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

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Appendix A: separation of variables

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

[EQUATION]

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

[EQUATION]

which could be solved for [FORMULA], but since the general solution is then [FORMULA] and taking the curl in Eq. (2), it is clear that the components [FORMULA] arising from [FORMULA] vanish, and although the component [FORMULA] is initially non-zero for [FORMULA], its effect vanishes upon taking the further curl to generate [FORMULA]. Hence we need only solve the homogeneous problem Eq. (A.1) where [FORMULA]. Writing [FORMULA], and separating variables [FORMULA]

[EQUATION]

but since the [FORMULA] dependence is isolated, this term must be constant, denoted [FORMULA], so that [FORMULA] and

[EQUATION]

In order that [FORMULA] 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

[EQUATION]

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

[EQUATION]

since [FORMULA] is excluded (being infinite at one pole, [FORMULA]). Thus the separation can be written [FORMULA] where [FORMULA] are surface spherical harmonics [FORMULA] or [FORMULA] since these functions are solutions to

[EQUATION]

and then the radial equation is simply

[EQUATION]

which with a change of variable [FORMULA] can be satisfied for [FORMULA] by combinations of Bessel functions written

[EQUATION]

related simply to the second kind of Bessel functions, because n is an integer, by

[EQUATION]

Finally using Eq. (A.7), the radial component of Eq. (7) simplifies to

[EQUATION]

and so the field transformation Eq. (7) is explicitly

[EQUATION]

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

[EQUATION]

where the boundary conditions [FORMULA], that is [FORMULA], [FORMULA] on S, can now be directly related to [FORMULA] in Eq. (18a) by Eq. (A.11). An element of the potential solution is (by inspection) using Eq. (17)

[EQUATION]

but in general there is a synthesis of such terms indexed by [FORMULA] to satisfy the boundary conditions, that is, each coefficient [FORMULA] is found from Eq. (12). Using Eq. (15) we define the radial function

[EQUATION]

the homogeneous equation is Eq. (26) and from Appendix A the eigenfunctions can be written as

[EQUATION]

or introducing some arbitrary phase coefficients in [FORMULA] these can be written

[EQUATION]

Again there is in general a superposition [FORMULA] 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

[EQUATION]

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

[EQUATION]

but substituting Eq. (25) and Eq. (26) into Eq. (B.7) gives an equation as

[EQUATION]

and using Eq. (B.2) and Eq. (B.5)

[EQUATION]

and the unknown coefficients [FORMULA] and [FORMULA] are found through some general orthogonality relations in given by Ozisik (1980), Table 3-2.9, that

[EQUATION]

where

[EQUATION]

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

[EQUATION]

where

[EQUATION]

[EQUATION]

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

[EQUATION]

whereas Eq. (B.12b) is

[EQUATION]

Ultimately we seek

[EQUATION]

and using Eq. (B.11),

[EQUATION]

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

[EQUATION]

(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)

[EQUATION]

where [FORMULA] is specified in Eq. (29c). Finally combining Eq. (B.2) and Eq. (B.18), and superposing all [FORMULA] 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 [FORMULA], writing

[EQUATION]

This must satisfy a generalised eigenfunction equation written as

[EQUATION]

where from Eq. (9) and Eq. (8),

[EQUATION]

Now [FORMULA] and so substituting into Eq. (C.3) and cancelling

[EQUATION]

also Legendre's associated equation provides that

[EQUATION]

which is also satisfied by [FORMULA]. Substituting into Eq. (C.4) and cancelling the [FORMULA] terms

[EQUATION]

and following the procedure from Eq. (A.8) a solution is

[EQUATION]

using Eq. (15) where the linear combination has been imposed to directly satisfy the eigenfunction requirement at [FORMULA]. However the eigenfunction must also vanish through [FORMULA], 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 [FORMULA] so that [FORMULA] (see Eq. (29b)), radial zeroes are found from Eq. (30), and most generally the problem Eq. (C.3) is di facto [FORMULA] 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

[EQUATION]

[EQUATION]

[EQUATION]

and the eigenfunctions are given by Eq. (37a,etc.). In forming the dot product [FORMULA], only the [FORMULA] terms survive the [FORMULA] 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

[EQUATION]

The following integrals of µ occur (given also that [FORMULA] now, but not yet assuming that [FORMULA])

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Now [FORMULA] can be integrated by parts and simplifies through [FORMULA] and [FORMULA]. Next using the Associated Legendre equation on the second integral then

[FORMULA]

It turns out that [FORMULA] and [FORMULA] only occur in this combination in expanding the dot product and so only [FORMULA] is non-zero. The integrals for [FORMULA] and [FORMULA] are zero for [FORMULA] since they can be written as [FORMULA] which vanishes at the end-points. For [FORMULA] we can integrate by parts to show that [FORMULA] and again this is the only combination encountered in the dot product. Thus only [FORMULA] is involved. Following these reductions, the only radial integrals of concern are written below

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

using [FORMULA] where the superscript [FORMULA] is implicit. The dot product integral (after removing [FORMULA] terms) becomes

[EQUATION]

where the square bracketed terms are associated with the two halves of the problem. Using the results on the µ integrals

[EQUATION]

The remaining task is to combine and evaluate the radial integrals:

[FORMULA]

[FORMULA]

where we have used [FORMULA] and [FORMULA] and [FORMULA] given by Eq. (B.13), Eq. (B.14). Hence

[EQUATION]

Following similar manipulations the denominator Eq. (33) is

[EQUATION]

Combining Eq. (D.6) and Eq. (D.7), the expansion becomes

[EQUATION]

Now writing out the field eigenfunctions Eq. (37a,etc.) and using the relationship Eq. (B.17) and the similar conversions

[FORMULA]

[FORMULA]

the field expression eventually simplifies to Eq. (42a,etc), Sect. 3b.

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Online publication: October 2, 2000
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