SIMULATIONS OF EMERGING MAGNETIC FLUX. II. THE FORMATION OF UNSTABLE CORONAL FLUX ROPES AND THE INITIATION OF CORONAL MASS EJECTIONS

The equations are presented here in Lagrangian form:

$$\begin{eqnarray} \frac{D\rho }{Dt} & = & -\rho \nabla.\mathbf {v}, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \frac{D\mathbf {v}}{Dt} & = & -\frac{1}{\rho }\left[\nabla P + \mathbf {j}\wedge \mathbf {B} + \rho \mathbf {g} + \nabla.\mathcal {S}\right], \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \frac{D\mathbf {B}}{Dt} & = & (\mathbf {B}.\nabla)\mathbf {v} - \mathbf {B}(\nabla.\mathbf {v}) - \nabla \wedge (\eta \mathbf {j}), \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \frac{D\epsilon }{Dt} & = & \frac{1}{\rho }\left[-P\nabla.\mathbf {v} + \varsigma _{ij}S_{ij} + \eta {j}^{2}\right], \end{eqnarray} \tag{ 4 }$$

where ρ is the mass density, v the velocity, B the magnetic field, and the specific energy density. The current density is given by j = ∇ × B/μ₀, μ₀ is the permeability of free space, and the resistivity η = 14.6 Ωm. The gravitational acceleration is denoted by g and is set to the value of gravity at the mean solar surface ($\mathbf {g}_{{\rm sun}} = 274 \,\hbox{m}\hbox{s}^{-2} \hat{\mathbf {z}}$). $\mathcal {S}$ is the stress tensor which has components $\mathcal {S}_{ij}=\nu (\varsigma _{ij}-({1}/{3})\delta _{ij}\nabla.\mathbf {v})$, with ς_ij = (1/2)((∂v_i/∂x_j) + (∂v_j/∂x_i)). The viscosity ν is set to 3.35 × 10³ kg(m s)⁻¹, and δ_ij is the Kronecker delta function. The gas pressure, P, and the specific internal energy density, , can be written as

$$\begin{eqnarray} p & = & \rho k_{B}T/\mu _{m}, \ {\rm and} \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \epsilon & = & \frac{k_{B}T}{\mu _{m}(\gamma -1)} \end{eqnarray} \tag{ 6 }$$

respectively, where k_B is Boltzmann's constant and γ is 5/3. The reduced mass, μ_m, is given by μ_m = m_fm_p where m_p is the mass of a proton, and m_f = 1.25.

The equations are non-dimensionalized by dividing each variable (C) by its normalizing value (C₀). The set of equations requires a choice of three normalizing values. We choose normalizing values for the length (L₀ = 1.7 × 10⁵ m), magnetic field (B₀ = 0.13 T), and gravitational acceleration (g₀ = g_sun = 274 m s⁻²). From these three values the normalizing values for the density gas pressure ($P_{0}=B_{0}^2/\mu _{0}= 1.34\times 10^{4}\, \hbox{Pa}$), density ($\rho _{0}=B_{0}^2/(\mu _{0}L_{0}g_{0})=2.9\times 10^{-4} \,\hbox{kg} \, \hbox{m}^{-3}$), velocity ($v_{0}=\sqrt{L_{0}g_{0}}=6.82\times 10^{3} \,\hbox{m} \, \hbox{s}^{-1}$), time ($t_{0}=\sqrt{L_{0}/g_{0}}$ = 24.9 s), temperature (T₀ = m_pL₀g₀/k_B = 5.64 × 10³ K), current density (j₀ = B₀/(μ₀L₀) = 0.609 A m²), viscosity ($\nu _{0} = B_{0}^{2}\sqrt{L_{0}/g_{0}}/\mu _{0}=3.35\times 10^{5} \,\hbox{kg}(\hbox{m} \, \hbox{s})^{-1}$), and resistivity ($\eta _{0} = \mu _{0}L_{0}^{({3}/{2})}g_{0}^{({1}/{2})} = 1.46\times 10^{3}\, \Omega$m) can be derived. With these values of normalization, and the values of ν and η given above, the Reynold's number and magnetic Reynolds number in this simulation are both 100.

The simulation grid is the same as used in Paper I, and is stretched in all three directions. In the vertical direction, z, the grid extends from −30L₀ to 210.45L₀ with a vertical resolution of 0.428L₀ at the bottom boundary and 1.99L₀ at the top boundary. In the horizontal directions, x and y, the grid is centered on 0 and has side boundaries at ±126.85L₀. The horizontal resolution at x = y = 0 is 0.658L₀, and at the side boundaries is 2.61L₀.

As in Paper I, at the boundary all components of the velocity are set to zero, and the gradients of magnetic field, gas density, and specific energy density are set to zero. The resistivity is also smoothly decreased to zero close to the boundary to eliminate diffusion of magnetic field at the boundary. This approach ensures as much as possible that the side boundaries are line-tied. In addition, the velocities are damped close to the horizontal side boundaries and above z = 180L₀ near the top z boundary, as described in Paper I.

The initial conditions consist of a hydrostatic background atmosphere which represents the upper 30L₀, or 5.1 Mm, of the solar convection zone, plus the photosphere/chromosphere, and the corona up to 210.45L₀, or 35.8 Mm, above the surface. The temperature gradient in the convection zone is equal to its adiabatic value (Stix 2004). The photosphere/chromosphere (0 < z < 10L₀) is isothermal with temperature T_ph = T₀, and the corona (z > 20L₀) is isothermal with temperature T_cor = 150T_ph. There is a transition region between the photosphere/chromosphere and corona (10L₀ < z < 20L₀) which has a power law profile

$$\begin{equation} T(z)=\left[\left(\frac{T_{{\rm cor}}}{T_{{\rm ph}}}\right)^{(\frac{\frac{z}{L_{0}}-10}{10})}\right]T_{{\rm ph}}. \end{equation} \tag{ 7 }$$

The magnetic field consists of a background dipole field that permeates the entire domain, and a twisted flux tube superimposed in the model convection zone. The dipole field is translationally invariant along y, the tube's axial direction, and is given by B = ∇ × A where A = A_ye_y and

$$\begin{equation} A_{y}(x,z) = B_{d}\frac{z-z_{d}}{r_{1}^{3}}, \end{equation} \tag{ 8 }$$

where $r_{1}=\sqrt{x^2+(z-z_{d})^2}$ is the distance from the source. We choose z_d to be −100L₀ so that the initial sub-surface flux tube is far from the source of the dipole field. The twisted flux tube is aligned along the y axis, at a height of z = z_tube = −12L₀. The flux tube axial field strength B_ax exponentially decays with radius from the center B_ax(r) ∼ exp (− r²/a²) where a = 2.5L₀. The tube field has a constant twist q = −1/a, with the twist field B_θ(r) = qrB_ax(r). The tube is perturbed such that it is buoyant at the center (y = 0) and neutrally buoyant at the ends (y boundaries). This magnetic configuration is the same as in Paper I, but the dipole field has the opposite orientation (B_x for the dipole field has the opposite sign). The background atmosphere and plasma β (2μ₀P/B²) along the z axis are shown in Figure 1, which also shows a 3D representation of the fieldlines associated with the initial convection zone flux tube and the dipole field. Above the flux tube axis, the horizontal field B_x changes sign at the separatrix between flux tube and dipole, and this separatrix is a favorable location for magnetic reconnection. We perform four different simulations with differing strengths of dipole (B_d). We can quantify the strength of the dipole field relative to that of the flux tube by comparing the azimuthal flux per unit length in y in the dipole above the tube

$$\begin{equation} \Phi _{{\rm dip}} = \int _{z_{{\rm sep}}}^{z_{{\rm top}}}B_{x}(x=0,y=0,z)dz, \end{equation} \tag{ 9 }$$

to that in the top half of the tube

$$\begin{equation} \Phi _{{\rm tube}} = \int _{z_{{\rm tube}}}^{z_{{\rm sep}}}B_{x}(x=0,y=0,z)dz, \end{equation} \tag{ 10 }$$

where z_sep is the intersection of the z axis and the separatrix between the tube's field and the dipole field, and z_top is the top of the vertical domain. As the horizontal field B_x is nearly antiparallel on either side of this separatrix, under favorable forcing, these two fluxes could reconnect until one of the fluxes is destroyed. However, previous flux emergence simulations show that as the flux tube emerges through the photosphere, only the fieldlines which are concave down and able to shed mass continue to emerge into the corona (e.g., Magara 2001). As in Paper I, we perform three different simulations, Strong Dipole (SD), Medium Dipole (MD), and Weak Dipole (WD), which have decreasing dipole strengths. In this paper, the dipole strengths are chosen such that Φ_dip/Φ_tube = 0.13, 0.1, 0.067, respectively. We also add results from a simulation where no dipole exists (Simulation ND, presented in Paper I).

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Figure 2 shows the partial emergence of the convection zone flux tube into the overlying dipole field for Simulation MD. Despite the orientation of the dipole being opposite to that of the dipole in the simulations of Paper I, the early emergence is quantitatively similar to the emergence in those simulations, as in the convection zone the magnetic field of the tube is much larger than that of the dipole. The sub-surface flux tube rises to the surface, experiences a significant horizontal expansion, which is primarily caused by the suppression of the rise by the convectively stable photosphere/chromosphere (e.g., Archontis et al. 2004), and then begins to emerge into the atmosphere via the magnetic Rayleigh-Taylor instability. Whereas in Paper I the dipole was aligned so reconnection was most favorable beneath the flux tube axis, in the simulations in this paper reconnection is more favorable above the axis, as can be seen in Figure 2 where some of the gray fieldlines reconnect with the emerging field above the tube's axis. This reconnection of dipole field and emerging field is a continuous process. As the flux of the tube is much larger than the flux of the dipole, reconnection has very little effect on the tube's rise in the convection zone. However, as the tube partially emerges into the corona, the relative amount of flux in the emerging field and dipole field becomes comparable, and magnetic reconnection between the two systems becomes important.

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This continued reconnection in the corona between dipole field and emerging field changes the connectivity of both flux systems. This connectivity change is shown in Figure 3, which shows a later stage in the emergence for Simulation MD at times t = 100t₀, 110t₀, and 120t₀. The gray dipole fieldlines, which originate at the lower boundary, reconnect with the emerging field, and leave the domain at the side y boundaries near the axis of the convection zone flux tube. These reconnected fieldlines of the dipole and flux tube create "lobes" on either side of the emerging structure, which can be seen in Figure 3, panels (b) and (c). Figure 3 also shows isosurfaces of |j/j₀| > 0.004 above z = 50L₀, where j is the current density and j₀ = B₀/(μ₀L₀).

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The view from above, in panels (d)–(f), shows how the connectivity of the field changes and how the structure of the side lobes is formed by the reconnection. The snapshots shown in Figure 3 also highlight how the reconnection acts to remove overlying field and allow further vertical expansion of the central emerging structure. Horizontal expansion is restricted by the creation of flux lobes on either side of the arcade

Figure 3, panel (b) in particular, shows how this reconnection creates a quadrupole structure above the surface, with a central arcade expanding vertically into the corona, an overarching dipole field, and side lobes caused by the reconnection of this central arcade and the dipole field. This is qualitatively similar to the magnetic field configuration used in the magnetic breakout CME model (e.g., Lynch et al. 2008) where a quadrupolar field is used as the initial magnetic configuration, with a null point separating the central arcade and the overlying dipole field. In the magnetic breakout CME model, the central arcade is sheared by surface motions to create magnetic field in the axial (y) direction, perpendicular to the plane of the arcade (and referred to as "shear field"). This sheared field drives an outward expansion, which deforms the original null point between the arcade and dipole field into a current sheet. Reconnection is most favorable directly above the central arcade, and reconnection at this current sheet above the central arcade, hereafter known as external reconnection, allows further vertical expansion. In the simulations in this paper, as in the simulations in Paper I, twist field (with a relatively small component in the y direction) emerges first. Later in time, as the axis of the flux tube emerges through the surface, an increasing amount of magnetic energy is present in the shear (y) component of the field. Thus the continued emergence of the flux tube creates a sheared arcade structure which is similar to the configuration created by shearing motions in the breakout model, as shown in Figure 3.

Figure 4 shows the state of the emergence for Simulations SD, MD, and WD at time t = 100t₀. The gray fieldlines show that not much of the dipole field has reconnected with the emerging structures. The orange, blue, and black fieldlines originate on circles centered on the flux tube axis at the side boundary at radii of 0.5L₀, L₀, and 2L₀, respectively. In all three simulations, we see the same sheared arcade formed, with reconnection between emerging field and dipole field creating the quadrupolar structure above the surface.

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The simulations in this paper show that the emergence of a sub-surface flux tube from the convection zone into a simple dipole field, orientated so as to favor reconnection with the upper twisted fieldlines of the tube, can create the shearing quadrupolar configuration used in the magnetic breakout CME model. The axis of the flux rope in this simulation is providing the role of the shear field, and the emergence of the tube from the convection zone brings this shear field into the lower atmosphere, which drives further reconnection between emerging field and dipole field, hence allowing further emergence into the corona. In this way the breakout model has been generalized by being driven by a more realistic emergence of the free energy required to drive a CME. We now investigate how the continued emergence process affects this quadrupolar structure, and whether it can create an unstable configuration which erupts.

As discussed in Fan (2009) and Paper I, the emergence of the flux tube into the atmosphere is partial in the sense that only certain portions of fieldlines expand into the corona, while other portions remain near the surface. The portions that do emerge expand and lengthen, thus reducing their twist per unit length, while the non-emerging portions retain the same twist per unit length. This can be seen in Figures 8 and 9 in Paper I and in Figure 13 of Fan (2009). This gradient in twist along fieldlines drives twist to propagation from the convection zone into the corona along the length of the fieldlines, which results in an apparent rotation of each sunspot about a central point, and the twisting up of the emerged coronal field.

The rotation of the sunspots is represented by the horizontal velocity vectors in Figure 5, panels (d)–(f) at times t = 70t₀, 85t₀, and 100t₀, respectively. In Figure 5 the black (purple) fieldline originates from the location of the convection zone flux tube axis on the y = max y (y = min y) boundary. At time t = 70t₀ these two fieldlines are coincident in space. As the twist equilibrates along the fieldlines, the sunspots appear to rotate, as indicated by the velocity arrows on the surface (z = 0), and shown in more detail in Paper I. The two axial fieldlines now diverge due to non-zero resistivity, and appear to wrap around a common point. As in Paper I, we designate a new axis by taking the fieldline which intersects the O-point of the in-plane magnetic field in the y = 0 plane, which the black and purple fieldline now twist around. This axis fieldline is indicated by the orange fieldline in Figure 6, which also shows the in-plane magnetic field (B_x, B_z) in the y = 0 plane. Early on in the emergence (t ⩽ 70t₀) there is one single O-point above the surface, which is originally intersected by the flux tube axis fieldlines (black and purple lines in Figures 5 and 6). During the emergence, these fieldlines twist around the O-point, and a different fieldline goes through this O-point (the orange line in Figure 6). The O-point rises as the central arcade expands vertically due to the external reconnection with the overlying dipole field. At time t = 120t₀, shown in Figure 6, panel (c), there are clearly two O-points in the in-plane field above the surface, which indicates internal magnetic reconnection is occurring. This will be discussed in more detail in the next section.

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Figure 7 shows the same snapshots in time for Simulation MD as Figure 6, but without the orange fieldline that intersects the O-point in the y = 0 plane. Figure 7 also shows isosurfaces of |j/j₀| > 0.2 localized beneath the O-point and above z = 10L₀. Note that the strong currents associated with the initial flux tube remain near the surface below z = 10L₀. Panels (a)–(c) in Figure 7 show a buildup in current density above z = 10L₀, which is caused by the vertical deformation of the magnetic field. The lines that follow (B_x, B_z) in the y = 0 plane indicate that this current is mainly j_y ∼ ∂B_z/∂_x and this is borne out by calculations of the individual contributions to the current density. By t = 120t₀ evidence of reconnection can be seen, in the form of the formation of an X-point in the y = 0 plane (white lines).

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The current sheet viewed from above in panels 7(d)–(f) resembles two structures which grow and combine in the center of the active region, as was also seen in the simulations in Paper I. Various observational studies suggest that the formation and coalescence of J-shaped loops occurs before the formation and eruption of coronal flux ropes (Canfield et al. 1999; Sterling et al. 2000; Liu et al. 2010). Whereas in the simulations of Paper I, no obvious evidence of magnetic reconnection, such as an X-point or outflows, was observed at the site of the current density buildup beneath the O-point, X-points are observed in the simulations in this paper. To demonstrate further evidence of reconnection observed in these simulations, Figure 8 shows a 2D slice of current density in the y = 0 plane, at three different times. Along with this slice is a line plot of the vertical velocity along the line x = y = 0. In panel (a), the faint structure above z = 30 is the flux rope, and the strong vertical structure beneath is the current sheet seen in Figure 7, but at a later time. In panel (b) the flux rope has rapidly expanded out of the field of view (this rapid expansion will be discussed in the next section), and the current sheet's extent in the vertical direction has increased. Panel (e) shows that there exists a bidirectional vertical flow along the x = y = 0 line, an indication that reconnection is occurring at a height of z = 21L₀. Panels (c) and (f) show that at a later time the site of reconnection has risen up to z = 30L₀, as the flux rope rapidly expands high into the model corona, and "drags" the current sheet with it. In panel (f), short loop-like structures below z = 30L₀ can be seen, and the structure of the current density closely resembles the classical CSHKP flare model (Carmichael 1964; Sturrock & Coppi 1966; Hirayama 1974; Kopp & Pneuman 1976).

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In the simulations of Paper I, the dipole was orientated so as to minimize external reconnection with the emerging structure, and so the vertical expansion of the emerging structure was not as pronounced as in the simulations in this paper. Furthermore, in the simulations of Paper I, there was little direct evidence of internal magnetic reconnection, such as the evidence seen in the simulations in this paper. Internal reconnection is pronounced here due to the strong vertical expansion, facilitated by the external reconnection between emerging field and dipole field. Of the previous flux emergence simulations that exhibit evidence of internal reconnection, some had a horizontal coronal field which favored external reconnection and hence increased vertical expansion of the emerging structure (e.g., Archontis & Hood 2012, and references therein). Others had no dipole field but had a stronger emerging field than that in Paper I (e.g., Manchester et al. 2004), which was able to expand vertically due its own magnetic pressure. Therefore we hypothesize that the likelihood of this internal reconnection is related to the extent that the emerging structure can expand vertically to form a strong current sheet beneath the O-point.

As mentioned in Paper I, there have been different proposed mechanisms for the formation of a coronal flux rope during the partial emergence of a convection zone flux tube. Fan (2009) suggested that a coronal flux rope can be formed when the equilibration of twist along fieldlines extending from the convection zone into the corona effectively twists up the field in the corona. During this process sections of fieldline which initially have a low level of writhe wrap around a new axial fieldline. This process was also observed in the simulations of Paper I and this paper. On the other hand, Manchester et al. (2004), Archontis & Török (2008), and Archontis & Hood (2012) suggest that the flux rope is formed by the internal reconnection that occurs when the emerging structure has expanded sufficiently in the vertical direction to create a current sheet. Reconnection at this current sheet converts arched field into twisted field in the corona. As Figure 7, panel (c) shows, it is during this period that two separate O-points can be seen in the y = 0 plane in Simulation MD. One could argue that a precise definition of the formation of a new coronal flux rope is when these two O-points can be identified, and that the new flux rope axis is the fieldline which goes through the upper of these O-points. This type of internal reconnection can also be seen in the simulations of Fan (2009) but was not suggested as a flux rope formation mechanism in that paper. Recent observations of active region CMEs suggest that the formation of a coronal flux rope is associated with a confined flare, which is a consequence of the magnetic reconnection associated with this formation (e.g., Patsourakos et al. 2013).

Despite these findings, it is not clear if these two suggested formation mechanisms are independent. Furthermore, we propose that they will both be seen in any simulation such as the simulations in this paper where sufficient vertical expansion of the emerging structure is observed.

Figure 9 highlights the changes in connectivity that lead to a topologically distinct flux rope in the corona. Panels (a), (b), and (c) show selected fieldlines at times of t = 100t₀, 122.5t₀, and 130t₀, respectively. The red and blue lines originate from the same point in the dipole field at the lower z boundary (the seed locations are denoted by colored spheres). The orange and purple fieldlines, which coincide in panel (a), originate on the y_min and y_max boundaries respectively, again, at the same seed point for each snapshot. They form part of the erupting flux rope, though they only intersect the O-point in the y = 0 plane at t = 100t₀, which demonstrates that the axis of the erupting flux rope is not the same throughout the simulations. As the emerging field interacts with the pre-existing dipole field, external reconnection causes fieldlines that were once dipole field to connect at one end to the flux tube, e.g., the red and blue fieldlines in panel (b). Later in time, in panel (c), the internal reconnection discussed above can reconnect these blue and red fieldlines so that now they connect from one region of dipole to another, but pass underneath the coronal flux rope, forming an "M" shape. The green line in Figure 9 is the only fieldline that is not the same in each snapshot (it originates from the point (0, 0, −1)L₀), but is present to show that there is flux tube field that is underneath these M-shaped dipole fieldlines. Thus, the erupting flux rope is topologically distinct to the flux tube field that remains near the surface. The topological separation of coronal flux rope from the original flux tube was originally shown in simulations by MacTaggart & Haynes (2014) of the emergence of a toroidal flux rope into a horizontal coronal field. In these simulations, the topological separation happens as a result of external reconnection between emerging field and coronal field, and then internal reconnection which reconnects dipole field underneath the coronal flux rope.

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Shortly after the start of the internal reconnection observed in Simulations SD, MD and WD, at approximately t = 120t₀, the new coronal flux rope rises rapidly into the corona. This is shown for Simulation MD in Figure 10, which highlights the opening up of the dipole field by reconnection with the emerging flux structure. The simulation is stopped when the erupting flux rope hits the top boundary damping region. The blue fieldlines in Figure 10 originate inside a circle of radius 10L₀ in the y = 0 plane, centered on the flux rope axis. Following one of these fieldlines from one polarity region to another, one can see that it completes two turns around the flux rope axis, indicating that the flux rope has significant twist as it erupts.

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Figure 11 shows the height of the original flux tube and coronal flux rope's axis for all four simulations. To calculate this height at each time, O-points in the y = 0 plane are located. For the majority of the time in the simulation, there is only one O-point. During the early emergence process, before t = 80t₀, this O-point coincides with the intersection of the original convection zone flux tube's axis fieldline with the y = 0 plane. At t = 80t₀, as a consequence of the twisting of the coronal field mentioned in the previous sections, there is a small but rapid jump in the height of the O-point. Following this, in Simulations SD, MD, and WD, there is another jump in the O-point at about t = 115t₀. This occurs during the period of strongest internal reconnection, as the emerged structure is able to expand vertically following external reconnection with the dipole field. During this period, two O-points are found in the y = 0 plane, as shown in Figure 7, and the higher one is used to indicate the coronal flux rope's height. For the simulation with no dipole field, ND, there is no such jump in the height of the O-point, and, as discussed in the previous section, there is little direct evidence for internal magnetic reconnection. The flux rope slowly rises and the height-time curve indicates that it may be asymptoting to an equilibrium. The three squares in Figure 11 show the approximate height of the flux rope shown in Figure 2 of Manchester et al. (2004), which reports on a simulations with no dipole field but a stronger magnetic flux tube strength. As discussed in the previous section, in those simulations, evidence of magnetic reconnection and the rise of the coronal flux rope was reported. The height of the flux rope in those simulations reached about z = 55L₀ at about t = 73t₀. In the conclusions of Manchester et al. (2004) it was proposed that the flux rope would not eventually erupt, but be confined by its own field. As the height of the flux rope in those simulations is less than the height reached in Simulation ND (∼100L₀), and as Simulation ND, which also has no dipole field, does not erupt, we agree with their conclusion. Reporting on simulations of the emergence of a flux tube into a field-free corona, Archontis & Török (2008) also conclude that the coronal flux rope is confined by its own ambient field. Hence we propose that the presence of the dipole field is vital for the eruption of the coronal flux rope.

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After this internal reconnection, for t > 120t₀ the coronal flux rope axis accelerates. Figure 12 shows the velocity of the flux rope axis for simulations SD, MD, WD and ND in the time interval [120t₀: 165t₀]. The peak in the rise speed for Simulation MD is 10v₀ = 68.2 km s⁻¹. This is similar to the peak rise speed in the simulations of Archontis & Hood (2012), where a horizontal coronal field was used rather than a dipole coronal field. For all simulations except ND, the speed increases with time, peaks, and then decreases, and the time at which the speed peaks is the time at which the flux rope envelope interacts with the upper boundary. Hence we cannot draw conclusions on the further acceleration of the flux ropes after this point. Figure 12 shows that there is an inverse relationship between the peak in the rise speed and the strength of the dipole. Note that as the flux rope in simulation ND appears to asymptote to a certain height at t = 400t₀, the speed of its rise falls to 0.028v₀ = 190 m  s⁻¹.

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The strength of the dipole not only affects the maximum rise speed in these simulations, but also the amount of the original convection zone flux tube that reconnects with the overlying dipole during the emergence and eruption process. The weaker the dipole the less flux reconnects and the larger the resulting erupting coronal flux rope is. For Simulation SD, at approximately t = 140t₀, almost all the flux tube has reconnected with the overlying field, and it is not possible to identify a coronal flux rope after this time, which is why the red curve in Figures 11 and 12 is halted there. The curves for Simulations MD and WD in Figure 11 continue until the O-point hits the damping region at z = 180L₀. Because the coronal flux rope in Simulation WD is larger than in Simulation MD, the interaction of the outer sections of the rope are affected by the damping region and boundary conditions when the rope axis is at a lower height than in Simulation MD. The decrease in O-point height at t = 175t₀ for Simulation WD in Figure 11 is a result of this interaction of the flux rope and the damping region and boundary.

To check for convergence of the solution with resolution, given the choice of resistivity in the model, and to test the robustness of these results, we additionally show results for a number of simulations with the same initial conditions as Simulation MD (which has a grid of 304³). Simulation MD2 uses a grid of 384³ but keeps the same domain and stretching algorithm. Thus the resolution in Simulation MD2 is everywhere 384/304 ∼ 1.26 better than Simulation MD. Simulation MD3 uses a grid of 416³ so has a resolution 1.36 better than Simulation MD. Both Simulation MD2 and MD3 use the same numerical parameters (η, ν etc.) as Simulation MD. We also present results for a simulation MD4 which has the same grid as Simulation MD, but where η is set to zero. The height-time plot for the coronal flux rope is shown for these additional simulations in Figure 13, along with that for Simulation MD. Increasing the resolution from Simulation MD (solid curve), through Simulation MD2 (dashed curve), to Simulation MD3 (dot-dashed curve) appears to give convergence, at least in terms of the height of the coronal flux rope. The dotted curve shows the solution for Simulation MD4, with numerical resistivity η/η₀ = 0.005 which exhibits an earlier eruption of the flux rope. As in Paper I, we estimate the numerical resistivity using typical Alfvén speeds and length scales in regions of strong current and find an effective numerical resistivity of 0.005η₀, thus effectively double the Lundquist number of Simulation MD. We postulate that having a lower effective resistivity in Simulation MD4, compared to Simulation MD, leads to larger current density buildup beneath the flux rope axis and faster internal reconnection, and so the required amount of internal reconnection to initiate the eruption is reached earlier in Simulation MD4, compared to Simulation MD.

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In Leake et al. (2010) and Leake & Linton (2013), we studied flux emergence into a dipole in 2.5D, and found that the amount of shear energy supplied to the corona was insufficient to drive an eruption. Figure 14 shows the amount of axial flux above a height of z = 0 and z = 10L₀, normalized to the total axial flux in the domain, in the 2.5D simulations of Leake et al. (2010) for a dipole of the same strength as Simulation MD. Also shown is the same calculation in the y = 0 plane for Simulation MD in this paper. Although in both the 2.5D and 3D simulations, a large amount of shear flux (almost all in the 2.5D simulation) gets above the surface (z = 0), only 40% gets above (and stays above) the z = 10L₀ level in 2.5D, whereas over 60% achieves this in the 3D simulations. This suggests that the eruption requires the emergence process to raise sufficient axial flux into the corona to a height at which it is unstable. In the 3D simulations presented in this paper, this is caused by the draining of plasma along fieldlines which allows axial flux to emerge into the corona. Further rise into the corona is driven by breakout reconnection, which occurs above the coronal flux rope axis and removes overlying dipole field.

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