The most common quantitative model of the global corona is the Potential Source Surface (PSS) model first introduced by Schatten et al. (1969) and Altschuler and Newkirk Jr (1969), and refined by subsequent investigators. The field is taken to be potential, , in the region between the photosphere and an outer shell at called the source surface (see Figure 24). The solenoidal condition, , means that the scalar potential must be harmonic, , within the range . The upper boundary, , is meant to represent the base of the true solar wind, in which all field is open and approximately radial. The appropriate boundary condition for this purpose is , which makes all field purely radial () on the surface. The general solution to Laplace’s equation satisfying this boundary condition is19. The coefficients and are fixed using the lower boundary condition from the photospheric magnetic field. This boundary condition can, for example, be formulated in terms of a sequence of magnetograms covering one complete solar rotation (Altschuler and Newkirk Jr, 1969)20.
In a complete magnetic field model the outside of the potential field from Equation (28) is matched to the inside of a heliospheric field. The latter typically consists of field lines tracing Parker spirals outward or inward along cones from , as shown in Figure 24. The radial field at the source surface, , therefore serves as the “source” of the heliospheric field. The outward and inward sectors are anchored where and , respectively.
The curve along which thereby defines the base of the heliospheric current sheet separating the heliospheric sectors: It is the sector boundary (see Figure 25). At radii approaching the source surface from below, Equation (28) becomes strongly dominated by its lowest poles, and , corresponding to the photosphere’s overall dipole and quadrupole. One consequence of its low-order nature is that tends to be very smooth, vanishing along a single closed curve dividing the sphere in two. Thus, at least at its base the heliospheric current sheet is not a discontinuity since passes continuously through zero.
The sector boundary is defined by the curve or curves on the source surface where , as shown in Figure 25. Were the field purely dipolar the sector boundary would be a great circle on the source surface, although it would only coincide with the rotational equator in the case . It is the influence of the quadrupole (and the higher poles to a lesser extent) which deforms the heliospheric current sheet, sometimes resulting in four sector crossings per rotation or even multiple distinct unipolar regions (Hoeksema et al., 1983). Examples of particularly complex configurations are shown in Figure 26.
Since both and vanish everywhere over the source surface, the sector boundary, , is a curve of genuine magnetic null points. Such a one-dimensional continuum of nulls is one of the constructions expressly discounted in the foregoing section (2.4) since it is non-generic. It owes its appearance now to the definition of the source surface as a surface on which and vanish simultaneously (in general circumstances, these functions would vanish on separate surfaces which would generically intersect transversally only along curves). These null points are neither positive nor negative but are all X-type nulls whose neutral direction (along the eigenvector corresponding to the eigenvalue) is parallel to the sector boundary curve. Each null point has two distinct separatrix curves extending downward into the current-free corona: one forward and one backward (see Figures 24 and 25). For each closed sector curve on the source surface the forward separatrix curves form a continuous separatrix surface mapping downward to negative polarity photospheric regions; this is the red curve on the right part of Figure 27. The backward separatrices form independent separatrix surfaces mapping to positive regions (the blue curve). These surfaces are the separatrices between open and closed magnetic field lines. Their footpoints trace photospheric curves which are the theoretical manifestation of coronal hole boundaries in the PSS model. The coronal field becomes increasingly complex at decreasing radii and can also contain isolated coronal null points at radii . Thus the separatrices leaving the smooth sector boundary may map to fairly complex coronal hole boundaries.
The topology of a heliospheric field model may be tested against several types of observation. The model was first developed by Schatten et al. (1969) for comparison to the magnetic sectors observed by spacecraft at 1 AU. This showed that setting gave a model whose predictions of mean field strength, sector duration and smoothness in was in good agreement with the average values observed by spacecraft. Hoeksema et al. (1982) compared observations to time histories of radial field direction (inward or outward) predicted by source surface models over 18 solar rotations (1976–1977). The observations showed four sector boundary crossings per rotation, implying a rather complex heliospheric current sheet. The PSS model produced a time history agreeing with the observations with a correlation coefficient of ; the maximum correlation occurred when . Burton et al. (1994) compared a PSS model to sector crossings of the ISEE-3 spacecraft and found that not only were the transitions correctly predicted, but so was the inclination of the current sheet at the crossings (with a correlation coefficient of 0.96). An independent test was provided by Bruno et al. (1984), who observed a correlation between the predicted location of the heliospheric current sheet and the location of maximum K-corona brightness observed from the Mauna Loa coronagraph.
Originally developed to model the heliospheric magnetic field, the PSS model has also proven useful in studies of large-scale coronal topology. The separatrices extending downward from the sector boundary, , form the boundary between open and closed coronal field lines. There is evidence in support of this interpretation and of the correspondence between open field lines and coronal holes. Coronal holes are defined observationally by their lower X-ray or EUV emission or by a lack of chromospheric network in He ii emission at 304 Å or 10,830 Å. Levine (1982) found very good agreement between open/closed boundaries (separatrices) in a PSS model and observed coronal hole boundaries, especially in the period near solar maximum.
The PSS model has been used to study the evolution of the large-scale corona including its coronal holes (Sheeley Jr et al., 1987; Wang and Sheeley Jr, 1993; Wang et al., 1996; Luhmann et al., 1998). Evolution of the photospheric field leads to changes in the overlying coronal field extending all the way to the source surface. The open/closed boundaries extend from the source surface, so the motion of their photospheric footprint need not coincide with photospheric motions. This was found to give a compelling explanation for the fact that coronal holes were observed to rotate rigidly in spite of the photosphere’s differential rotation (Levine, 1982; Sheeley Jr et al., 1987). The implication of this model is that field lines anchored to the photosphere undergo reconnection in order to open or close as the separatrix sweeps over them. While the need for such reconnection is predicted by the PSS model, its dynamics cannot be studied in a quasi-static model (Wang and Sheeley Jr, 2004). There has been observational confirmation of reconnection occurring at coronal hole boundaries (Madjarska et al., 2004). Luhmann et al. (1998) propose that CMEs are a more dramatic manifestation of the need to open magnetic fields.
A still more global consequence of this application is its prediction of how the net open flux responds to the photospheric field. The open flux in the PSS model can be calculated by integrating over the entire source surface. The field at the source surface will evolve over the solar cycle, principally in response to the lowest moments of the photospheric flux. As a consequence tends to reach its maximum when the dipole moment is largest some time near solar minimum (Wang et al., 2000). This is in close agreement with observations (King, 1979), which must infer the total flux from a few point measurements as discussed above. Once again the quasi-static model cannot be used to determine how the open flux is changed, only to predict that it will change. One possibility for increasing the open flux, discussed further below, is that coronal mass ejections leaving the corona drag open previously closed field lines (Gosling, 1975). The subsequent reduction in open flux after solar minimum could occur only through the reconnection of open field lines to produce more closed flux.
© Max Planck Society and the author(s)