In recent years, a large body of 3D MHD simulations (e.g. Fan, 2001b; Magara and Longcope, 2001, 2003; Magara, 2004; Manchester IV et al., 2004; Archontis et al., 2004, 2005, 2006; Galsgaard et al., 2005, 2007; Murray et al., 2006; Magara et al., 2005; Magara, 2006, 2007, 2008; Archontis and Hood, 2008; Archontis and Török, 2008; Archontis et al., 2009; Fan, 2009b) 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, connecting to 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 a couple of Mms 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 immediately 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; Manchester IV, 2007). 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; Manchester IV, 2007). Newly developing active regions on the photosphere often exhibit such a shear flow pattern (Zirin, 1988; Strous et al., 1996). Since the subsurface flux tubes are assumed to be significantly twisted in these simulations, the two polarities of the emerging region on the photosphere initially emerge as north-south oriented, and then they undergo shearing motion 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; Fan, 2009b).
It is further found (Magara, 2006; Fan, 2009b) that, as the two polarity flux concentrations become separated, a prominent rotational or vortical flow develops within each polarity, centered on the peak of the vertical magnetic flux concentration (see Figure 36), reminiscent of sunspot rotations that have been observed in many events preceding X-ray sigmoid brightening and the onset of eruptive flares (e.g Brown et al., 2003; Zhang et al., 2008).
The vortical motions are counter-clockwise (clockwise) for a left-hand-twisted (right-hand-twisted) emerging flux tube. Fan (2009b) showed that the vortical motions in the two polarity flux concentrations twist up the inner emerged field lines, causing them to rotate in the atmosphere from an initial normal configuration (i.e. arching over the polarity inversion line from the positive to the negative polarity) into an inverse configuration (directed from the negative polarity to the positive polarity over the neutral line) (see the movie associated with Figure 37).
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In this manner, a flux rope with sigmoid-shaped dipped core fields forms in the atmosphere, with the center of the flux rope, as represented by the O-point of the magnetic field in the central cross-section, rising into the corona (see left panel of Figure 36).
Fan (2009b) showed that the rotational or vortical motions centered on the two polarity flux concentrations are a manifestation of non-linear torsional Alfvén waves propagating along the flux tube, consistent with what has been predicted by an earlier idealized analytical model of Longcope and Welsch (2000). Due to the rapid stretching of the emerged magnetic field in the atmosphere and corona, the magnitude of along the coronal field lines, which is a measure of the rate of twist per unit length, drastically decreases. As a result, a gradient of is established along the field lines of the flux tube from the interior into the atmosphere with the interior portion having a much higher magnitude of (Figure 38).
This gradient of drives torsional Alfvén waves along the flux tube, transporting twist from the interior highly twisted portion into the expanded coronal portion (Longcope and Welsch, 2000). The rotational motion will continue until the coronal equilibrates with the interior along the field lines. The time scale for establishing the equilibrium is on the order of the Alfvén transient time along the interior flux tube, which means that the rotational motion can persists for a few days after the initial emergence. Magara and Longcope (2003) and Fan (2009b) found that the helicity flux due to flux emergence is dominating only for a brief initial period, and then horizontal shearing and rotational motions at the footpoints of the emerged fields provide the dominant and steady source of helicity flux into the atmosphere. These results suggest that the observed sunspot rotations are due to non-linear torsional Alfvén waves naturally occurring during the emergence of a twisted flux tube, and is an important means whereby twist is transported from the interior into the corona, driving the development of a coronal flux rope as a precursor structure for solar eruptions. The horizontal vortical motion and its subsurface extension corresponding to the torsional Alfvén waves may be detectable by local helioseismology techniques (see review by Gizon and Birch, 2005).
The simulations consistently show that the subsurface twisted flux tube does not 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. While the upper parts of the helical field lines of the twisted flux tube expand into the atmosphere and the corona, the U-shaped parts of the winding field lines remain largely trapped at and below the photosphere layer, and the center of the original tube axis ceases to rise a couple of pressure scale heights above the photosphere. Nevertheless in the end, a twisted coronal flux rope with sigmoid-shaped concave upturning field lines threading under a central axis that rises into coronal heights is found to develop in many simulations (Magara and Longcope, 2001, 2003; Magara, 2004; Manchester IV et al., 2004; Magara, 2006; Manchester IV, 2007; Archontis and Hood, 2008; Archontis and Török, 2008; Archontis et al., 2009; Fan, 2009b). By Larangian tracking of the evolution of the emerged field lines as their footpoints are undergoing shearing and vortical motions (see Figure 37), Fan (2009b) found that the inner emerged field lines (including the original tube axis) rotate in the atmosphere into an inverse configuration, and that is how the sigmoid-shaped, dipped core fields of the new coronal flux rope come into being. Driven by the continued vortical motions at the footpoints of the emerged field lines, the newly formed coronal flux rope accelerates upwards, and a current sheet of an overall sigmoid morphology develops in the lower atmosphere, extending up to the base of the corona (Manchester IV et al., 2004; Magara, 2004, 2006; Archontis and Török, 2008; Archontis et al., 2009; Fan, 2009b). Figure 39 shows such an example.
It is found that the field lines going through the current sheet all show a sigmoid-shape. Thus the heating associated with the current sheet may cause these sigmoid-shaped field lines to preferentially brighten up in soft X-ray, giving rise to the observed X-ray sigmoid loops in an active region (e.g. Rust and Kumar, 1996; Canfield et al., 1999).
Archontis et al. (2009) further examined the detailed evolution of the (overall) sigmoid-shaped current layers in their simulation of an emerging twisted flux tube, and found that it agrees qualitatively with the complex features and evolution of a coronal sigmoid observed by the XRT of Hinode, which is characteristic of many evolving sigmoids that result in eruptions (see Figure 40).
The simulation shows that earlier in the emergence, the current layers and the associated bald-patch separatrix field lines form two “J”-like structures. With time the electric current becomes more rich in structure, with additional thin current layers forming. The current layers and the associated reconnected field lines form an overall sigmoid-shaped structure. Then in the last phase, a central current sheet develops in the middle of the sigmoid structure, accompanied by the rapid rise of the coronal flux rope and the appearance of post-reconnection loops under the current sheet. Observations of evolving coronal X-ray sigmoids often show a similar transition from an initial morphology of two “J”-like bundles, to the development of a central brightening in the middle of the sigmoid at the onset of an eruptive flare.
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 tube have been investigated in a series of work (Archontis et al., 2004, 2005, 2006; Galsgaard et al., 2005, 2007; Archontis and Török, 2008). 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 90°, the simulations show immediate and substantial magnetic reconnection, producing collimated high-speed and high-temperature jets from the reconnection site. However, the cases that have a more parallel orientation of the flux systems show very limited reconnection and none of the associated features. Most recently, Archontis and Török (2008) found that rapid reconnections with a pre-existing horizontal coronal magnetic field that is nearly anti-parallel with the top of the emerging flux system can greatly enhance the eruption of the newly formed coronal flux rope. The reconnections remove flux from both the pre-existing field and the outer emerged fields, which are arcade-like fields acting to confine the sigmoid core fields of the newly formed coronal flux rope. Thus the coronal flux rope shows an enhanced acceleration, reaching a maximum speed of about 240 km s–1. In contrast, in the case of emergence into a field-free atmosphere, the coronal flux rope that forms generally reaches a maximum rise speed of a few tens of km s–1 (e.g. Manchester IV et al., 2004; Archontis and Török, 2008; Fan, 2009b).
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 spatial 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 ( 150 km), the 3D simulations of the emergence of twisted flux tubes from the interior into the solar atmosphere and the corona described above are done for a domain size of up to a few tens of Mm, significantly smaller than the size of a typical active region. The implicit assumption is that the qualitative 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, these simulations, which focus on understanding how twisted active region flux tubes emerge dynamically into the atmosphere and the corona, typically ignore convection in the interior layer. They also grossly simplify the treatment of the thermodynamics by assuming an ideal gas equation of state and an adiabatic evolution of the plasma, which are not appropriate for the atmosphere layers considered. Thus these simulations of flux emergence into the atmosphere do not study the magneto-convection process of sunspot formation, which depends critically on the convective and radiative energy transport at the photospheric layer. The assumed adiabatic expansion of the tube plasma emerging into the atmosphere results in unphysically low temperatures. Radiative heat exchange, thermal conduction, and the still uncertain process of coronal heating driven by the mechanical energy of convective motions, all play an important role in the thermal energy evolution of the plasma in emerging flux regions, and in maintaining the observed temperature profile of the atmosphere layers.
Progress is being made towards building a full radiation MHD model of active region flux emergence, encompassing both the magneto-convection process of the formation of sunspots from rising flux tubes and the emergence of the active region flux into the stably stratified solar atmosphere and the corona with realistic treatment of the thermodynamics. Prototypes of such numerical models include, for example, Abbett (2007), Amari et al. (2008), and Martínez-Sykora et al. (2008, 2009). The simulations of Martínez-Sykora et al. (2008, 2009) are at present the most sophisticated in terms of the physics included. They include realistic magneto-convection in the interior layer, with a realistic treatment of radiative energy transport in the interior and the overlying atmosphere layers, and taking into account thermo-conduction along magnetic field lines. The hot corona is self-consistently maintained by the magnetic energy dissipation driven by the stresses applied to the fields due to the photospheric convective motions (Gudiksen and Nordlund, 2002). In these simulations, a quiet-sun 3D domain is first initialized which contains a convecting interior layer that realistically represent the top layer of the solar convection zone, a photosphere and chromosphere, a transition region, and a hot corona, and with a pre-existing magnetic field of varying mean strength treading through the layers. A horizontal flux tube with varying degree of twist and field strength is then transported though the lower boundary into the domain, and its subsequent rise through the convection zone and the emergence into the atmosphere layers is studied. The domain sizes considered in these simulations are similar to the idealized simulations described above. The rise of the twisted flux tube through the interior layer and its emergence at the photosphere are in agreement to what have been found in Cheung et al. (2007). Similar to previous findings, a large amount of horizontal flux is retained at the photosphere, and with greater twist and/or greater field strength, more flux crosses the photosphere into the chromosphere and the corona. Compared to previous ideal MHD simulations which ignored convection in the interior layer, the rising flux tube is more fragmented, and the dynamic expansion of magnetic flux into the upper chromosphere and corona is more patchy and slower. Plasma of chromosphere temperature is found to be lifted into great heights (far greater than that expected for hydrostatic equilibrium) due to magnetic support by the emerging fields. A low density structure of the emerged fields is found to form in the corona with an underlying high density structure of chromospheric plasma supported by the magnetic fields. These are suggestive of the formation of a filament, although the scales of the structures in the simulations are far smaller than their realistic counterparts (due to the limitation in numerical resources which prevent this type of simulations from being carried out on realistic scales of active regions and filaments). Associated with the high density structure, low density inclusions or voids are found to rise in the high density structure as shown in the synthetic Ca II images at the limb, which resembles the dynamics observed in quiescent prominences with the Hinode Spacecraft (Berger et al., 2008). New results are also found with regard to the spatial temporal evolution of Joule heating during flux emergence, and how it affects the energetics of the chromosphere and corona, and changes the emission of various transition region and coronal lines. Further advancement in computing power in the next decade is likely to enable such realistic radiation MHD simulations of flux emergence on a domain size-scale that is approaching a realistic active region.
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