4.3 Modeling of energy-release processes

The ejection of a flux rope to the interplanetary space has widely been investigated using various kinds of models. Coronal mass ejections (CMEs) are one of the direct manifestation of this ejection. Since the ejection occurs in the corona with low plasma beta, the main forces exerting on a flux rope are the gravitational force, gradient force of magnetic pressure, and magnetic tension force. Losing a balance among them causes the ejection of a flux rope.

4.3.1 Bipolar system (single flux domain)

Several models explain ejection in bipolar system. The so-called mass-loaded model assumes that the drain of plasma from a flux rope reduces the gravitational force, leading to the ejection of a flux rope via enhanced magnetic buoyancy (Low and Hundhausen, 1995; Low, 1996; Wu et al., 1997). The flux-cancellation model shows that the dissipation of magnetic flux at the surface (known as flux cancellation) reduces the magnetic tension force of the overlying field that confines a flux rope so that the upward magnetic pressure dominates at some point of the evolution and an eruption occurs (the configuration has a low beta plasma and magnetic force dominates) (Linker et al., 2003Jump To The Next Citation Point, see Figure 37View Imagea). A similar mechanism has been proposed in the so-called the tether-cutting model (Sturrock et al., 1984; Moore et al., 2001Jump To The Next Citation Point, see Figure 37View Imageb). If a flux rope is composed of highly twisted field lines, then the kink instability might develop, causing the ejection of it (Sturrock et al., 2001; Fan and Gibson, 2003Jump To The Next Citation Point; Török and Kliem, 2005; Inoue and Kusano, 2006, see Figure 37View Imagec). The basic physics related to the kink instability, or more precisely the toroidal magnetic force, has also been studied in early analytical models (Mouschovias and Poland, 1978; Chen, 1989; Vršnak, 1990, see also references therein).

The so-called loss-of-equilibrium model suggests that there is a critical height of a flux rope, beyond which no neighboring equilibrium state exists, so if a flux rope reaches this height, then a dynamic transition inevitably occurs to cause eruption (Forbes and Isenberg, 1991Jump To The Next Citation Point; Amari et al., 2000Jump To The Next Citation Point; Roussev et al., 2003; Lin, 2004; Isenberg and Forbes, 2007, see Figure 37View Imaged). In fact, the flux cancellation model mentioned above demonstrates the dynamic process of the loss of equilibrium.

Recently, it has been proposed that the torus instability plays a key role in the ejection of a flux rope (Kliem and Török, 2006). They suggest that the spatial distribution of the magnetic field overlying a flux rope is an important factor in controlling the dynamic state of a flux rope. The loss-of-equilibrium and the torus instability are in fact two different views of the same physical process (Démoulin and Aulanier, 2010).

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Figure 37: Models for the eruption of a flux rope. (a) Flux-cancellation model (from Linker et al., 2003). (b) Tether-cutting model (from Moore et al., 2001). (c) Kink instability (from Fan and Gibson, 2003). (d) Flux cancellation model (from Amari et al., 2000). (e) Loss-of-equilibrium model (from Forbes and Isenberg, 1991).

4.3.2 Multi-polar system (two-step reconnection)

In multi-polar systems, the following models have been proposed to explain the ejection of a flux rope. In the model called breakout (Antiochos et al., 1999aJump To The Next Citation Point, Figure 38View Image), a current sheet is first formed at the interface between the inner and overlying flux domains. Magnetic reconnection then occurs at this current sheet, reducing the confining tension force of the overlying field. The inner domain then starts to expand, inside which another current sheet is formed and the second, much more energetic reconnection than the previous one, occurs to produce a flare. As a result of the second reconnection, a flux rope is formed, which eventually erupts into the interplanetary space.

Another model is called the emerging flux trigger model (Chen and Shibata, 2000Jump To The Next Citation Point, Figure 39View Image), in which newly emerging magnetic field interacts with the preexisting field that contains a flux rope. That interaction leads to the formation of a current sheet at the interface between those two fields. Magnetic reconnection then occurs in this current sheet, destabilizing the flux rope, which erupts via the second reconnection that is similar to the one explained in the breakout model. In these two models, the two-step reconnection is a key mechanism for producing a flare and the ejection of a flux rope. This mechanism also works in the helicity annihilation model (Kusano et al., 2004Jump To The Next Citation Point, Figure 40View Image).

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Figure 38: Destabilizing mechanism for the eruption of a flux rope, known as the breakout model presented in Antiochos et al. (1999a).
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Figure 39: Destabilizing mechanism for the eruption of a flux rope, known as the emerging flux trigger mechanism presented in Chen and Shibata (2000). The solid lines correspond to the magnetic field, the arrows to the velocity, and the color map to the temperature. At the top panels, the emerging field appears just on the polarity inversion line of the preexisting field, while it appears at one side of the inversion line at the bottom panels.
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Figure 40: Destabilizing mechanism for the eruption of a flux rope, known as the reversed magnetic-shear model developed in Kusano et al. (2004). Typical plasma flows are illustrated by thick arrows. The green surface in (b) represents an isosurface on which Vz = 0.1VA. The color maps on the side and bottom boundaries represent the flux density of sheared and vertical component of magnetic field.


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