List of Footnotes

1 By convention, we will use the term forward to denote the direction pointed by the magnetic field vector.
2 The inclusion of vector components in the ignorable direction is referred to as two-and-a-half dimensions.
3 This representation is analogous to the use of Clebsch variables, also called a Clebsch transformation, to express vortex lines in hydrodynamics (Lamb, 1932).
4 While α and β are more frequent choices for denoting these, α is used for too many other things in MHD already.
5 The initial condition B(x,0) must be divergence-free in order that ∇ ⋅B = 0 for all time.
6 The term generic and the related term structurally stable will be used repeatedly in a very precise mathematical sense. A full definition, as given in, e.g., Guckenheimer and Holmes (1983Jump To The Next Citation Point), is not possible here. The terms basically refer to situations whose qualitative form (i.e. topology) will not be destroyed by small changes. It is invoked to rule out specially constructed cases such as a perfect symmetry, to which a given statement will not apply.
7 An alternative term, B-type null, first appeared in the magnetospheric literature (Yeh, 1976) and has also been used in the solar physics literature (Greene, 1988Jump To The Next Citation Point; Lau and Finn, 1990Jump To The Next Citation Point). The term “positive null” is used here since it is more descriptive than “B-type”. Field lines within the fan surface are directed outward from the null as from a positive charge.
8 There may, nevertheless, be some utility in adopting some definition whereby field lines evolve in time. Hesse and Schindler (1988Jump To The Next Citation Point), and Priest et al. (2003Jump To The Next Citation Point) present such definitions.
9 This term is in common use in solar physics, although it risks confusion with the contradictory usage in unbounded systems.
10 Many of these details can, however, prove to be important in modeling magnetic topology.
11 Some theoretical investigations of hypothetical magnetic monopole particles, most notably by P.A.M. Dirac, treat them as the termini of semi-infinite solenoids. This artificial construction is remarkably reminiscent of the actual configuration of solar flux tubes.
12 The term “field-free” refers to the absence of the normal component of the magnetic field in these regions. There can be non-vanishing horizontal field, at least when defined as the limit from above.
13 In this respect, the photospheric surface, z = 0, actually represents the merging layer at which the pressure-confined flux tubes expand into a volume-filling coronal field.
14 According to the foregoing definition of topological equivalence not all fan surfaces are separatrices since some separate field lines having the same footpoints which thus lie in the same domain.
15 Since infinity is a balancing source, the rows will sum to zero demonstrating that there are not S independent relations.
16 Longcope and Klapper actually excluded unbroken fans along with the sources and domains they enclosed. Beveridge and Longcope (2005Jump To The Next Citation Point) added them back in, and made certain less restictive assumptions.
17 The actual squashing degree is ∘ ------ (Q + Q2 − 4)∕2.
18 It is important to recall that this definition of topology arises from the high-energy electrons which have a large but not infinite mean free path.
19 In practice the expansion includes a pre-factor ∼ (RS ∕R⊙)ℓ, and P mℓ (x) is defined using the Schmidt normalization in order to keep all coefficients of roughly comparable magnitude. Equation (28View Equation) is intended only to indicate the basic form of the potential.
20 Ideally this would require knowledge of the field over the full solar surface, including the North and South poles, at one instant. Information from the “back” side is obtained by letting solar rotation bring it to the front; information about the poles is more difficult to obtain (see Hoeksema et al., 1982Jump To The Next Citation Point).