The Hyper-inflation Stage in the Coronal Mass Ejection Formation: A Missing Link That Connects Flares, Coronal Mass Ejections, and Shocks in the Low Corona

The two CMEs analyzed here were selected because they exhibited a bubble-like appearance in the EUV. An EUV "bubble" is a spherical-like dark (in the EUV) structure surrounded by a bright rim (first noted by Dere et al. 1997). As discussed in Patsourakos et al. (2010a), such structures are the low coronal counterpart of the white-light CME cavity and are strong indications of an impulsive event (i.e., events that exhibit a rapid acceleration low in the corona). Bubble-like CMEs are, hence, primary candidates for hyper-inflation studies. The two CMEs occurred on 2011 March 7 and 2011 June 7.

Both events were observed near simultaneously from different vantage points with the best available temporal cadence. This allows us to derive the low coronal 3D kinematics and volume of the expanding structures in great detail. We use images from AIA on board SDO and from the Extreme Ultra-Violet Imager (EUVI) on board STEREO-A/B. We select images in the EUV channels in which waves are best observed: 195 Å (EUVI) and 193, 211, and 335 Å (AIA). These channels capture plasmas with peak response temperatures in the range 1.0–2.5 MK (O'Dwyer et al. 2010). For the events analyzed here, the AIA data have a 12 s temporal cadence, while the EUVI data are acquired at 2.5- and 5-minute cadence for STA and STB, respectively. To better identify and distinguish the wave and the driver features, we use wavelet-processed images with the technique described by Stenborg et al. (2008) that brings out the fainter EUV structures. Fits files for the wavelet-processed EUVI images in all four wavelengths are readily available at http://solar.jhuapl.edu/Data-Products/EUVI-Wavelets.php. The AIA data are also processed with the same technique, optimized for the AIA image characteristics.

We analyze the low coronal CME evolution via 3D reconstructions of both the CME and the accompanying wave based on forward-modeling techniques that take advantage of multi-viewpoint observations. As in the majority of CME analyses, we assume that the CME is an erupting magnetic flux rope (MFR), reminiscent of a croissant. We use the widely known graduated cylindrical shell (GCS) model (Thernisien et al. 2009, 2011; Thernisien 2011) to determine the 3D geometry of the CME and the ellipsoid model (Kwon et al. 2014; Kwon & Vourlidas 2017) for the wave. We rely on images from three viewpoints (AIA and EUVI-A/B) to determine the best-fit parameters.

Figure 1 provides a visual description of the relevant quantities used for the 3D kinematic characterization of the CME (top) and wave (bottom). The superscripts denote the parameters of interest for the flux-rope (F) and ellipsoid (E) models. The subscripts t and c represent the measured features, i.e., the top (leading edge) and center, respectively. Thus, the heliocentric distances of the CME top and center are indicated by ${H}_{t}^{F}$ and ${H}_{c}^{F}$, respectively, while the lateral distances (i.e., half-widths) from the main axis of symmetry in the face-on and edge-on views are indicated by ${W}_{f}^{F}$ and ${W}_{e}^{F}$, respectively. Similarly for the ellipsoid model (bottom panels of Figure 1), the heliocentric distances of the center and top of the ellipsoid are indicated by ${H}_{c}^{E}$ and ${H}_{t}^{E}$, respectively. Two of the semi-axes (a and c) are depicted in Figure 1(d), while the semi-axis b is into the page. Following the notation used for the flux-rope widths, the semi-axes a, b, and c are indicated by ${W}_{a}^{E}$, ${W}_{b}^{E}$, and ${W}_{c}^{E}$. The expansion of the ellipsoid in the radial and lateral directions is given by the temporal variation of ${W}_{c}^{E}$ and ${W}_{a,b}^{E}$ (i.e., the semi-axes c and a—or b), respectively. The ellipsoid is oriented so that the semi-axis a is equivalent to the CME face-on width, while the variation of c gives a measure of the expansion in the radial direction. The flux-rope model has a circular cross section along its main body, and hence there are only two directions for the expansion. In the ellipsoid model, the third semi-axis (b) can be different from the other two and is equivalent to the CME edge-on width.

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We perform 3D reconstructions for both the CME and the wave at each time stamp. Because of the impulsive nature of the events, the accuracy of the reconstruction depends on the simultaneity of the EUVI-AIA images. Thankfully, AIA images are available at 12 s cadence, which enables us to obtain 3D reconstructions at the maximum available EUVI cadence of 2.5 minutes at the time.

After we determine the 3D location and size of the CME and the wave, we can proceed to derive their 3D kinematic and dynamic profiles as they form in the low corona. This is traditionally performed via analytical curve fitting, i.e., linear, quadratic, or exponential fits (see, e.g., Gallagher et al. 2003). More recent studies adopt a model-independent approach using regularization techniques (Temmer et al. 2010), optimized moving-average smoothing (Podladchikova et al. 2017; Veronig et al. 2018), or more traditional splines (Patsourakos et al. 2010b; Majumdar et al. 2020). A model-independent approach is more suitable for our work, since the impulsive acceleration phase of CMEs cannot easily be described by a specific analytical function. Starting from Kontar et al. (2004), Kontar & MacKinnon (2005), and Temmer et al. (2010), we implemented the approach briefly described below.

Assume a function y(t) continuous over the interval [t₀, t_N] such that

$$\begin{eqnarray}&&{h}_{i}-y({t}_{i})\lt \delta {h}_{i},\end{eqnarray} \tag{ 1 }$$

where the measurements h_i have finite uncertainties δ h_i .

The problem consists of finding the best smooth representation for the velocity, v(t), and acceleration, a(t), defined as

$$\begin{eqnarray}&&v(t)={dy}(t)/{dt}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&a(t)={d}^{2}y(t)/{{dt}}^{2}.\end{eqnarray} \tag{ 3 }$$

Kontar & MacKinnon (2005) and Temmer et al. (2010) tackled this problem by treating the derivative as an inverse problem (Craig & Brown 1986). Following this approach, we solve the system using an integral operator instead of a derivative operator. The integral operator minimizes the errors that are greatly amplified when estimating the derivatives using, for instance, finite differences. The regularization technique, described in Tikhonov et al. (1995), allows us to write the problem as

$$\begin{eqnarray}&&| | {\boldsymbol{h}}-{\boldsymbol{S}}{\boldsymbol{a}}| {| }^{2}+\lambda | | {\boldsymbol{L}}{\boldsymbol{a}}| {| }^{2}=\min ,\end{eqnarray} \tag{ 4 }$$

where S is the double integral operator, L is the constraint operator (first-order derivative), and λ is the regularization parameter that controls the smoothness of the solution. The system in Equation (4) is solved via the singular value decomposition (SVD) technique, where the parameter λ is adjusted to fulfill the condition ∣∣ h − S a ∣∣ ∼ ∣∣δ h_i ∣∣. The velocity v is then obtained by integrating a . The solution to the system provides model-independent smooth functions for the velocity and acceleration profiles that take into account the uncertainties in the data. This solution is a piecewise function (Hanke & Scherzer 2001).

The CME on 2011 March 7 originated from NOAA Active Region 11164 (N30W48). In Figure 2, we show a sequence of four pairs of running-difference images (AIA on the top and EUVI-A on the bottom) of the early evolution of the CME and the associated wave. The source region was located near the AIA west limb (Figure 2, top) and near the EUVI-A east limb (Figure 2, bottom). The eruption occurred behind the EUVI-B limb (not shown). This CME was accompanied by an M3.7 X-ray flare, peaking at around 19:43 UT. The eruption was associated with a metric type II radio burst (Table 1), which indicates the formation of a coronal shock. According to the Sagamore Hill station ⁵ of the Radio Solar Telescope Network (RSTN) operated by the US Air Force, the burst lasted from 19:54 to 20:15 UT, drifting from 180 down to 25 MHz with a derived speed of 1133 km s⁻¹. A decimetric type II burst was also detected at lower frequencies (16 MHz–200 KHz) by the Wind and STA radio experiments, starting at 20:00 UT and lasting until 08:00 UT on March 8. We discuss the radio observations in Section 5.

To study the formation of the associated EUV wave and its relation to the early evolution of the CME, we first take a look at the sequence of EUV images from single viewpoints. The first indication of the eruption/CME onset comes from the observations of a rising filament indicated by the arrow in Figure 2(a)) between 19:38 and 19:40 UT. Starting around 19:44 UT, the CME front becomes more visible, and by 19:48 UT it can be distinguished in both AIA and STA images (Figure 2(b), top and bottom, respectively). At 19:50 UT (Figure 2(c)) the wave is seen to separate from the CME boundary at the flanks, more clearly in STA images (southeast direction).

We take advantage of the high-cadence sequence of images provided by AIA and construct stackplots from 5-pixel-width slices extracted (1) along the direction marked with the yellow dashed line in Figure 2(a) and (2) parallel to the solar limb at a height of 1.27 R_⊙ as indicated in Figure 2(d). The stackplots are shown in the top left and right panels of Figure 3. The plots are composites of two AIA channels to provide a rough thermal history of the event: 171 Å (green), nominally representing 0.7 MK plasma, and 211 Å (red), representing hotter 2 MK plasma. We used base-difference images.

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In Figure 3(a), we indicate the filament, F, and the high-lying loops, which correspond to the top of the driver D (i.e., the rising CME). We extract the distance values over time (panel (c)) and derive the projected speed profile (panel (e)) for the two features marked as diamonds (filament) and triangles (driver). The right panels in Figure 3 show the evolution of the CME as it crosses a slit parallel to the limb at a distance of 1.27 R_⊙. A similar representation was used by, e.g., Cheng et al. (2012) and is useful to visualize the expansion of the CME. In Figure 3(b) we identify the driver D (CME) and the diffuse wave W. We measure the distances of these features from the center (central position angle, CPA) of the CME (diamonds and squares for the northern and southern flanks of the driver, respectively) and the wave (southern flank; squares), as shown in Figure 3(d). From these measurements we can derive the projected expansion speed profiles shown in Figure 3(f). The derived speed and acceleration profiles in both directions, lateral and radial, are obtained via the regularization technique described in Section 2.3. We assume errors of 2% in heights and widths (Veronig et al. 2018).

The radial analysis reveals that the CME front reaches a speed of about 300 km s⁻¹ in the first 10 minutes. The speed then abruptly increases, surpassing 1000 km s⁻¹ within AIA's field of view. The filament, on the other hand, attains a speed of about 500 km s⁻¹.

In the lateral direction, we can distinguish the wave from the CME in one direction only where the projected speeds are larger (bottom right panel), after ∼19:50 UT. We use purple and orange lines to denote the driver and the wave, respectively, following the color convention in Figure 2. We measure a peak projected speed of ∼900 and ∼1100 km s⁻¹ for the driver and the wave (presumably a shock at that point), respectively. The peak in the driver speed occurs a few minutes before the wave top speed. This is the behavior seen in previous analyses and is one of the defining characteristics of EUV waves as fast magnetosonic waves (e.g., Patsourakos & Vourlidas 2012; Downs et al. 2021). The evolution of the lower CME flank in the two channels also indicates shock formation. The lower flank is primarily red, indicating hot 2 MK plasma at the wave front, which, when it separates from the CME, reveals a cooler (green) component.

We note that the sharp change in the radial and lateral kinematics occurs at the same time, at around 19:48 UT. However, these one-dimensional measurements provide only an approximation of the actual 3D kinematics. It is not immediately obvious, for example, why the shock appears only on the south side of the CME. Is the northern flank expanding at a lower rate, or is the wave obscured by a line-of-sight effect? On the other hand, a cut at a given height does not follow the same location along the CME front, since we cannot be sure that the event is expanding self-similarly relative to its expansion center. For these reasons, it is necessary to expand the analysis into three dimensions by taking advantage of our multi-viewpoint observations.

The near-quadrature AIA and EUVI-A views (separation angle of 88°) are favorable for analyzing the 3D evolution of the CME and formation of the wave. We model the 3D shape of both the driver (CME) and the wave using AIA and EUVI-A/B images with a 2.5-minute cadence. A sequence of three snapshots showing the evolution of the CME (top row) and wave (bottom row) in the low corona is shown in Figure 4, with the corresponding 3D models overlaid. The details of the evolution of the parameters derived from these 3D reconstructions are presented in Section 4.

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Figure 5 presents a sequence of four pairs of images (AIA on the top and EUVI-A on the bottom) with the early evolution of the CME and associated wave of the 2011 June 7 CME. The eruption originated in NOAA AR 11226 (S21W64) on the western solar limb as seen from Earth, on the east limb as seen from EUVI-A, and behind the west limb for EUVI-B. The event was visible in the field of view of the EUV instruments from 06:20 to 06:30 UT. A GOES M2.5-class flare began on 2011 June 7 at 06:16 UT and peaked at 06:30 UT. The onset of the CME coincides with the development of the solar flare. Both the Learmonth and Culgoora radio spectrographs report a metric type II burst in association with the flare. Culgoora, however, reports two metric type II bursts ⁶ in close succession. We suspect that they may be the same burst, but this requires further analysis that is beyond our current scope. We consider here the parameters of the first burst, which is close in time to the one reported by Learmonth. The burst lasted from 06:26 to 06:40 UT and was detected between 140 and 30 MHz, with a derived speed of 850 km s⁻¹. The burst was also detected at lower frequencies (16 MHz to 250 KHz) by both the Wind and STA radio experiments between 06:45 and 18:00 UT, indicating that the shock, or subsequent shocks, lasted into the inner heliosphere. We discuss the radio observations in Section 5.

First, we inspect the sequence of EUV images. A filament is seen to slowly rise from 06:15 to 06:21 UT, as well as the faint overlaying loops. At 06:23 UT, the front of the CME becomes clearer, particularly in the AIA images. Between 06:25 and 06:30 UT, the CME undergoes significant lateral expansion. These stages can be appreciated in the time-slice plots in Figure 6 that show the height evolution along the direction marked by the yellow dashed line in Figure 5(a) and the lateral distance measured parallel to the limb at a fixed height of 1.02 R_⊙, respectively. We mark the driver D and filament F in Figure 6(a). From their distance–time measurements (panel (c)) we determine the projected radial speed shown in Figure 6(e) (squares and triangles for driver and filament) using the regularization technique in Section 2.3. In Figure 6(b) we identify the diffuse wave W and the driver D. The distances from the center of the CME are extracted and marked as triangles (driver) and squares (wave) in Figure 6(d) and used to determine the expansion speeds shown in Figure 6(f). For this event, the CME slowly reaches a speed of ∼100 km s⁻¹ as it rises during the first 8 minutes and then increases abruptly to 1000 km s⁻¹ within 4 minutes during the impulsive phase. The expansion speed of the CME, on the other hand, reaches a peak of ∼700 km s⁻¹ during the same phase.

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We next investigate the 3D evolution of both the CME and the wave at heliocentric distances close to the Sun. For this, we take triplets of EUVI and AIA images to fit the GCS and ellipsoid model to the CME and the wave, respectively. Figure 7 shows three snapshots with the reconstruction of the CME and wave overlaid to AIA images. We analyze the evolution of the deprojected heights and widths derived from these reconstructions in the next section.

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First, we study the outward/radial propagation of the CME and the wave for both events, as shown in Figure 8. The top panels show in black the time variation of the heliocentric distances to the CME top and center (${H}_{t}^{F}$ and ${H}_{c}^{F}$). The orange symbols and lines represent the heliocentric distances to the wave top and center (${H}_{t}^{E}$ and ${H}_{c}^{E}$). The symbols mark the values from the 3D reconstruction at each time stamp, along with their respective error bars. Following Kwon et al. (2014), we assume errors in the heights of ±7% for the flux rope and ±8% for the ellipsoid. The solid curves correspond to the fits from the regularization technique described in Section 2 (CME/wave front, solid lines; CME/wave center, dashed lines). The speed and acceleration profiles also from the regularization technique are shown in panels (b) and (c) and panels (d) and (e), respectively. For clarity, we plot the speed and acceleration profiles for the front and the center of the flux rope and the wave separately. The shaded bands represent the corresponding 95% CI intervals.

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We find that at heliocentric distances below 5 R_⊙ the CME and the wave evolve very similarly (top panels of Figure 8) for both events, with the wave front lying close to the top of the CME (${H}_{t}^{F}$ ≃ ${H}_{t}^{E}$). At these low heliocentric distances, the speed of the CME top ranges between about 1500 and 2000 km s⁻¹ (panel (b), black line). We note that when measuring the heights of the top of the CME, the measurements include the contribution of the expansion in the radial direction in addition to the bulk speed (i.e., measured by the motion at the center of the structure). These measurements are still useful and included here because they allow for the comparison with the equivalent projected quantities shown in Section 3. The propagation of the CME, however, is more appropriately characterized by the bulk motion, i.e., the time variation of ${H}_{c}^{F}$. For this, we find that the peak speed of the CME center (or peak bulk propagation speed) ranges between 1000 and 1200 km s⁻¹ (panel (c)).

Although the impulsive acceleration phase in the CME propagation is not fully captured with the 3D reconstruction (Figure 8(d)), we see that the acceleration of the CME front decreases from ∼1 to ∼0.5 km s⁻² within 5 R_⊙, in both events. The bulk acceleration, on the other hand, exhibits much less variability with values closer to 0.5 km s⁻², indicating that much of the variation in speed of the front comes from the CME expansion rather than propagation (Figure 8(e)).

Next, we analyze the lateral expansion evolution of the CME and the wave for both events (Figure 9). We consider two main directions for the CME expansion: across and along the main axis of the MFR, i.e., the face-on and edge-on views. The top panels plot the time variation of the CME edge-on (${W}_{e}^{F}$) and face-on (${W}_{f}^{F}$) half widths in magenta and blue, respectively. We include in orange the parameters from the ellipsoid model for the wave.

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Since the cross section of the GCS model for the CME is circular, there is no difference between the expansion in the radial direction and the lateral direction for the edge-on view, which are parallel and perpendicular to the direction of propagation, respectively. For the wave, however, we can consider the expansion in three directions given by the time variation of the ellipsoid semi-axes. These are the "lateral" expansion in the two directions perpendicular to the outward motion of the CME (given by a and b) and the expansion in the radial direction (given by c). The time variations of the a and b semi-axes are shown in orange.

We plot separately the speed profiles from the distance–time fits via the regularization technique for the edge-on (in panel (b)) and face-on (in panel (c)) half widths of the CME. We then compare them with the equivalent quantities from the ellipsoid model, i.e., the semi-axes b for the edge-on view and a for the face-on view. Similarly, we plot the corresponding acceleration profiles in panels (d) and (e).

We find that in both events the wave expansion speed overtakes the CME expansion speed early in the event evolution, at heliocentric distances below 2 R_⊙ as we infer when comparing with Figure 8(a)).

In the March event, the three semi-axes are similar to the edge-on and face-on half widths of the CME until about 20:07 UT (Figure 9(a)). Both CME and wave are accelerating as seen in panels (b) and (c). Between 20:09 and 20:24 UT, however, the wave continues to accelerate in the lateral directions while the CME decelerates. As a result, the wave expands rapidly, reaching a peak speed of 2000 km s⁻¹ at 20:24 UT while the CME lateral growth slows down, having achieved a top speed of 1000 km s⁻¹ between 20:10 and 20:11 UT. We note three interesting behaviors in the left panels: (i) the divergence in the CME/wave speeds precedes by about 2−3 minutes the separation of the wave and CME fronts, (ii) the speed and acceleration behavior of the wave follows closely the behavior of the face-on width of the CME, and (iii) the CME face- and edge-on speeds and accelerations are similar. Observations (i) and (ii) are the expected behaviors for a wave−driver pair (Lulić et al. 2013). Namely, the CME lateral expansion leads to compression of the ambient plasma and the creation of a wave front that stays close to the CME flank until the driver begins to decelerate (after about 19:55 UT). The wave (most likely a shock by that time) escapes with enough energy to reach 2000 km s⁻¹ and begins to decelerate after 20:24 UT. This is also the time when the CME widths reach a constant, and similar, speed of about 700 km s⁻¹.

The June event exhibits similar characteristics, albeit at lower levels, reflecting the lower impulsiveness of this event. In this case, the wave evolution seems to track the CME edge-on width slightly closer than the CME face-on width, but the behaviors are not significantly different than for the March event. The wave begins to separate from the CME flank from 06:30 UT onward (panel (a)). This is the same time that the CME face-on width begins to decelerate. By 06:32 UT, both CME widths are decelerating. The CME expansion (for both the face-on and edge-on widths) reaches ∼600 km s⁻¹, with the wave expansion speeds almost twice as much.