## Appendix A: separation of variablesSubstituting Eq. (7) into Eq. (1), the radial component yields an identity, whilst the and components together imply a scalar Grad-Shafranov equation where which could be solved for , but since the general solution is then and taking the curl in Eq. (2), it is clear that the components arising from vanish, and although the component is initially non-zero for , its effect vanishes upon taking the further curl to generate . Hence we need only solve the homogeneous problem Eq. (A.1) where . Writing , and separating variables but since the dependence is isolated, this term must be constant, denoted , so that and In order that be single-valued,
which is simply Legendre's associated differential equation,
provided ; since is excluded (being infinite at one pole, ). Thus the separation can be written where are surface spherical harmonics or since these functions are solutions to and then the radial equation is simply which with a change of variable can be satisfied for by combinations of Bessel functions written related simply to the second kind of Bessel functions, because
Finally using Eq. (A.7), the radial component of Eq. (7) simplifies to and so the field transformation Eq. (7) is explicitly ## Appendix B: scalar field eigenfunction analysisIt is required to find a "series" solution to Eq. (9) with operator specified by Eq. (9) and Eq. (8) and subject to the decomposition Eq. (24). The complementary problem Eq. (25) is explicitly where the boundary conditions ,
that is ,
on but in general there is a synthesis of such terms indexed by to satisfy the boundary conditions, that is, each coefficient is found from Eq. (12). Using Eq. (15) we define the radial function the homogeneous equation is Eq. (26) and from Appendix A the eigenfunctions can be written as or introducing some arbitrary phase coefficients in these can be written Again there is in general a superposition
of such functions besides the
explicit radial expansion associated with the index for every term . Now it turns out
that the operator but substituting Eq. (25) and Eq. (26) into Eq. (B.7) gives an equation as and using Eq. (B.2) and Eq. (B.5) and the unknown coefficients and are found through some general orthogonality relations in given by Ozisik (1980), Table 3-2.9, that Separating the and
parts of Eq. (B.9),
integrating in Eq. (B.12a) can be calculated as Ultimately we seek but using Eq. (B.14), the combination can be written as (notice the change in the subscript of where is specified in Eq. (29c). Finally combining Eq. (B.2) and Eq. (B.18), and superposing all elements, the series solution is Eq. (27). ## Appendix C: the determination of zeroesConsider an eigenfunction like Eq. (B.4) but for now with an open specification of the radial function , writing This must satisfy a generalised eigenfunction equation written as where from Eq. (9) and Eq. (8), Now and so substituting into Eq. (C.3) and cancelling also Legendre's associated equation provides that which is also satisfied by . Substituting into Eq. (C.4) and cancelling the terms and following the procedure from Eq. (A.8) a solution is using Eq. (15) where the linear combination has been imposed
to directly satisfy the eigenfunction requirement at
. However the eigenfunction must
also vanish through , and this zero
is independent of ## Appendix D: integration and orthogonalityThis appendix details the calculation of the field eigenfunction expansion from Sect. 3.2 associated with the Laurence et al. formula Eq. (33). From Eq. (39a) and Eq. (40a,etc.), the difference function is and the eigenfunctions are given by Eq. (37a,etc.). In forming the dot product , only the terms survive the integration, and also combinations of sines and cosines vanish (justifying the simplification of the decomposition into two distinct eigenfunctions in Sect. 3b and a solution by superposition), because The following integrals of Now can be integrated by parts and simplifies through and . Next using the Associated Legendre equation on the second integral then
It turns out that and
only occur in this combination in
expanding the dot product and so only
is non-zero. The integrals for
and
are zero for
since they can be written as
which vanishes at the end-points.
For we can integrate by parts to
show that and again this is the
only combination encountered in the dot product. Thus only
is involved. Following these
reductions, the only using where the superscript is implicit. The dot product integral (after removing terms) becomes where the square bracketed terms are associated with the two halves
of the problem. Using the results on the The remaining task is to combine and evaluate the radial integrals:
where we have used and and given by Eq. (B.13), Eq. (B.14). Hence Following similar manipulations the denominator Eq. (33) is Combining Eq. (D.6) and Eq. (D.7), the expansion becomes Now writing out the field eigenfunctions Eq. (37a,etc.) and using the relationship Eq. (B.17) and the similar conversions
the field expression eventually simplifies to Eq. (42a,etc), Sect. 3b. © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |