A Model for Energy Buildup and Eruption Onset in Coronal Mass Ejections

The formation of the filament channel is shown in Figure 1 at times early (t = 1000 s, when the vortical flows attain their peak speed) and late in the evolution, (t = 80,000 s, not long before the onset of eruption). Panels (d) and (f) show the shear component of the magnetic field, B_ϕ, color-shaded against the sky at the central meridian (ϕ = 0). The alternating red and blue contours in Figure 1(d) mark the weakly twisted structures generated initially by the individual vortical cells. Over time, these alternating bands of shear field strengthen and reconnect with their neighbors, transporting the shear flux toward the outer boundary of the minority-polarity region, i.e., the PIL. Figure 1(f) shows the resulting concentration in a localized region above the PIL, consistent with previous modeling of helicity condensation (e.g., see Figure 4 in Knizhnik et al. 2015). The three-dimensional filament channel penetrates the meridional plane twice, yielding two oppositely signed regions of B_ϕ. Viewed from above in Figure 1(e), the channel is seen to be quasi-circular and to envelope the whole PIL. It consists of an arcade of highly sheared, low-lying flux (orange) beneath weakly sheared, high-lying flux (red, above the southern segment of the PIL).

A key point is that the simple driver of our system consists solely of small-scale, compact, close-packed, vortical flows. If the system evolution were purely ideal, these flows would never produce the global-scale filament channel evident in Figure 1. The combination of the flows with helicity-conserving coronal magnetic reconnection, however, produces a very highly sheared filament channel whose properties are consistent with solar observations. As Figure 1(f) indicates, the resulting self-organized structure consists of a balance of forces between an outward magnetic pressure exerted by the out-of-plane field component and an inward magnetic tension exerted by the stretched in-plane field components. We discuss below how this force balance is disrupted.

The buildup of the filament channel shown in Figure 1 strongly perturbs the overlying magnetic field and sets the stage for the later eruption. As shown in Figure 2, between times t = 0 (a) and t = 59,500 s (b), the expansion of the separatrix dome moves the coronal magnetic null point outward and slightly equatorward. This motion also compresses the null to form a current sheet, which allows slow magnetic-breakout reconnection to begin there. Evidence for this can be seen in Figure 2(b): the two innermost green field lines that initially connected into the southern hemisphere now connect into the minority-polarity region in the north. This has removed flux that overlay the separatrix dome and the southern portion of the PIL by transferring it to overlie the northern portion of the PIL, weakening the restraining tension force above the dome and enabling further expansion. Additional evidence of breakout reconnection is shown by the orange field lines, which now connect to the southern hemisphere rather than to the minority-polarity region in the north as they did previously.

By time t = 89,000 s in Figure 2(c), the eruption is well underway. All but two of the green field lines now connect into the minority-polarity region and no longer overlie the southern portion of the PIL, where a section of the strongly sheared filament channel (bundle of orange field lines) is erupting outward at high speed, v_r > 1000 km s⁻¹ (dark red shading). At time t = 90,000 s in Figure 2(d), the ejected sheared flux has more than doubled its height above the surface, and reconnection has commenced below where the fluxes from each side have closed back together in the wake of the CME. This is evidenced by the green field lines that have reestablished their connections to the southern hemisphere at this time. Paired upflows (red) and downflows (blue) highlight the flare reconnection jets beneath the high-speed ejecta.

The flare reconnection not only restores all of the green field lines to their original connections to the southern hemisphere, as seen in Figure 2(e), but also creates entirely new connections of blue field lines to the southern hemisphere, whereas previously they connected only into the minority polarity in the north. This process weakens the field overlying the northern portion of the PIL, so now that segment of the sheared filament channel also erupts at high speed in a "sympathetic" ejection. (The orange sheared field lines that are drawn participate only in the first eruption, not the second.) This second eruption is deflected equatorward by the overlying field, as can be seen at time t = 93,000 s in Figure 2(f). The flare reconnection jets below the second CME are also clearly visible in this image. Therefore, although the eruptive event begins in the filament channel above the southern portion of the PIL, it eventually encompasses the entire channel and PIL encircling the minority-polarity region.

To clarify the mechanism for eruption onset, we display the global energetics and other properties of our simulation in Figure 3. Full time histories of the total, volume-integrated magnetic (U_M; black curve) and kinetic (U_K; blue curve) energies are shown in Figure 3(a). Filament channel formation occurs up to time t ≈ 88,300 s, marked by the vertical dashed line in the plot, when the eruption begins. During the formation phase, the vortical surface driving builds up the free magnetic energy at a nearly constant rate, until this energy eventually reaches U_M ≈ 6 × 10³² erg, which is about 20% of the initial potential-field energy. After a brief startup phase, the kinetic energy U_K exhibits two extended intervals of quasi-steady values: ≈2 × 10²⁹ erg early, and ≈5 × 10²⁹ erg later, with a transition occurring between t ≈ 50,000 s and t ≈ 60,000 s. This transition coincides with the onset of breakout reconnection in the corona, as discussed and illustrated in the preceding Section 3.2 and Figure 2. The early behavior is associated primarily with the imposed vortical flows and the pervasive reconnection outflows that follow from the helicity-condensation process; the late evolution has an additional kinetic-energy contribution due to the slow outward expansion of the filament channel and the overlying separatrix dome, along with some reconnection outflows near the null point. All of this flow energy remains small, amounting to only about 0.1% of the stored magnetic free energy, during the buildup phase.

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Also shown in Figure 3(a) is the maximum longitudinal field strength (${B}_{\phi }^{\max };$ red curve) within the computational domain. This quantity provides an approximate measure of the maximum magnitude of the shear field in the filament channel. The initial value in the potential field is about 30 G. As the vortical surface driving twists the magnetic field lines within the minority-polarity region, the maximum strength increases, peaking at about 70 G at time t ≈ 20,000 s. Thereafter, although the amount of shear flux continues to increase as the helicity condenses to build the filament channel, the upward expansion increases the cross-sectional area of the channel fast enough that the strength of the shear field declines. This result is expected from the classic arguments of the Aly–Sturrock free-energy limit (Aly 1991; Sturrock 1991). We note that there is a subtle, but perceptible, steepening of the slope of the B_ϕ^max curve simultaneous with the increase in kinetic energy, associated with the onset of slow breakout reconnection as noted above.

The initiation of the filament channel eruption at t ≈ 88,300 s (dashed line) in Figure 3(a) is accompanied by steep changes in all three quantities. The kinetic energy rises by more than two orders of magnitude, reaching a large fraction (about 10%) of the total stored magnetic free energy. This indicates a sudden transition from very low-speed flows to very high-speed flows, whose average Alfvén Mach number is about 0.2. The maximum longitudinal field strength nearly doubles very quickly as the shear flux below the eruption is compressed by the downward contracting flare loops (e.g., Zucarello et al. 2017). Energy is extracted from the magnetic field to power the high-speed flows and the local B_ϕ compression, as indicated by the approximately 50% drop in magnetic energy during the eruption.

In order to understand in more detail the transition from channel formation to eruption, we expanded the small interval indicated by the yellow shading in Figure 3(a) for display in Figure 3(b). There we show the rates of change of the total magnetic (dU_M/dt; black) and kinetic (dU_K/dt; blue) energies, along with that of the total flare-reconnected flux dΦ_flare/dt; green), which we measured by tracking discontinuous changes in the lengths of magnetic field lines rooted in the minority-polarity region. This plot shows that strong flare reconnection begins first, and is followed by initiation of the rise in kinetic energy simultaneous with the decline in magnetic energy. This sequence demonstrates conclusively that reconnection plays the primary role in initiating the explosive eruption of the filament channel, and is fully consistent with previous work on magnetic breakout for both CMEs/EFs (Karpen et al. 2012) and coronal jets (Wyper et al. 2017, 2018).

The reconnection dynamics that accompany the filament channel formation and eruption are illuminated by examining the topological evolution of the system in Figure 4. Figure 4(a) depicts the initial magnetic connectivity. The solid-shaded medium blue region in the north maps to the southern hemisphere (diagonal dark blue shading), while the solid-shaded medium red region in the north maps to the minority-polarity flux in the active region (diagonal dark red shading). The black contour is the intersection of the separatrix dome, which encloses all of the flux connecting to the minority-polarity region, with the solar surface. All field lines outside of the minority-polarity region are line-tied and fixed ( v_⊥ = 0), and only those lines are used to compute the connectivity maps. Hence, all changes in the connectivity shown in Figure 4 occur due to reconnection.

As the minority-polarity flux is twisted by the vortical motions and then accumulates concentrated shear via helicity condensation, magnetic pressure builds up within the separatrix dome and deforms the coronal null to form a current sheet. Over time, the resulting reconnection changes the connectivity of the flux at the solar surface. By time t = 59,000 s, some of the majority-polarity flux that initially closed within the separatrix dome (medium red, Figure 4(a)) has reconnected and now closes to the southern hemisphere (light blue, Figure 4(b)). Correspondingly, an equal amount of the majority-polarity flux that initially closed to the southern hemisphere (medium blue, Figure 4(a)) now closes to the minority-polarity region (light red, Figure 4(b)). Thus, the breakout reconnection has the effect of transferring the flux that overlies the southern segment of the PIL (the straight portion of the dark red minority-polarity boundary) so that it overlies the northern segment (the curved portion of that boundary).

This process continues and expands the area of changed connectivity until the flux overlying the southern segment of the PIL weakens sufficiently to allow eruption to occur. Figure 4(c) shows the topology at t = 89,250 s, shortly after flare reconnection onset and initiation of the eruption above the southern segment of the PIL (Figure 2(c)). Additional flux has been removed from above the southern segment of the PIL (see in particular the light blue regions in the southern half of the active region, where a small area corresponds to a comparatively large amount of flux due to the high field strength). The flux that has undergone closed–closed flare reconnection within the separatrix dome is shown in green. This region corresponds to the classic flare ribbons. It maps out a region along the southern segment of the PIL and, due to the shear concentrated there, the kernel of reconnected flux south of the PIL is displaced to the east. We note also that the separatrix boundary has become highly fragmented and fractal in character, possessing structure down to the grid scale.

As the eruption proceeds, the connectivity continues to evolve in structure and complexity. Subsequent maps are shown at times t = 90,000 and 91,000 s in Figures 4(d) and (e) for comparison with the preceding Figures 2(d) and (e). As the flare reconnection progressively processes flux in the green and blue flux systems drawn in Figure 2 at these times, the connectivity of the field lines switches from the minority-polarity region in the northern hemisphere to the southern hemisphere. This change is indicated by the new light blue regions adjacent to the northern separatrix boundary in Figure 4, especially at the later time (Figure 4(e)). The strong reduction in the flux overlying the northern segment of the PIL enables that section of the filament channel to erupt, as we noted above in Section 3.3. In the wake of that secondary ejection, flare reconnection above the northern segment of the PIL then acts to restore the connections of the blue field lines from the southern hemisphere back to the north, as shown in Figures 2(e) and (f). This is reflected in the change in the connectivity-map colors from light blue to medium red in Figures 4(e) and (f).

After the conclusion of the two episodes of CME ejection and flare reconnection, above first the southern and then the northern segments of the PIL, the system contains subcritical amounts of helicity and magnetic free energy and relaxes toward a new equilibrium state. The connectivity at the end of the simulation is shown in Figure 4(f). The separatrix dome has taken on a configuration that rather closely resembles the initial state, and all of the connectivity changes at this time are localized near the original separatrix boundary. Ongoing helicity condensation now can rebuild the free energy and helicity in the channel until it once again attains the critical condition for a new eruption.