Astron. Astrophys. 361, 743-758 (2000)

Appendix A: separation of variables

Substituting Eq. (7) into Eq. (1), the radial component yields an identity, whilst the and components together imply a scalar Grad-Shafranov equation

where L is given by Eq(9). The arbitrary function is set to zero because a general solution to Eq. (A.1) comprises a complementary function , the solution to the homogeneous equation, and a "particular solution" which satisfies

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, m must be an integer since the full azimuthal range is included. The remaining equation has terms separate in r and µ and so equal to a constant C leaving

which is simply Legendre's associated differential equation, provided ; n is taken to be a real positive number following Demoulin & Priest (1992), but the resulting series solution would not converge at the poles unless n is an integer. The solution is then simply

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 n is an integer, by

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 analysis

It 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 S, can now be directly related to in Eq. (18a) by Eq. (A.11). An element of the potential solution is (by inspection) using Eq. (17)

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 v. The eigenvalues are given by Eq. (30). Now Eq. (24) (in generalised notation) can be substituted into Eq. (9) writing

for every term . Now it turns out that the operator L can be commuted across the sum (see Clegg et al. 2000b) and this is explicitly

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

where

Separating the and parts of Eq. (B.9), integrating in r using Eq. (B.10a) gives

where

Eq. (B.12a) can be calculated as
which is simply

whereas Eq. (B.12b) is

Ultimately we seek

and using Eq. (B.11),

but using Eq. (B.14), the combination can be written as

(notice the change in the subscript of g). Substituting Eq. (B.13) and Eq. (B.17) into Eq. (B.16) and using Eq. (B.10b)

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 zeroes

Consider 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 m, i.e. the vth radial zero is affected only by the order n of the Bessel function equation Eq. (C.7). Hence it is appropriate to interchange so that (see Eq. (29b)), radial zeroes are found from Eq. (30), and most generally the problem Eq. (C.3) is di facto as used in the text.

Appendix D: integration and orthogonality

This 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 µ occur (given also that now, but not yet assuming that )

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 radial integrals of concern are written below

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 µ integrals

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
helpdesk.link@springer.de