There is a general consensus that solar eruptions, which lead to coronal mass ejections (CMEs) and are frequently associated with solar flares, are a consequence of instability of coronal structures, most often considered to be coronal magnetic arcades embedding a helically twisted magnetic flux rope (for a review see, e.g., Schmieder et al., 2015; Green et al., 2018). Basically, there are three different scenarios that can explain the presence of the flux rope within the eruptive structure: (i) an already-formed flux rope emerged from the subphotospheric layers; (ii) rope is formed gradually by a series of reconnections within a sheared arcade; (iii) it forms during the eruption itself (e.g., Green et al., 2018, and references therein). In this paper a situation where the flux rope exists already prior to the eruption (i.e., the mentioned first two options) will be studied to get an insight into physical background of the evolution of the pre-eruptive arcade/flux-rope structure.

Most of eruptions show three basically different stages: a gradual pre-eruptive stage, main acceleration stage, and the propagation stage (e.g., Vršnak, 2001; Zhang et al., 2001, 2004; Chen, 2011; Mierla et al., 2013; D'Huys et al., 2017). The pre-eruptive stage, which is the main objective of this paper, most briefly can be described as externally-forced evolution of the pre-eruptive system through a series of equilibrium states until a stage when the system comes to the point when no equilibrium of forces is possible anymore (e.g., Priest, 1982; Vršnak, 1990; Green et al., 2018, and references therein). After that the system finds itself in an instability regime, dynamically erupting in trying to find a new equilibrium state. Such type of evolution is usually denoted as loss-of-equilibrium scenario. A critical height where the system losses equilibrium and starts rapidly accelerating is usually comparable with the flux-rope footpoint half-separation(for the observatins see, e.g., Vršnak, 1990; Chen et al., 2006, for the theoretical aspect see, e.g., Vršnak 1990; Chen and Krall 2003; Chen et al. 2006).

From the observational point of view, the pre-eruptive stage, often called also a gradual-rise phase, is characterized by a number of different signatures. Frequently, the new magnetic flux emergence, shearing/twisting motions, and flux cancelation are observed (e.g., Schmieder et al., 2015; Green et al., 2018, and references therein). At the same time the pre-eruptive structure slowly rises at a low almost-constant velocity, and shows signatures of swelling (e.g., Tandberg-Hanssen, 1974; Vršnak et al., 1993; Maričić et al., 2004; Veronig et al., 2018), appearance of helical structures within the prominence (e.g., Tandberg-Hanssen, 1974; Sakurai, 1976; Vršnak et al., 1988, 1991, 1993; Rompolt, 1990; Romano et al., 2003; Ali et al., 2007), and mass-draining down the footpoints (e.g., Tandberg-Hanssen, 1974; Rust et al., 1975; Vršnak et al., 1987, 1993).

In this paper we analyze the influence of these processes on the evolution of the pre-eruptive system, focusing on the effects of the flux rope twisting, electric current increase, and the mass loss. The following analysis of these effects is based on the analytical semi-toroidal flux-rope model employed by Vršnak (2008), Vršnak (2016), and Green et al. (2018), where mainly a quantitative analysis of the acceleration stage of eruptions was considered. In contrast, this paper is focused on a quantitative study of characteristics of the pre-acceleration stage.

In Figure 1 the kinematics of a limb CME that erupted on May 15, 2001 is presented (for details see, Maričić et al., 2004), to illustrate a typical example of the height–time evolution of the eruption (see, e.g., Tandberg-Hanssen, 1974, and references therein). The height–time measurements of the frontal rim, the top of the cavity, and the top of the embedded prominence are displayed in Figure 1A, whereas the corresponding velocity–time graph is shown in Figure 1B. In both graphs the gradual-rise phase and the impulsive acceleration stage are indicated.

In Figure 2 the gradual-rise phase of the prominence is shown enlarged. The measured velocity slowly increases from 5 to 15 km s⁻¹, within Δt ~ 2 h, which corresponds to a very weak acceleration on the order of ~ 1 m s⁻². In the following, effects of the flux-rope twisting, external flux emergence, and mass loss will be considered, to find out if these processes can explain the described flux-rope behavior.

For the previously mentioned purpose, let us apply the semi-toroidal flux-rope model (Figure 3) proposed by Vršnak (1990) and elaborated by Vršnak (2008), Vršnak (2016), and Green et al. (2018), to study the effects of the increase of the flux-rope twist, the increase of the flux-rope electric current, and the mass leakage through the flux-rope legs (the former two processes are illustrated schematically in Figure 4). A simplified equation of motion, relevant for the processes which are essential for comprehending processes that govern the gradual-rise phase, can be expressed in the form that defines the force per unit mass (i.e., acceleration):

where Λ, R_t, and Z are the length of the flux rope axis, major radius of the torus, and the height of its summit, all normalized with respect to the footpoint half-separation d [Λ = λ/d, Rt=R^t/d, Z = z/d; for symbols see Figure 3; see also (Vršnak, 2008; Green et al., 2018)], whereas X represents the ratio of the poloidal and axial flux-rope field X ≡ B_φ/B_∥ at the flux-rope surface. The first two terms in brackets on the right-hand side of Equation (1) represent the so called “hoop force” (Chen, 1989), whereas the third term is due to the diamagnetic effect (Kuperus and Raadu, 1974). The last term represents the Lorentz force related to the background arcade magnetic field.

The expressions for the parameter C_L and C_c read:

respectively. Here, I_∥ and M represent the axial electric current and total mass within the flux rope, respectively, B_c is the background arcade field and μ₀ stands for the permeability.

Note that a, C_L, and C_c are expressed in m s⁻², and that in Equation (1) the drag force and gravity are neglected. Details of derivation of Equation (1) can be found in Vršnak (2008) and Green et al. (2018).

The geometrical quantities Λ, R_t, and Z are mutually related, i.e., Λ ≡ Λ(Z) and R_t ≡ R_t(Z). Taking approximately that the flux-rope axis remains a part of the circle with fixed footpoint separation 2d, and inspecting Figure 3, where the angle ζ is introduced, one finds simple parametric relationships: R_t = 1/cosζ, Z = (1+sinζ)/cosζ, and Λ = (π+2ζ)/cosζ. Thus, at given values of C_L and X, Equation (1) in fact defines a as a function of Z, a ≡ a_{CL, X}(Z). For low values of X and C_L, the structure is stable (da/dZ < 0) and has only one equilibrium height Z_s, where a_{CL, X}(Z_s) = 0 (e.g., black curve in Figure 5A). As X or C_L increase, the equilibrium height rises and the shape of a(Z) transforms, which is illustrated in Figure 5. To distinguish what are the effects of increasing X and C_L, the graphs a(Z) are shown separately for the case when the evolution of the system is driven by increasing X, and analogously, by increasing C_L. The former option is presented in Figures 5A,B, where the graph in Figure 5B shows an enlarged part of Figure 5A around the stable-equilibrium. Analogous graphs are displayed in Figures 5C,D for the effect of increasing C_L, where the increase of C_L can be either due to the increasing current I_∥ or decreasing mass M [see Equation (2)]. In Figure 5D, the stable-equilibrium heights are explicitly denoted as Z₁ – Z₄, and the height at which the system looses equilibrium as Z^* (red curve). In the insets of Figures 5B,D the change of the stable-equilibrium height Z_s(X) and Z_s(C_L) is presented, respectively, all the way to the state when the system looses equilibrium (marked by a square symbol).

Inspecting Figure 5, one finds that by increasing C_L and/or X, first the a(Z) curve evolves into a shape where another equilibrium point occurs Z_u. However, this one is unstable (green, yellow, and blue curves in Figures 5A,C), since da/dZ > 0. Thus, in this stage the structure is metastable, since if pushed from a stable equilibrium height Z_s to a height above the unstable-equilibrium height Z_u, it will erupt, because beyond this height the acceleration becomes a(Z) > 0 for any Z > Z_u.

In other words, Figure 5 shows that the increase of C_L or X implies rising of Z_s, and in the following this will be considered as a cause of the gradual rise of the pre-eruptive structure. Furthermore, one finds out that the unstable equilibrium height Z_u descends, so the distance between Z_s and Z_u decreases, implying that the structure becomes more and more unstable, since weaker and weaker push is required to move the structure from the stable to the unstable equilibrium point. Eventually, at a given critical combination of C_L and X, the stable and unstable heights merge, Z_u = Z_s, meaning that there is no equilibrium existing any more, i.e., equilibrium state is lost and the structure erupts (a(Z) > 0 for any Z; see red curve in Figures 5A–D).

In the following, Equation (1) will be employed to inspect the effects of the increasing C_L and X, as generally illustrated in Figure 5, by specifically considering the two processes depicted in Figure 4. These are twisting motions at the flux-rope footpoints (e.g., Török et al., 2013, and references therein) and emerging-flux process (e.g., Schmieder et al., 2015, and references therein). The former process leads to the increase of X, whereas the latter one directly induces an increase of the current I_∥, i.e., increases the value of the parameter C_L. It should be noted that the change of X causes also a change of I_∥, whereas the change of I_∥ causes also the change of X, i.e., the behaviors of X and C_L are tightly physically related. Finally, note that from the point of view of Equation (1), the mass loss related to the material draining down the prominence legs is equivalent to the increase of axial electric current I_∥ related to emerging flux and/or twisting processes, since the mass loss also increases the value of C_L [see Equation (2)].

In the presented analysis three processes relevant for the evolution of the pre-eruptive structure during the gradual-rise phase are considered. In particular, we focused on the height–time and velocity–time evolution, to compare it with typical behavior usually observed during the gradual-rise stage.

The first process is twisting motion at the flux-rope footpoint, which leads to the increase of the flux-rope twist, indirectly enhancing also the axial electric current and minor radius of the flux rope at its summit, causing the rise of the pre-eruptive structure. All of these signatures drive the pre-eruptive system toward a point where the structure losses its equilibrium and erupts. However, the considered twisting process is not effective enough to cause a sufficiently fast rise of the pre-eruptive structure, i.e., its rise is for at least an order of magnitude slower than observed. Note that the three examples presented in Figure 6 cover typical values coming from observations, and even increasing the parameter-values related the poloidal flux injection (including much too-high values) could not result in sufficiently high pre-eruptive rising speeds.

The second process is related to the emergence of a new magnetic flux beneath the flux rope. This causes an increase of the magnetic flux encircled by the flux rope, which induces an increase of the electric current flowing along the rope. The increase of the current causes also the increase of the flux-rope twist and its minor radius. The analysis has demonstrated that this process results in the rise of the pre-eruptive structure that is faster than in the case of twisting for almost two orders of magnitude, and is compatible with observations. This process is more effective than the footpoint twisting, since the emerging flux occurs on the spatial scale comparable with the flux-rope length, whereas the footpoint twisting is restricted to the footpoint radius, and that is generally an order of magnitude smaller than the flux-rope length. This results in an order of magnitude larger change of poloidal flux caused by the emerging flux (see Table 1). Note also that the processes like flux cancellation and other various forms of reconnection beneath the flux rope, as well as the converging motions of the arcade footpoints, can result in the same effects as described emerging flux process.

The third process, the mass leakage down the flux-rope legs, results in the pre-eruptive rise similar to that caused by the emerging flux process, except that the mass loss does not cause the increase of the flux-rope twist. Although the draining of the cold plasma from the embedded prominence is most prominent during the acceleration stage of the eruption, it is frequently observed already during the gradual-rise phase. This is caused by stretching of the flux rope as it rises, which makes the magnetic dips containing dense prominence plasma successively shallower, i.e., certain fraction of the prominence material is no longer supported against the gravity. Thus, the rise of the pre-eruptive structure caused by emerging flux is likely to be additionally enhanced by the mass-loss effect, making the pre-eruptive rise even faster than shown in Figure 7.

To conclude, the enhancement of the flux-rope electric current, the increase of the twist, and the mass loss, are tightly related phenomena, expected to occur during the gradual pre-eruptive phase of an eruption. Combination of these three processes is sufficiently effective to explain the nature of the gradual rise of the pre-eruptive structure and its evolution to the loss-of-equilibrium point.

The author confirms being the sole contributor of this work and has approved it for publication.