Density and temperature are heavily structured in the highly dynamic chromospheric environment so that the relative strength of the plasma pressure and magnetic forces also varies strongly with position, at a given height. The height at which the magnetic forces start to dominate over others (i.e., where \(\beta \ll 1\)) is expected to be strongly corrugated relative to the solar surface. In the QS, that height is expected to vary between \(\approx 800\hbox km\) and 1.6 Mm above the photosphere (Rosenthal et al. 2002). In ARs, this height is likely to be lower, as shown by Metcalf et al. (1995). They used chromospheric vector magnetic field measurements inferred from observations in the Na i 5896 Å spectral line to test the relative contribution of the plasma pressure and magnetic forces in an AR. They found that the atmosphere above that AR could be considered to be force-free from \(\approx 400\hbox km\) above the solar surface. Gary (2001) was able to confirm that finding by combining a plasma pressure and magnetic field model to estimate the pattern of interchanging dominance of plasma and magnetic pressure with height in the solar atmosphere (see Fig. 5). He concluded that the magnetic forces above sunspots should start to dominate from relatively low heights (\(\gtrsim 400\hbox km\) above the photosphere). Above plage regions, the model results suggest this to be true from \(\gtrsim 800\hbox km\) above a photospheric level upwards. In summary, ARs can be considered to be force-free in most of the chromosphere (in contrast to quiet-Sun areas; see Sect. 5.2.5).
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The solar magnetic field vector is measured routinely with high accuracy only in the photosphere, e.g., by SDO/HMI at a constant resolution of \(1^\prime \prime \) over the whole solar disk. Under reasonable assumptions, we can extrapolate these photospheric measurements into the higher solar atmosphere, where direct magnetic field measurements are more challenging (see Sect. 2.1). So, which assumptions are reasonable in the solar atmosphere? A key to answer this question is the comparison of magnetic and non-magnetic forces and in particular the plasma-\(\beta \). While the plasma-\(\beta \) is around unity in the photosphere it becomes very small (about \(10^-4\) to \(10^-2\)) in the corona (at least in ARs; see Fig. 5 and Sect. 1.2.3 for details). Consequently, non-magnetic forces can be neglected in the low \(\beta \) corona, and the coronal magnetic field can be modeled as a force-free field (the Lorentz-force vanishes). The electric current density,
A full understanding of the physical processes in the upper solar atmosphere requires the knowledge of the plasma that populates the investigated magnetic structures. Deriving these properties in the outer solar atmosphere, however, remains a challenging task. Most commonly used models for a self-consistent description of the plasma and magnetic field are based on the MHD approximation. Interestingly, even though the MHD approximation is strictly valid only in collisional plasmas, the collision-free coronal plasma is often modeled using such an approach. More sophisticated, collisionless kinetic models cannot be applied to model large-scale structures in the solar corona since the considered scales are several orders of magnitude larger than the relevant (microscopic) scales which have to be resolved in kinetic simulations (e.g., the gyro-radius or Debye-length). This approach, however, is frequently applied to model the solar wind plasma (see review by Marsch 2006).
Mathematically simpler, and computationally much faster, is the subclass of MS models, which are based on the assumption that electric currents flow on spherical shells perpendicular to gravity (resulting in horizontal,i.e., parallel to the lower boundary, currents in cartesian geometry). This approach allows linearizing the MS equations and solving them with a separation ansatz (see Low 1991; Bogdan and Low 1986; Neukirch 1995, for one cartesian and two spherical approaches, respectively). Because of the linearity of the underlying equations, a field-parallel electric current can be superposed (for a constant value of \(\alpha \)). The final current distribution consists of two parts: a LFF one and another one that compensates non-magnetic forces such as pressure gradients and gravity. These classes of MS equilibria require only LOS photospheric magnetograms as boundary conditions, which are relatively easy to implement and allow the specification of two free parameters (the force-free parameter \(\alpha \) and additionally a parameter which controls the non-magnetic forces). The limitations on \(\alpha \) are similar to those discussed for LFF modeling approaches (see Sect. 2.2). In these models, plasma pressure and density are computed self-consistently to compensate the Lorentz-force. Above a certain height the corresponding configurations become almost force-free, which in principle allow it to model a forced photosphere and chromosphere, together with a force-free corona above. A limitation of MS equilibria is that the two free parameters are globally constant and the method does not guarantee a positive plasma pressure and density. To ensure positive values of these quantities, one either has to add a sufficiently large background atmosphere (which may lead to unrealistically high values of the plasma-\(\beta \)), or is limited to small values of the parameter controlling the non-magnetic forces. Note that, as force-free approaches, MS models are only snapshots of the coronal field and the temporal evolution of such configurations can only occur as a series of equilibria, in response to temporally changing boundary conditions.
Pevtsov (2000) tested the importance of the chirality (handedness) of active-region magnetic fields for the formation of TEL systems. The results suggested that in roughly two-thirds of the cases the connected active-region fields were of the same handedness. Recently, Chen et al. (2010) examined the twist of a larger number of TELs (a subset of the samples analyzed by Chen et al. 2006). They found that the ones that linked ARs displayed an obvious sigmoidal shape and were related to a flaring activity stronger than C-class (i.e., peak SXR fluxes of \(>10^6\hbox W\hbox m^-2\)). They calculated the ratio \(\tildeL/D\), where \(\tildeL\) is the apparent length of the TEL system and \(D\) is the apparent distance between the locations where the TEL system seems rooted at the solar surface. \(\tildeL\) was measured by tracing the length of the coronal loops at the outer edges of the sigmoidal loop system, where they are well distinguished from the faint emission from the (quiet Sun) background. Higher values of that ratio \(\tildeL/D\) indicate a more pronounced sigmoidal shape and thus imply a stronger twisting of the associated field lines. They found that most of the TELs possess only weak sigmoidal shapes, indicating a low degree of non-potentiality. It appears that flares above C-class preferentially originate from structures of a specific amount of twist \((\tildeL/D\approx 1.4)\). It is an important future task to model the associated cross-equatorial 3D coronal magnetic field and its evolution with the help of global force-free and time-dependent MHD models to reveal the importance for eruptions to occur (see also Sect. 4.4).
It has long been puzzling why especially the loops seen in coronal images do not show a significant variation of their width with height in the atmosphere. Given magnetic flux tubes expand with height in the solar atmosphere, one would naturally expect this to be reflected in form of a clear height dependence of the emission observed from the thin threads which compose the flux tubes (DeForest 2007). Instead, an apparently constant cross section and more or less constant brightness along the loops, but no significant expansion was observed at the two wavelengths mentioned above. This is the result of the analysis of coronal loops seen in Yohkoh/SXT images (Klimchuk et al. 1992), and EUV observations with TRACE (DeForest 2007) as well as with SDO/AIA (Aschwanden and Boerner 2011). It has been argued that this might just reflect the fact that the coronal loops are entities of a constant diameter (Klimchuk 2000), although force-free magnetic field models do not support such an interpretation.
a Magnetic topology of AR 10365 on 27 May 2003, estimated from a current-free field model. The white arrow in a indicates the possible location of a coronal null point, based on the associated current-free field configuration. Selected field lines that outline the basic magnetic field configuration are displayed on top of a near-in-time \(\hbox H\alpha \) image taken at the Solar Observatory Tower Meudon. Strongest emission in a outlines the location of flare ribbons (that is, where newly reconnected field lines are line-tied to the low atmosphere). b The same model configuration, shown on top of the SOHO/MDI LOS magnetogram (gray scale, where positive/negative polarity corresponds to white/black areas). Sample magnetic field lines, connecting the AR and its periphery are shown in yellow. Green, light and dark blue, as well as red field lines connect the observed flare ribbons to other regions within the AR. (Adapted from Figure 8 of Chandra et al. 2009. With kind permission from Springer Science and Business Media.) c TRACE 1,600 Å emission (bright ribbon and kernels) associated to a C-class flare on 11 November 2002, overlaid on a SOHO/MDI photospheric LOS magnetic field (gray-scale background; white/black represents positive/negative polarity) of AR 10191. d Sample field lines outlining the potential field reconstruction of the associated coronal field. The red, yellow and blue lines indicate a coronal null-point topology. (Adapted from Figures 2 and 3 of Masson et al. 2009. AAS. Reproduced with permission) 2ff7e9595c
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