This subsection gives a short overview about the modelling done on the coronal heating problem and the corona in general. This is based on Peter (2007).
2.4.1 1D models
The first models developed to investigate the heating of closed coronal loops were one di-mensional models. The 1D approach was required because of the limited computational power at that time, but such an approximation of a coronal loop is none-the-less a good simplification. Since the plasma is frozen-in into the magnetic field, nearly no flow across the magnetic field is allowed. The flow of plasma and the energy transport take place almost exclusively along the field-lines, and therefore one field line can be considered isolated from neighbouring lines.
Based on this principle of using the magnetic field line as a flow channel, a large amount of successful 1D models were produced. One set of these are the RTV models, which were mentioned in Sect. 2.2. Advances in computational capacity and power allowed for a more complete treatment of the coronal loops by including additional physics, such as radiative transfer and ionization. Very high resolutions are reached with the use of adaptive mesh refinement. This led to very thin transition regions, which is a result of the inefficient heat conduction at lower temperatures, which is compensated by a high
temperature gradient to accommodate the energy flux into the photosphere via heat con-duction.
Besides closed coronal loops, the corona contains open field-lines. This means that the field-lines do not connect back in the nearby vicinity, but either far away or to the inter-planetary field. These open field-lines generally start from a funnel-like-structure in the corona. These originate from a concentrated magnetic field at the bottom of the corona and then fan out as a results of the difference in the pressure balance between magnetic and thermal pressure (Reeves 1976; Gabriel 1976). These funnels are believed to be the source of the solar wind (Tian et al. 2009). The emission measure, EM = R
VnedV, cal-culated from funnel models were unable to reproduce the observed emission measure.
This was a result of the extreme thin TR, which resulted in a too low emission at lower temperature. This lead to the proposal by Dowdy et al. (1986) of a "magnetic junk yard", a region of short and cool coronal loops which can account for the missing emission.
Despite the high resolution and high level of included physics of these 1D models, single coronal magnetic field-lines do not exist in a vacuum. Changes within one single strand has influence on neighbouring strands and vice-versa.
2.4.2 3D coronal box models
Although the energy transport and mass balance is modelled well in a 1D set-up, these models fall short in the heating mechanisms. Especially in Parker’s braiding model the heating is a function of the 3D-configuration of the magnetic field. Therefore 1D models (as well as 2D models) have to rely on an ad-hoc parametrization of the heating, which often takes the form of an exponentially decreasing heating function.
The ability of Parker’s field line braiding model to maintain a hot corona was demon-strated by Gudiksen and Nordlund (2002, 2005b,a). They developed a 3D MHD model which includes the photosphere and lower corona. By solving the full energy equation, including the field-aligned Spitzer heat conduction and optical thin radiative losses, the evolution of the corona is solved self-consistently. The model is driven by an evolving magnetic field at the bottom boundary, with the purpose of entangling the magnetic field-lines. This braiding results in the formation of current structures. These are assumed to dissipate and convert into thermal energy. This energy is sufficient to keep the coronal regions of the model at temperatures of the order of 1 million degrees. Statistical analysis of the results of these models show a good match with actual observations of e.g. emis-sion structures and Doppler shifts (Peter et al. 2004, 2006).
The work presented in this thesis is based on the model developed by Bingert and Pe-ter (2011) which follows the concept originally developed by Gudiksen and Nordlund (2005a). Also in this numerical experiment the full MHD equations, are solved. The ma-jor difference between the earlier work is the inclusion of magnetic network elements.
This type of model has proven itself successful in reproducing several observational con-strains. Figure 2.2, taken from Zacharias et al. (2011), shows the Dopplershifts as a function of formation temperature. The diamonds indicate the calculated Dopplershifts from Bingert and Peter (2011). The dashed line is the trend derived from actual obser-vations (Peter and Judge 1999). The observed red-shifts of several transition region lines are reproduced by the model. However, the observed blue shifts at higher temperature are not reproduced. This could be explained by the closed top boundary, which would
Figure 2.2: The Doppler shifts as a function of formation temperature of an active region.
The diamonds are the calculated Doppler shifts from a 3D MHD simulation. The dashed line indicates the trend as derived from observations (Peter and Judge 1999). Image taken from Zacharias et al. (2011)
constrain significant up-flows. Follow-up models at higher resolution and an extended vertical range do find blue shifts at high temperature (P. Bourdin, 2013, Priv. Comm.).
Also the emission measure (EM), as derived from the model follows, the same trend as in the observations. Synthetic observations derived from synthesised emission found constant cross-sections of intensity for coronal loops as well as a similar intensity profile along the loop (Peter and Bingert 2012). Statistical analysis of the energy release shows a consistent scale-invariant distribution that is consistent with a nano-flare heated corona.
These similarities to observations make the results of both models very robust.
2.4.3 Dopplershifts
The cause of the observed blue and red-shifts on the Sun are still under debate. Athay (1984) proposed that these observation could be explained by the down flows of plasma draining of cooling coronal loops. An alternative suggestion was proposed by Boris and Mariska (1982), which involved syphon flows along the coronal loops.
Spadaro et al. (2006); Hansteen et al. (2010) suggested that a localized heating would push mass up and down, away from the point of heating. This would cause lower lying, and this cooler, plasma to move downward, whereas the the higher, and hotter, plasma would move upward. This could explain the excess of observed redshift for cooler plasma, a blue shifts for hotter plasma.