen et al. 2002Jump To The Next Citation Point, 2005cJump To The Next Citation Point; see Vršnak and Cliver 2008Jump To The Next Citation Point for a review), and numerical simulations of magnetic reconnection (e.g., Chen et al., 1999) did not show a blast wave from the flaring loop, even when chromospheric evaporation is reproduced (Yokoyama and Shibata, 2001). This question definitely deserves further clarification.

Finally, we would like to comment on the formation of the piston-driven shock straddling over the CME. It was often mentioned that the shock forms when the velocity of the ejecta becomes larger than the local Alfvén wave or fast-mode wave speed. We want to emphasize that this is not necessary since a compressible simple wave may steepen and form a shock wave nonlinearly (Vršnak and Cliver, 2008), especially for a simple wave with a short wavelength, e.g., in the case when a CME is suddenly accelerated.

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