3.2 Field extrapolation

Topological coronal models generally use magnetic fields extrapolated into the corona from photospheric data or an assumed photospheric distribution. The models themselves are independent of the extrapolation technique: They describe topological properties common to all magnetic fields. Nevertheless, since much of the literature is cast in terms of specific extrapolation methods, we briefly review their basics below. A typical procedure is to extrapolate the field upward from photospheric values after making some assumption about the state of the coronal field. (For concreteness we phrase the following discussion in terms of Cartesian coordinates with $z = 0$ being the photospheric plane; most of the discussion can be translated into spherical coordinates with only minor complications.)

The most common assumption made in extrapolation, that the coronal magnetic field is in force-free equilibrium

$B × ( \~/ × B) = 0, z > 0, (13)$

is motivated by the corona’s general calm and relatively small values of $b =_ 8pp/B2$ , at least above the chromosphere. Under the least restrictive assumption, Equation (13

) may be satisfied by any solution of

$\~/ × B = aB, (14)$

where $a(x)$ is arbitrary except for the requirement $B . \~/ a = 0$ , in order to preserve $\~/ .B = 0$ .

Equation (14 ) is nonlinear since both $B$ and $a$ are formally unknown, and is therefore difficult to solve for arbitrary boundary conditions. It is almost never used except in large-scale numerical solutions (see McClymont et al., 1997 , for a review of these techniques). Making the additional restriction that $a$ is spatially uniform leads to a special case called the linear force-free field or the constant- $a$ field (Nakagawa and Raadu, 1972 ; Chiu and Hilton, 1977 ; Gary, 1989 ). This additional restriction can be justified by an appeal to minimization of energy and conservation of helicity (Woltjer, 1958 ), but it is most often adopted simply for expediency. Governed by the linear Helmholtz equation, $2 2 ( \~/ + a )B = 0$ , the constant- $a$ field is significantly easier to find, although it can behave unphysically in unbounded domains (Nakagawa and Raadu, 1972 ).

The system can be made easier still by assuming $a = 0$ , which is equivalent to assuming the coronal field contains no current density. This ultimate simplification leads to the so-called potential field model which is by far the simplest, most frequently used, and most often criticized. For a potential field $\~/ × B = 0$ and $\~/ .B = 0$ , making the magnetic field a direct analog of an electrostatic field in a charge-free region. This analogy is exploited by writing the magnetic field in terms of a scalar potential, $B = - \~/ x$ , which can be found directly from the boundary data (Schmidt, 1964 ). For a planar photosphere, unbounded above, the scalar potential is

by analogy to Coulomb’s law. In spherical geometry one uses a spherical harmonic expansion to solve Laplace’s equation, $\~/ 2x = 0$ , in the region $r > R o.$ . The inner boundary, $r = Ro .$ , is constrained by magnetograms; source surface models impose the homogeneous Dirichlet condition, $x = 0$ , at the outer boundary $r = RS$ , making the magnetic field purely radial there.

In the potential field model the normal component $Bz(x, y,0)$ is the boundary data necessary and sufficient for unique solution. This means the photospheric horizontal field, $Bx(x, y,0)$ and $By(x, y,0)$ , can be found from $Bz(x, y,0)$ . If these differ from measurements of those components (and they almost always will) then the field is evidently not potential. It is not so easy to know how much data is necessary for a unique solution of the less restrictive models, constant- $a$ or general force-free equilibria.

In the case of a potential field the normal photospheric field plays the role of a surface magnetic charge density, analogous to an electrostatic surface charge. A localized magnetic region, such as a sunspot, therefore appears as a magnetic charge, and the leading order in their multipole expansion (Jackson, 1975 ) will be their monopole term. It is commonly held that there are no actual magnetic charges in the universe, and the present situation does not contradict this belief. Rather each localized photospheric region is the end of a sub-photospheric flux tube (Parker, 1955 ) which only appears as a charge in the coronal half-space $z > 0$ ¹¹ . The magnetic charge of a given source is proportional to the flux in the tube, $Qmag = P/(2p)$ . Nor is the concept of magnetic charge unique to potential field extrapolation. Since coronal field lines are anchored at the photosphere, the photospheric normal field $B (x,y, 0) z$ is the source of field lines, regardless of what form the coronal field takes.