2.1 How to obtain the force-free parameter α

Linear force-free fields require the LOS magnetic field in the photosphere as input and contain a free parameter α. One possibility to approximate α is to compute an averaged value of α from the measured horizontal photospheric magnetic fields as done, e.g., in Pevtsov et al. (1994Jump To The Next Citation Point), Wheatland (1999), Leka and Skumanich (1999), and Hagino and Sakurai (2004Jump To The Next Citation Point), where Hagino and Sakurai (2004) calculated an averaged value ∑ ∑ α = μ0Jzsign(Bz)∕ |Bz |. The vertical electric current in the photosphere is computed from the horizontal photospheric field as ( ) Jz = μ1 ∂B∂yx-− ∂B∂xy- 0. Such approaches derive best fits of a linear force-free parameter α with the measured horizontal photospheric magnetic field.

Alternative methods use coronal observations to find the optimal value of α. This approach usually means that one computes several magnetic field configurations with varying values of α in the allowed range and to compute the corresponding magnetic field lines. The field lines are then projected onto coronal plasma images. A method developed by Carcedo et al. (2003Jump To The Next Citation Point) is shown in Figure 4View Image. In this approach the shape of a number of field lines with different values of α, which connect the foot point areas (marked as start and target in Figure 4View Image(e)) are compared with a coronal image. For a convenient quantitative comparison the original image shown in Figure 4View Image(a) is converted to a coordinate system using the distances along and perpendicular to the field line, as shown in Figure 4View Image(b). For a certain number of N points along this uncurled loop the perpendicular intensity profile of the emitting plasma is fitted by a Gaussian profile in Figure 4View Image(c) and the deviation between field line and loops are measured in Figure 4View Image(d). Finally, the optimal linear force-free value of α is obtained by minimizing this deviation with respect to α, as seen in Figure 4View Image(f). The method of Carcedo et al. (2003) has been developed mainly with the aim of computing the optimal α for an individual coronal loop and involves several human steps, e.g., identifying an individual loop and its footpoint areas and it is required that the full loop, including both footpoints, is visible. This makes it somewhat difficult to apply the method to images with a large number of loops and when only parts of the loops are visible. For EUV loops it is also often not possible to identify both footpoints. These shortcomings can be overcome by using feature recognition techniques, e.g., as developed in Aschwanden et al. (2008a) and Inhester et al. (2008) to extract one-dimensional curve-like structures (loops) automatically out of coronal plasma images. These identified loops can then be directly compared with the projections of the magnetic field lines, e.g., by computing the area spanned between the loop and the field line as defined in Wiegelmann et al. (2006bJump To The Next Citation Point). This method has become popular in particular after the launch of the two STEREO spacecraft in October 2006 (Kaiser et al., 2008). The projections of the 3D linear force-free magnetic field lines can be compared with images from two vantage viewpoints as done for example in Feng et al. (2007b,a). This automatic method applied to a number of loops in one active region revealed, however, a severe shortcoming of linear force-free field models. The optimal linear force-free parameter α varied for different field lines, which is a contradiction to the assumption of a linear model. A similar result was obtained by Wiegelmann and Neukirch (2002) who tried to fit the loops stereoscopically reconstructed by Aschwanden et al. (1999). On the other hand, Marsch et al. (2004) found in their example that one value of α was sufficient to fit several coronal loops. Therefore, the fitting procedure tells us also whether an active region can be described consistently by a linear force-free field model: Only if the scatter in the optimal α values among field lines is small, one has a consistent linear force-free field model which fits coronal structures. In the generic case that α changes significantly between field lines, one cannot obtain a self-consistent force-free field by a superposition of linear force-free fields, because the resulting configurations are not force-free. As pointed out by Malanushenko et al. (2009) it is possible, however, to estimate quantities like twist and loop heights with an error about of 15% and 5%, respectively. The price one has to pay is using a model that is not self-consistent.

View Image

Figure 5: Low and Lou’s (1990Jump To The Next Citation Point) analytic nonlinear force-free equilibrium. The original 2D equilibrium is invariant in φ, as shown in panel a. Rotating the 2D-equilibrium and a transformation to Cartesian coordinates make this symmetry less obvious (panels b–d), where the equilibrium has been rotated by an angle of π π φ = 8 ,4, and π 2, respectively. The colour-coding corresponds to the vertical magnetic field strength in G (gauss) in the photosphere (z = 0 in the model) and a number of arbitrary selected magnetic field lines are shown in yellow.

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