## 2. Bench Mark SolutionsWe aim here to solve for the linear force-free field in a spherical shell using the scalar potential approach. This both provides a useful solution in its own right, and also provides a benchmark to test the vector eigenfunction solutions. ## 2.1. General field in a spherical shellIn spherical co-ordinates, Eq. (2) is explicitly The equilibrium problem Eq. (1), with Eq. (7), reduces to solving a Grad-Shafranov equation and subject to appropriate boundary contraints on where , and and are surface spherical harmonics (see for example Sneddon (1979), their Eqs. (23.1) and (23.3)) The particular problem of a field between inner and outer spherical
shell boundaries at and
with given radial field on the
boundaries (which is directly proportional to and using standard orthogonality relations (see for example Sneddon (1979), their Eqs. (23.2),(23.4),(23.5)) the coefficients are found to be where with
for
, otherwise
, and
is found from Eq. (12) by
replacing It is then simply a matter of comparing Eq. (10) at
with Eq. (11) to find an
expression for the coefficients in
terms of and the resulting
combinations of the Bessel functions at the boundaries. Following some
manipulations, the final solution for and is obtained by interchanging
and abbreviate various combinations of sinusoidal functions like Now the field can be obtained directly by transforming Eq. (13) through Eq. (2), although this includes an assumption on commutability, to give and is obtained from
Eq. (19a) by replacing with
, while
is obtained from Eq. (19b) by
interchanging Field lines obtained from the solution Eqs. (18a,18b,18c) are shown in Fig. 1 for a particular solar magnetic field. In this example, boundary conditions were taken from SOHO-MDI photospheric magnetograms centred on 27 August 1996 when an extended low-latitude coronal hole was present. A uniform radial field strength has been imposed across the boundary at , selected from trial values to ensure that the solution closely resembles SOHO-LASCO observations (as discussed in the introduction). Different magnetic helicities are shown corresponding to different values of as detailed in the caption. Note that the coronal hole lies alongside the western (right-hand) edge of the N-S aligned arcade which can be seen near the centre of the disc.
## 2.2. Simplified boundary conditions
The solution Eq. (13) does, with a -
(i) take the outer shell boundary, , (for now) to be perfectly conducting so that all related coefficients vanish. -
(ii) take only axisymmetric fields , and so coefficients also vanish, -
(iii) admit only an mode inner boundary condition, , specified by .
Now (see the r-component of
Eq. (A.12) in the Appendix), whereas Eq. (13) reduces to
so that
, and so the Notice that there is a pole when the denominator vanishes, that is, for values of where and for this model with and , then and . The , vector potential, found from and similarly the simplified model field is obtained from
which can then be shown directly to
satisfy the requirements of Eq. (1) without the need to assume a
commutability. A marginally less simplified case can be used to model
an idealised corona and is obtained by first inferring the solution
with only an inhomogeneous The interesting question of how the field structure changes as crosses the first eigenvalue, and the topology of fields at higher eigenvalues, is discussed at length in Cantarella et al. (2000) (see their Sect. VII and Fig. 4 & Fig. 5) and also in Dixon et al. (1990). Such fields have "islands" of magnetic field detached from the boundary surfaces (such as the photosphere). At an eigenvalue, the field is fully detached but of infinite magnitude. In practice a minimum energy state must always have below the first eigenvalue because of the infinite magnetic energy barrier there, and it is anyway true that only relatively small (and non-uniform) values of helicity have been measured in the solar corona. It is then better to model the solar corona by assuming a "partially relaxed" magnetic field whereby if enough helicity could be injected it would be a weighted average of that approaches the first eigenvalue (Kitson & Browning 1990). A final equilibrium (if one exists) then depends on both the initial and boundary conditions and so the model of Fig. 2 is too simplified in practice although it does elucidate the magnetic form. The model does however provide some insight into the paradox, as presented by say Eq. (4), since the curl of this field solution can be compared directly with the partial sums of the right-hand side of Eq. (4) (the coefficients are obtained in Sect. 3.3). It turns out that Eq. (4) is indeed an erroneous expansion but nevertheless the coefficients obtained through Eq. (5), an equation also closely related to the paradox, do provide a correct expansion of the field in Eq. (3) (as discussed in Clegg et al. 2000b and Laurence et al., in preparation).
© European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |