FILAMENT INTERACTION MODELED BY FLUX ROPE RECONNECTION

We construct the initial magnetic field by superposing two TD configurations (Figure 2). The flux ropes are oriented along the y direction and placed at x = ±1.6. Guided by the footpoint locations of the filaments before and after their interaction (Figure 1), we choose the axial magnetic field B_y > 0 (<0) for the eastern, x < 0, (western, x > 0) flux rope. To account for the observed sign of photospheric flux surrounding the middle sections of the filaments, we choose the magnetic charges located between the TD tori to be positive. Hence, to enable equilibrium, the flux rope current has to flow from north to south (south to north) in the eastern (western) rope, yielding left-handed twist for both ropes. The normalized parameters are chosen identical for both TD tori: R = 2.75, a = 0.7, d = 1.75, |L| = 1, and |q| = 2.33, yielding a twist of ≈1.35 windings (averaged over the torus cross-section), which is below the threshold of the helical kink instability (Török et al. 2004).

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In order to facilitate the diverging flows described in Section 3.2, we shift the positive charges away from y = 0, to y = −1 (1) for the eastern (western) flux rope. Furthermore, to obtain more compact photospheric potential field sources for the given TD parameters, we place all charges at a depth z = −d/2. To assure an approximate equilibrium after these modifications, we finally replace the above-mentioned values of q by 0.7 and −1.4 for the positive and negative charges, respectively. The final configuration (after numerical relaxation) is shown in Figure 2(a). The potential field is open at large heights and contains a null point at (0, 0, z = 4.7).

As in previous simulations of the TD model (e.g., Török & Kliem 2005), we integrate the zero β compressible ideal MHD equations, neglecting thermal pressure and gravity. The equations are discretized on a non-uniform Cartesian grid of size [ − 5, 5] × [ − 8, 8] × [0, 8], with almost uniform resolution Δ ≈ 0.04 where the flux ropes are located. The plane {z = 0} corresponds to the photosphere. The equations are normalized by the initial axis apex height of the TD tori, R − d, the maximum initial field strength, $B_{0_{\rm max}}$, and the Alfvén velocity, $v_{a0_{\rm max}}$, at the position of $B_{0_{\rm max}}$, (x, y, z) = (±2.64, 0, 0), and derived quantities. The Alfvén time is $\tau _a=(R-d)/v_{a0_{\rm max}}$. The boundary conditions are as in Török & Kliem (2003), to which we refer for further numerical details. The initial density distribution is ρ₀(x) = |B₀(x)|^3/2, such that v_a slowly decreases with the distance from the flux concentrations.

We first relax the system for 114 τ_a. The resulting configuration (Figure 2(a)) shows no indication of dynamic behavior. We now have to bring the ropes into contact at their middle sections, for which we employ quasi-static flows at the bottom boundary.

Since the potential field between the flux ropes is unipolar and only weakly sheared (Figure 2(a)), driving the ropes toward each other would yield a pile-up of flux between them, hindering the ropes from coming into contact and reconnecting. Therefore, we first apply a diverging flow that moves the positive potential field polarities out of the area between the ropes. It also weakens the flux between them, as suggested by the observations (Section 2). To this end, we impose the following flow pattern in z = −Δz, within the area (|x| < 1.25, |y| < 6):

$$\begin{eqnarray*} u_{y}(t)& =& h(t)\,u_m\,\cos (2 \pi x/5)\,\mathrm{sign}(y)\\ &&\times {\left\lbrace {\begin{array}{@{}ll@{}}\mathrm{sin}(\pi |y|/2),& 0 \le |y| < 1 \\ 1,& 1 \le |y| < 5 \\ \mathrm{sin}(\pi [|y|-4]/2),& 5 \le |y| < 6. \end{array}}\right.} \end{eqnarray*}$$

Here, u_m = 0.02 and u_x,z(−Δz, t) = 0. The flow decreases smoothly to zero toward the edges of the area. Outside it, u_x,y,z(−Δz, t) = 0. We use h(t) to linearly ramp the velocities up and down for 10 τ_a before and after the constant driving, respectively. The latter is applied for 110 τ_a. Afterward, the system is relaxed for 113 τ_a and time is reset to zero. The resulting configuration is shown in Figure 2(b).

In a second driving phase, we impose a similar flow pattern in z = −Δz, now within the area (|x| < 4, |y| < 1.5):

$$\begin{eqnarray*} u_{x}(t) & =& -h(t)\,u_m\,\cos ({2 \pi y/6})\,\mathrm{sign}(x)\\ && \times {\left\lbrace {\begin{array}{@{}ll@{}}\mathrm{sin}(\pi |x|),& 0 \le |x| < \frac{1}{2} \\ 1,&\frac{1}{2} \le |x| < 3 \\ \mathrm{sin}(\pi [|x|-2]/2),& 3 \le |x| < 4. \end{array}}\right.} \end{eqnarray*}$$

Here, u_m = 0.02 and u_y,z(−Δz, t) = 0. Again, all velocities are zero outside the area. The flow is ramped up and down for 10 τ_a, respectively, and is constantly driven for 80 τ_a. This yields a converging flow of the negative potential field polarities, which slowly moves the flux rope middle sections toward each other. Afterward, the system is finally relaxed for 65 τ_a.

During the diverging flow phase, the magnetic field between the flux ropes weakens and becomes increasingly sheared. As a consequence, the ropes expand slightly and lean toward each other (Figure 2(b)), and a localized vertical current layer forms between them, extending up to the region around the null point (Figure 4(b)). As the driving is stopped, the system relaxes toward a new numerical equilibrium in which the forces exerted by the ambient field on the flux rope currents and the repelling forces of these currents balance each other. The rope currents are very close to each other at their middle sections, divided by the current layer. Note that this layer is distinct from the flux rope currents. The current in the layer is directed primarily vertically, due to a change in sign across the layer of the horizontal component of the fields (Figure 2(b)). In contrast, the flux rope currents are directed primarily horizontally, parallel to the rope axes.

Figure 5 shows the evolution of the current densities in the lower section of the layer (where the flux ropes approach each other) during the driving phase and subsequent relaxation. The current densities grow exponentially, at an approximately constant rate, until the diverging flows are switched off, and do not change significantly afterward. They remain relatively weak, with the maximum current density in the layer reaching only about 15% of the maximum current density inside the flux ropes.

We checked different durations of the divergence flow phase. For too short durations, the field between the flux ropes does not get sufficiently weakened, and the converging flows applied later on lead to a strong pile-up of positive flux between the ropes, hindering their reconnection. Diverging flow phases longer than the one described here do not further weaken the field significantly, and the system evolves toward a configuration similar to the one shown in Figures 2(b) and 4(b). For the model settings considered here, converging flows hence appear essential for triggering reconnection.

The modeled filaments shown in Figure 2 seem to widen strongly during the diverging phase. This is due to the fact that dips typically form a thin vertical sheet within a flux rope (e.g., Dudík et al. 2008), and that the flux ropes here have oblong, rather than circular, cross sections. As the ropes lean to the side, so do the sheets of dips, which therefore appear to widen when seen from above. This effect might contribute to the frequently observed widening of filaments shortly before their eruption (see, e.g., Figure 3 in Gosain et al. 2009), if the flux rope carrying the filament material slowly rises in a direction inclined from the observer's line of sight.

As the converging flows are applied, the evolving ambient field continuously pushes the flux ropes against each other, and the lower parts of the ropes are additionally pushed toward the box center by the imposed flows. However, although the flux ropes are already very close to each other before the onset of the converging flows (Figure 4(b)), they do not reconnect significantly with each other for about almost 60 Alfvén times (Figure 3(a)). This slow onset of reconnection can be explained by the fact that the field lines at the opposing rope surfaces are initially still almost purely azimuthal, hence almost parallel to each other, and therefore resist to reconnect. However, since the ropes are continuously driven together, they deform and the contact angle between the field lines slowly increases. This allows reconnection to set in, and the driving ensures that it continues even though the field line contact angle is initially small. Then, as the almost purely azimuthal outer field lines reconnect away, the contact angle between successively reconnecting field lines quickly increases and the reconnection accelerates, leading to the full splitting and reconfiguration of the ropes within ≈10 τ_a (Figure 3(b) and 3(c)). Since the flux ropes thereby fully exchange their footpoint connections, this reconnection is of slingshot type (Linton et al. 2001). Reconnection continues in the volume afterward, but in a less impulsive manner and without further affecting the flux rope connectivities.

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The evolution of the current densities in the lower part of the vertical current layer can be inferred from Figure 5. As the converging flows are applied, the current densities start to grow exponentially, now with a significantly larger growth rate than in the diverging flow phase. At t ≈ 60, around the time when the reconnection between the flux ropes accelerates, the maximum current density inside the layer has become about five times larger than the maximum value inside the flux ropes. Between t ≈ 50 and the onset of accelerated flux rope reconnection, the current densities undergo a phase of additional increase, which may be attributed to flux pile-up between the flux ropes as they fail to "adjust" to the continuous pushing by further deformation (a similar behavior of the current densities was observed in the simulations of kink-unstable flux ropes by Török et al. 2004, where the flux rope in the downward kinking case continuously pushed the current layer toward the rigid bottom boundary; see their Figures 3 and 4). After t ≈ 70, when the ropes have fully reconnected, the current densities in the layer slowly decrease as the system relaxes toward a new equilibrium.

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We find slingshot reconnection between flux ropes with the same sign of twist. This contrasts with Linton et al. (2001), who found that ropes with a strongly azimuthal field at the surface and oppositely directed axial field bounce, without significant reconnection. This can be explained by the fact that the ropes in Linton et al. (2001) were not continuously driven together. Therefore, the reconnection did not have time to fully develop. In later simulations, however, with a weaker azimuthal field at the flux rope surfaces, and a correspondingly larger initial field line contact angle, Linton (2006) indeed found that slingshot reconnection can occur for flux ropes of the same sign of twist.

Figure 3 shows that the model filaments follow the morphological evolution of the flux ropes, hence qualitatively reproducing the observations (Figure 1). Mimicking the filaments by dip locations is actually not appropriate during the highly dynamic reconnection phase. However, since the system evolves almost quasi-statically before the reconnection, and relaxes toward a new equilibrium afterward (no indications of flux rope instabilities were found after the reconnection), calculating the dips before and after reconnection is sufficient to show that the model filaments exchanged their footpoint connections.

Here, we have used flux ropes with strong azimuthal fields at their surfaces to model the filament-carrying magnetic fields. We expect qualitatively similar results (i.e., merging and breaking) if suitably oriented sheared arcade fields (or weakly twisted flux ropes) were chosen instead, since the occurrence of slingshot reconnection is primarily due to the oppositely oriented axial fields in the (weakly twisted) cores of our reconnecting flux ropes. It can be expected, though, that the reconnection would set in much earlier for sheared fields, since the initial bouncing due to the almost parallel surface fields occurring in our simulation would be absent or much less pronounced. Accordingly, the evolution of the reconnection should be more strongly correlated with the photospheric driving, hence less impulsive and "slingshot-like." Sheared arcade fields may, however, not yield a sufficient number of field line dips after their reconnection to account for the observed filament pattern. Therefore, more sophisticated simulations (including a proper treatment of the coronal thermodynamics) may be required to test the ability of sheared arcade fields to reproduce the observed filament dynamics.