Recently, a series of 3D MHD simulations (Fan, 2001b; Magara and Longcope, 2001, 2003; Magara, 2004; Manchester IV et al., 2004; Archontis et al., 2004, 2005, 2006; Galsgaard et al., 2005; Murray et al., 2006; Magara et al., 2005; Magara, 2006) have been carried out to model the emergence of a twisted magnetic flux tube through a multi-layered atmosphere that includes a polytropic layer representing the top of the solar convection zone, an isothermal layer representing the photosphere and chromosphere, and another isothermal layer representing the million-degree corona. In these simulations, a twisted flux tube is initially placed in the convection zone at a depth of about a few times the photospheric pressure scale height and the central segment of the twisted tube rises buoyantly towards the photosphere. When the top of the tube enters the photosphere the rise velocity at the upper boundary of the tube first slows down as it encounter the stable stratification of the photosphere. As a result, magnetic flux begins to pile up at the photosphere and a steep magnetic pressure gradient is established. It is found that subsequently, the magnetic flux undergoes a run-away expansion into the atmosphere due to the non-linear growth of the magnetic buoyancy instability. Archontis et al. (2004) and Murray et al. (2006) quantitatively examined their simulation data using the criterion for the onset of the undulatory magnetic buoyancy instability (Newcomb, 1961; Thomas and Nye, 1975; Acheson, 1979):et al. (2006) found that if either the field strength or the twist of the subsurface flux tube is too low, the magnetic pressure build-up at the photosphere may not achieve the critical condition given above and flux emergence into the atmosphere may fail to take place.
During the run-away expansion of the magnetic flux tube into the solar atmosphere, strong diverging downflows are found along the emerged field lines, forming shock fronts just above the photosphere (e.g. Fan, 2001b; Magara and Longcope, 2003; Archontis et al., 2004). Due to the magnetic tension force associated with the twisted field lines, a shear flow pattern develops on the photosphere with the plasma on the two sides of the polarity inversion line moving oppositely in the east-west direction (Manchester IV, 2001; Fan, 2001b; Magara and Longcope, 2003; Manchester IV et al., 2004; Archontis et al., 2004). The effect of the tension force driven shear flow is to transport the axial magnetic flux upward into the expanding portion of the flux tube in the solar atmosphere (Manchester IV et al., 2004). Newly developing active regions on the photosphere often exhibit such a shear flow pattern (Zirin, 1988; Strous et al., 1996). The two polarities of the emerging region on the photosphere initially emerge as north-south oriented, and over time they appear to shear along the polarity inversion line and separate in the east-west direction (e.g. Fan, 2001b; Magara and Longcope, 2003; Manchester IV et al., 2004; Archontis et al., 2004; Magara, 2006). A detailed study of both the apparent and plasma motions in the photosphere resulting from the emergence of a twisted flux tube (with different degree of twist for the initial subsurface flux tube) is carried out by Magara (2006). It is found that the apparent rotational motion exhibited by the two polarities are opposite to the direction of the actual rotational motion of the plasma.
The simulations also suggest that it is difficult for a twisted flux tube to rise bodily into the corona as a whole due to the heavy plasma that is trapped at the bottom concave (or U-shaped) portions of the winding field lines. Independent simulations by Fan (2001b) and Archontis et al. (2004), considering nearly identical initial conditions both find that that as the upper parts of the winding field lines of the twisted flux tube expand into the atmosphere and the corona, the axial field line of the tube settles to an equilibrium not much above the photosphere, and the lower U-shaped parts of the field lines largely remain trapped below the photosphere. On the other hand, Manchester IV et al. (2004) considered the case where the length of the emerging portion of the tube is shortened to a half of that in Fan (2001b), such that the field lines only perform about one turn about the axis over the emerging segment. In this case it is found that after the growth of the magnetic buoyancy instability which causes the expansion of the flux tube into the solar atmosphere, a secondary eruptive phase sets in where the upper emerged part of the flux tube begins to “pinch off” from the lower mass-laden part via magnetic reconnection, forming a new flux rope with a new central axial field line that rises well into the corona. Manchester IV et al. (2004) explains that the cause of the eruptive phase is the shear flow driven by the magnetic tension force, which continually transport axial flux and magnetic energy to the upper expanded portion of the magnetic field, driving its eruption. It is found that a sigmoid shaped current sheet develops in the chromospheric heights during the eruptive phase. Simulations by Magara and Longcope (2001), Magara (2004) Magara et al. (2005), and Magara (2006) also found formation of current concentration of similar sigmoid morphology in the chromospheric heights (see e.g. Figure 25), and the ascent of a flux rope structure into the atmosphere with an O-point of the magnetic field in the central cross-section reaching the base of the corona. It is found that the emerged field lines with footpoints rooted in regions of high electric current density in the chromosphere display an inverse-S shape for a left-hand-twisted emerging tube, in agreement with the X-ray sigmoid morphology and the sense of active region twist in the northern hemisphere (see Figure 25 and Figure 4).
The effects of the presence of a simple horizontal coronal magnetic field and the interaction and reconnection of the pre-existing coronal field with the emerging flux rope have been investigated in a series of work (Archontis et al., 2004, 2005, 2006; Galsgaard et al., 2005). It is found that the dynamic interaction and the rate of magnetic reconnection depends sensitively on the relative orientations between the upcoming emerging flux and the pre-existing coronal magnetic field. When the two flux systems coming into contact have a relative angle above , the simulations show immediate and substantial magnetic reconnection, producing collimated high-speed and high-temperature jets from the reconnection site. As a result of the reconnection, most of the flux in the subsurface flux rope end up connecting to the coronal magnetic field. The interaction with the pre-existing coronal field tends to slow down the rise speed of the upper-boundary of the emerging flux tube into the corona, compared to the case where the flux tube emerges into a field-free atmosphere (Archontis et al., 2005). Isobe et al. (2005) performed high resolution 3D simulations of emerging flux and its reconnection with pre-existing magnetic field in the corona. In this calculation the emerging flux develops from an initial horizontal magnetic flux sheet (with uni-direction field) situated in the top of the convection zone. Due to the growth of the three-dimensional magnetic buoyancy instability of modes with high wave number in the direction perpendicular to the field (or magnetic Rayleigh–Taylor instability), undulating flux bundles with fine spacial scales rise into the corona and reconnect with the pre-existing coronal magnetic field in a spatially intermittent way. This results in the formation of filamentary structure resembling the observed arch filament systems in emerging flux regions (EFRs). The spatially intermittent reconnection and heating also explain the coexistence of many hot and cold loops and the jets being ejected from the loop footpoints in EFRS observed in EUV by the TRACE satellite.
Due to the need to resolve the photosphere pressure scale height (), the 3D calculations of twisted emerging flux tubes described above are done for a domain size of up to a few tens of Mm, the size of a small active region. The implicit assumption is that the basic dynamical behavior of the smaller emerging flux tubes modeled in these simulations is representative of that of the larger active region scale emerging tubes. Furthermore, adiabatic evolution of an ideal gas is assumed in these simulations, which is not appropriate for the atmosphere layers considered. The adiabatic expansion of the tube plasma emerging into the atmosphere resulting in unphysically low temperatures. Radiative heat exchange, thermal conduction, and the still uncertain sources of coronal heating all play an important role in the thermal energy evolution of the plasma in emerging flux regions.
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