## 4. DiscussionThe paper has concentrated on a detailed mathematical analysis of a linear force-free field in a spherical shell with inhomogeneous conditions over the bounding surfaces. There have been two motivations: first, to obtain a solution that could be used to model the solar or a stellar corona, and second, to obtain and compare different equivalent formulations of the solution. Of particular interest has been the validation of a new formula to obtain the coefficients in an eigenfunction expansion. The linear force-free field is in fact somewhat inappropriate as a model of the global solar coronal plasma (non-linear force-free models are more realistic but have so far only been used over local domains in the corona, for example Clegg et al. 2000a) but nevertheless it is of interest to compare the overall effects of magnetic helicity with the often used potential field approximation (Clegg et al. 1999a). In this respect, boundary conditions were obtained from SOHO-MDI data centred on 27 August 1996, a time associated with the meridian passage of a large N-S extended coronal hole. This entailed a suitable transformation of the line-of-sight data from an temporal set of magnetograms to obtain a radial field component boundary condition extrapolated over the entire solar surface. Some field results were shown in Fig. 1 and Fig. 4 and further discussion on the boundary conditions and the realism obtained by adding the magnetic helicity can be found in Clegg et al.(1999b). The general solution might also be applied to other areas of astrophysics with spherical domains. The second objective, to demonstrate and validate a new method, centred on the ability to obtain the solution in several ways. One method involved solving the problem in terms of a scalar then transposing to the magnetic field. A solution in terms of special functions was found, and through deriving an identity this could be converted into a series solution. The series could also be obtained directly by an eigenfunction expansion and we have shown that two different formulae attributed to Jensen and Chu, and Laurence et al. respectively yield the same result, although we feel that the latter formula for the expansion coefficients is the more convenient. The reader is referred to companion papers for the derivation and discussion of the two formulae (see Clegg et al. 2000b; Laurence et al. in preparation). © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |