8.2 Open/closed boundaries

Locating the photospheric source of the solar wind requires some kind of magnetic model extending through the low solar corona, where β ≪ 1, into the super-Alfvénic solar wind. The key topological elements in any such model are the boundaries between open and closed field lines. Yeh (1978Jump To The Next Citation Point) presented a general discussion of the possible topologies of the open/closed separatrices. We neglect at this point those separatrices between different regions of closed fields since they are topologically identical to cases without open field discussed at length in the foregoing sections. In the remaining cases, Yeh concluded that separatrices could either separate open field lines from closed field lines, or they could separate the two polarities of open field lines. A given closed flux region can either have a compact footprint or its footprint can surround one or more regions of open field lines (compare Panels a and b in Figure 23View Image). A compact footprint (one which includes no open field regions) will be enclosed by two separatrices, connected by a separator contacting the photosphere at two points, as in Panel a. When the closed region surrounds a section of open flux, as in Panel b, it will form a “crater-like” enclosure, with separatrices inside and outside intersecting along a closed coronal separator (Yeh, 1978Jump To The Next Citation Point).
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Figure 23: Possibilities for topological boundaries with open field lines. Lower panels show the photospheric field in relation to the PIL (dark solid curve) and separatrices (dashed). In each case, the positive field (+s) forms a compact region separated from the surrounding negative region by the PIL. Top panels are elevation views of the field lines (thin curves) and separatrices (dark curves) anchored to the positive and negative regions denoted with +s and −s. Panel a: a closed separatrix enclosing a compact region of closed field. Panel b: a “crater-like” enclosure where the closed field region surrounds open field (reproduced from Yeh, 1978).

The most common quantitative model of the global corona is the Potential Source Surface (PSS) model first introduced by Schatten et al. (1969Jump To The Next Citation Point) and Altschuler and Newkirk Jr (1969Jump To The Next Citation Point), and refined by subsequent investigators. The field is taken to be potential, B = − ∇ χ, in the region between the photosphere and an outer shell at r = RS called the source surface (see Figure 24View Image). The solenoidal condition, ∇ ⋅ B = 0, means that the scalar potential must be harmonic, 2 ∇ χ = 0, within the range R ⊙ < r < RS. The upper boundary, r = RS, 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 χ(RS, 𝜃,ϕ) = 0, which makes all field purely radial (B 𝜃 = B ϕ = 0) on the surface. The general solution to Laplace’s equation satisfying this boundary condition is

∑∞ ∑ℓ [ ( )ℓ+1 ( ) ℓ] χ (r,𝜃,ϕ) = RS- − -r- P m (cos𝜃)[gm cos(m ϕ) + hm sin (m ϕ)], (28 ) ℓ=1 m=0 r RS ℓ ℓ ℓ
where P mℓ (x ) is the associated Legendre polynomial19. The coefficients gmℓ and hmℓ 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.
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Figure 24: A schematic depiction of a source-surface model as viewed from above the Sun’s North pole (the sense of rotation is indicated by a semi-circular arrow). A dashed circle shows the source surface at r = RS. The field is made purely radial at this surface by setting B𝜃 = B ϕ = 0. Between the source surface and the solar surface, r = R ⊙, the magnetic field is potential: B = − ∇ χ. Field lines anchor to the photosphere in a negative region (shaded segment) and positive regions. Outside the source surface the field is swept back in a Parker spiral. Two null points (X-points) are shown as circles on the source surface. The upward and downward separatrices are shown in blue and red, respectively. The sector boundaries are shown as green curves.

In a complete magnetic field model the outside of the potential field from Equation (28View Equation) 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 r = RS, as shown in Figure 24View Image. The radial field at the source surface, BS (𝜃,ϕ) = Br(RS, 𝜃,ϕ), therefore serves as the “source” of the heliospheric field. The outward and inward sectors are anchored where BS > 0 and BS < 0, respectively.

The curve along which BS = 0 thereby defines the base of the heliospheric current sheet separating the heliospheric sectors: It is the sector boundary (see Figure 25View Image). At radii approaching the source surface from below, Equation (28View Equation) becomes strongly dominated by its lowest poles, ℓ = 1 and ℓ = 2, corresponding to the photosphere’s overall dipole and quadrupole. One consequence of its low-order nature is that BS (𝜃,ϕ ) = − ∂ χ∕∂r |r=RS tends to be very smooth, vanishing along a single closed curve dividing the r = RS sphere in two. Thus, at least at its base the heliospheric current sheet is not a discontinuity since Br passes continuously through zero.

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Figure 25: A schematic depiction of the sector boundary defined by the potential source surface model. On the left is a view of the coronal field, R⊙ < r < RS, in the meridional plane using the same scheme as Figure 24View Image. The X-points at the source surface are shown by circles. From them originate one downward separatrix (red) and one upward separatrix (blue). The map on the right is the source surface itself (r = RS) plotted as sine of latitude versus longitude. The dark curve is the sector boundary, BS = 0, separating the outward sector (BS > 0, white) from the inward sector (BS < 0, grey). The dashed vertical lines at 0∘ and 180∘ correspond to the two meridional slices on the left.

The sector boundary is defined by the curve or curves on the source surface where BS = 0, as shown in Figure 25View Image. 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 g11 = h11 = 0. 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 26View Image.

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Figure 26: Examples of particularly complex sector boundaries. These are contour plots of B (𝜃,ϕ) S at the source surface, R = 2.5R S ⊙, versus Carrington longitude (horizontally) and latitude (vertically) similar to the format of Figure 25View Image. Dark solid curves show the sector boundary, BS = 0, separating inward sectors (partially grey with red contours) from outward sectors (blue contours). The top panel is from September 1999, as the Sun’s dipole is reversing and the quadrupole moment is dominant. There are four sector boundary crossings along the solar equator. The bottom panel is from January 2002, when the Northern hemisphere is mostly inward but includes a patch of outward flux enclosed by a second sector boundary. (Courtesy of J.T. Hoeksema and Wilcox Solar Observatory).

Since both B𝜃 and Bϕ vanish everywhere over the source surface, the sector boundary, Br = 0, 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 B𝜃 and Bϕ 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 λ2 = 0 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 24View Image and 25View Image). 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 27View Image. 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 R⊙ < r < RS. Thus the separatrices leaving the smooth sector boundary may map to fairly complex coronal hole boundaries.

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Figure 27: A schematic depiction of the coronal hole boundaries defined by the source surface model. The left is the same meridional view as in Figure 25View Image. The right part now shows the solar surface (r = R ⊙) plotted as sine of latitude versus longitude. The blue and red curves are the footprints of the upward and downward separatrices, respectively; the black curve is the PIL. The shaded regions are (from top to bottom) the outward coronal hole (light grey), positive closed flux (white), negative closed flux (grey) and inward coronal hole (dark grey). The same colors on the left part indicate how these regions might appear on the disk.

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 R = 1.6R S ⊙ gave a model whose predictions of mean field strength, sector duration and smoothness in Br(t) 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 ≃ 0.63; the maximum correlation occurred when RS = 2.35R ⊙. 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, BS = 0, 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 (1982Jump To The Next Citation Point) 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., 1987Jump To The Next Citation PointWang and Sheeley Jr, 1993Wang et al., 1996Luhmann et al., 1998Jump To The Next Citation Point). 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, 1982Sheeley 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 Φopen responds to the photospheric field. The open flux in the PSS model can be calculated by integrating |BS | 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 Φopen tends to reach its maximum when the dipole moment is largest some time near solar minimum (Wang et al., 2000Jump To The Next Citation Point). This is in close agreement with observations (King, 1979Jump To The Next Citation Point), 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, 1975Jump To The Next Citation Point). The subsequent reduction in open flux after solar minimum could occur only through the reconnection of open field lines to produce more closed flux.

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