In the solar atmosphere, if the dynamics of the system are dominated by the plasma motions, which twist and drag the magnetic field lines while forcing them into highly non-potential configurations. If the opposite situation occurs, that is, the magnetic field is not influenced by the plasma motions. In this case, the magnetic field will evolve into a state of minimum energy which happens to coincide with a potential configuration (see Chapter 3.4 in Priest, 1982). Therefore, many works throughout the literature focus on the plasma- in order to study the potentiality of the magnetic field. Here, we will employ our results from the inversion of spectropolarimetric data in Section 2.2 to investigate the value of the plasma- parameter in a sunspot. Figure 16 shows the variation of the azimuthally averaged plasma- (along ellipses in Figure 10) as a function of the normalized radial distance in the sunspot . This figure displays at four different optical depths, from the deep photosphere (yellow) to the high-photosphere (blue). This figure shows that above and, thus, the magnetic field can be considered to be nearly potential (or at least force-free) in these high layers. At (referred to as continuum) and, therefore, the magnetic field is non-potential. At the intermediate layer of (around 100 kilometers above the continuum) the magnetic field is nearly potential in the umbra, but it cannot be considered this way in the penumbra: .
In Figure 16 the gas pressure was obtained under the assumption of hydrostatic equilibrium (Section 1.3.3), which we know not to be very reliable in sunpots. A more realistic approach was followed by Mathew et al. (2004, and references therein), where an attempt to consider the effect of the magnetic field in the force balance of the sunspot was made. Their results for the deep photosphere () obtained from the inversion of the Fe i line pair at 1564.8 nm are consistent with our Figure 16 (obtained from the inversion of the Fe i line pair at 630 nm), with close to the continuum everywhere in the sunspot. Similar results were also obtained by Puschmann et al. (2010c, see their Figure 4), who performed an even more realistic estimation of the geometrical height scale, considering the three components of the Lorenz force term (; Equation (17)). In Figure 17 we reproduce their results, which further confirm that the in the deep photospheric layers of the penumbra.
These results have important consequences for magnetic field extrapolations from the photosphere towards the corona, because they imply that those extrapolations cannot be potential. In addition, as pointed out by Puschmann et al. (2010c) the magnetic field is not force-free because in many regions the current density vector and the magnetic field vector are not parallel. Unfortunately, extrapolations cannot deal thus far with non-force-free magnetic field configurations. Considering that it has now become possible to infer the full current density vector , developing tools to perform non-force free magnetic field extrapolations will be a necessary and important step for future investigations. These results also have important consequences for sunspot’s helioseismology, because of the deep photospheric location of the region, which is the region where most of the conversion from sound waves into magneto-acoustic waves takes place.
In the chromosphere of sunspots, the magnetic field strength is about half of the photospheric value (see Figure 4 in Orozco Suarez et al., 2005). Therefore, the magnetic pressure in the chromosphere is only about 25% of the photospheric value. However, the density and gas pressure are at least 2 – 3 orders of magnitude smaller. Thus, the chromosphere of sunspots is clearly a low- () environment, which in turn means that the magnetic field configuration is nearly potential.
Living Rev. Solar Phys. 8, (2011), 4
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