The basic link between coronal holes and open magnetic field was found using PFSS extrapolations from synoptic magnetogram observations. While the open field regions in such extrapolations produce pleasing agreement with observed coronal hole boundaries (Levine, 1982; Wang et al., 1996; Neugebauer et al., 1998; Luhmann et al., 2002), our understanding has been enhanced by coupling coronal magnetic models with photospheric simulations, in the following ways:
We outline here the studies that fall into the second category. Firstly, a series of papers by Wang, Sheeley, and co-workers, coupling surface flux transport to PFSS extrapolations, has significantly added to our understanding of coronal hole dynamics. By isolating different terms in the flux transport model, Wang and Sheeley Jr (1990) demonstrate how each of the terms in the flux transport model play their roles in producing the global pattern of coronal holes. Supergranular diffusion spreads out active region flux so as to create unipolar areas containing open flux, and also facilitates the build up of trailing polarity holes in each hemisphere after Solar Maximum by annihilating leading polarity flux at the equator. Differential rotation helps to accelerate the formation of axisymmetric polar holes by symmetrising the active region flux distribution. Meridional flow (i) concentrates flux at the poles – preventing the polar holes from spreading to lower latitudes, (ii) hastens the decay of low-latitude holes by transporting them to mid-latitudes where differential rotation is more efficient, and (iii) impedes cancellation across the equator, thus reducing the flux imbalance in each hemisphere.
A striking observation arising from Skylab was that coronal holes rotate differently from the underlying photosphere. This is exemplified by the Skylab “Coronal Hole 1”. Nash et al. (1988) were able to reproduce and explain the behaviour of this hole using the coupled flux transport and PFSS model (see also Wang and Sheeley Jr, 1993). The mechanism is illustrated by their model of a single bipolar active region in a dipolar background field (Wang and Sheeley Jr, 1990; Wang et al., 1996, reproduced here in Figure 15). There are two important points. The first is that coronal hole boundaries in the PFSS model are determined by the global magnetic structure of the coronal field. This is because the source surface field depends only on the lowest order spherical harmonics. In particular, the open field regions in Figure 15 are not simply a superposition of those that would be obtained from the bipole and background fields individually (Wang et al., 1996). Hence, it need not be surprising that coronal holes are observed to rotate differently from the footpoints of individual field lines. The second point is that the rotation rate is determined by the non-axisymmetric component only. In other words, the “coronal hole extension” in Figure 15 must rotate approximately with the bipole giving rise to it. It follows that the rotation rate of the coronal hole matches that of the photospheric latitude where the bipole is located, not necessarily the latitude where the open field lines have their footpoints. This explains why the northern and southern coronal hole extensions in Figure 15 behave differently: the bipole flux causing the northern hole extension is located in a narrow band close to the equator, so that the hole rotates rigidly with the equatorial 27-day period. On the other hand, the flux causing the southern hole extension is spread between 0° and 40° latitude, so that this hole is considerably sheared by differential rotation. The same idea explains why rigidly rotating coronal holes are more prevalent in the declining phase of the solar cycle: during this phase, the non-axisymmetric flux is concentrated more at lower latitudes.
More recently, new insights into coronal hole evolution have started to come from coupling global MHD models (Section 3.4) to surface flux transport. For example, an important implication of the PFSS model for coronal magnetic fields is that continual magnetic reconnection is necessary to maintain the current-free field (Wang and Sheeley Jr, 2004). This has been tested in a global MHD model by Lionello et al. (2005), who applied differential rotation to a configuration consisting of a single bipole in a dipolar background field. They confirm the results of Wang et al. (1996) that the coronal hole extension rotates nearly rigidly without significant change, even when the surface field is significantly sheared by differential rotation. The dominant reconnection process was found to be interchange reconnection, with continual reconnection opening field on the eastern hole boundary and closing it on the western boundary. In some cases, when field line footpoints pass over the coronal hole boundaries multiple times, multiple openings and closing occur. However, it should be noted that these simulations used a uniform resistivity orders of magnitude greater than that in the real corona, so that reconnection may be less efficient in reality. Fisk (1996) has proposed an alternative scenario where open field lines continually circulate in latitude and longitude as their footpoints rotate differentially. This process was applied to explain the origin of high latitude Solar Energetic Particles (SEPs) seen by Ulysses (Keppler et al., 1995; Maclennan et al., 1995), which are thought to originate in low latitude corotating interacting regions (CIRs, McDonald et al., 1976). However, in a test with their global MHD model and an initial tilted dipole field, Lionello et al. (2006) found that although field lines did move in latitude, the coronal hole boundaries rotated in a manner consistent with the extrapolation models, and not perfectly rigidly as predicted by Fisk (1996). This meant that the latitudinal excursion of around 25° found in the simulations was insufficient to explain that required by Fisk (1996).
Recently, Linker et al. (2011) have applied their global MHD model to test another hypothesis, namely, that the total amount of open flux is conserved. This has been proposed by Fisk and co-workers based on heliospheric observations (Fisk and Schwadron, 2001). They propose that open flux is transported by interchange reconnection (between open and closed field lines), but not created or destroyed through other forms of reconnection. A feature of this model is that open field lines may be transported through closed field regions in a random manner during the reversal of the polar field. In their simulations, Linker et al. (2011) start with a synoptic magnetogram for CR1913, but add two small bipoles of varying orientation inside one of the large equatorward coronal hole extensions. They then advect the bipoles through the coronal hole from the open to closed field regions. The idea is to test (a) whether open flux associated with the bipoles can be transported out of the coronal hole and into the closed field region (as required by the interchange model; Fisk and Zurbuchen, 2006), and (b) whether isolated coronal holes can exist in each hemisphere. The results (Figure 16) seem to contradict the requirements of the interchange model, and to uphold the conjecture of Antiochos et al. (2007), namely that what appear to be isolated coronal holes in each hemisphere, are in fact, always connected (by very thin corridors) of open flux to the polar holes in each hemisphere. These thin corridors may intersect with strong field regions that form part of the streamer belt. Thus, they may have a non-negligible contribution to the amount of open flux, which may help resolve the discrepancy between the amount of open magnetic flux obtained in models and that directly observed through IMF measurements (Section 4.1). Following this, Titov et al. (2011) constructed a topological model for the open flux corridors that connect coronal holes. Further studies by Edmondson et al. (2009, 2010) considered idealised case studies of how magnetic reconnection affects the boundaries of coronal holes when either a bipole is (i) advected across the coronal boundary (Edmondson et al., 2010) or (ii) stressed by rotational motions (Edmondson et al., 2009). In line with the conjecture of Antiochos et al. (2007) and quasi-steady model results of Linker et al. (2011), the authors see a smooth transition of the open flux where the open field regions are never isolated. Such evolution of coronal hole boundaries along with the translation and opening/closing of magnetic field lines has important consequences for the solar wind. A full discussion of this is beyond the scope of the present review but details can be found in Cranmer (2009) and Antiochos et al. (2012).
Living Rev. Solar Phys. 9, (2012), 6
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