3. Eigenfunction expansion for the benchmark fields
3.1. Series solution using a scalar parameter
The calculation now moves on to find some eigenfunction series expressions to compare with the results from the previous section. First note that there is an equivalent (radial) series form for the scalar parameter P written in Eq. (13), and so a series for can also be found in this way. To see this we introduce the decomposition
Here the effective eigenvalues are (see Eq. (30)), rather than , being independent of m as explained in Appendix C. The eigenfunctions necessarily form a complete set but one has to be sure that all eigenfunctions have been found (a construction like Chandrasekhar-Kendall (1957) may not yield a full set). It is because the azimuthal components are determined by the boundary conditions that the problem reduces to finding a series for only the radial part of Eq. (13). The detail of the calculation is given in Appendix B, dependent on the orthogonality between combinations of Bessel functions, but the solution then involves replacing and from Eq. (13) by and , respectively, i.e.,
Thus the vacuum part of the solution, , corresponds to the terms multiplying and in Eq. (27), using Eq. (28), while the homogeneous part corresponds to the terms in the series over v. Explicitly, the potential parts, radial eigenfunctions and scalar expansion coefficients are respectively
which can be numerically evaluated by a Newton-Raphson method beginning with an approximation from an asymptotic expansion (Abromovitz & Stegun 1970, Eq. 9.5.27). Thus an explicit scalar series Eq. (27) has been found to compare with Eq. (13).
with a complementary expansion obtained upon interchanging a and b. The convergence of Eq. (31) is demonstrated numerically in Fig. 3.
3.2. Eigenfunction Field expansion using the new method of Laurence et al.
Notice that there is an auxiliary variable structure (see dash marks), that is, our calculation will show explicitly that cross-terms in the formula do vanish by orthogonality. In addition, it will be demonstrated that there is no contribution from , a result that extends to all simply-connected geometries (Clegg et al. 2000b).
By the principle of superposition, the problem Eq. (32), Eq. (33) can be decomposed into the sum of two solutions formed respectively from the boundary condition involving the sine terms and those from the cosine terms. To obtain the field eigenfunctions , we start from the scalar eigenfunctions specified by
so that each one of two possible eigenfunctions can be considered, represented by a zero coefficient E or F in Eq. (34) to allow for either a vanishing sine or cosine eigenfunction term in each half of the problem (we will show that crossed terms, for example , vanish in the coefficient integral and so the partition does not effect the result). The complementary radial eigenfunction term does not need to be considered because the two are linearly related through Eq. (B.14) and Eq. (B.17) as
where follows by interchanging a and b in Eq. (41).
Now the problem is to take Eq. (37a,etc.), Eq. (39a) and Eq. (40a,etc.) and using the standard orthogonality conditions given in Sneddon (1979) (their Eqs. (23.2),(23.4),(23.5)), find the effective coefficients Eq. (33). Details of the integration are given in Appendix D (so as not to further obscure the results), but the final result using the earlier notation, and with given by Eq. (40a,etc.) is
The form of the expansion Eq. (32) demands identical scalar [coefficients], as seen in Eq. (42a,etc.), i.e. they are isotropic when the limits of the sum are
If instead of the direct vector expansion approach Eq. (32) the problem is approached by transforming the scalar series Eq. (27) into a vector series through Eq. (2), then it becomes evident that, unlike in Eq. (42a,etc), the expansion coefficients differ between the component directions, i.e. they are tensors (Clegg et al. 2000b). This arises naturally because a term is generated that has to be recombined into the homogeneous part of the solution (which explains the appearance of tensor coefficients). In fact this tensor coefficient series is simply an alternative representation of Eq. (42a,etc.) noting that the former series is only summed over an infinite half-space . To see this, the series Eq. (42a,etc.) can be consolidated into a half-space sum by combining pairs of eigenvalues and , noting that and are all invariant under the mapping , but , and the quotient are not invariant. The result is that
that is the half-space series does have a tensor character to its coefficients since some of the terms differ by a factor , except at the (first) eigenvalue where this factor is 1. There is also a third way by which we have obtained this result, by exploiting the prior knowledge of the exact solution Eq. (18a,etc.), and finding by a manipulation of Eq. (3), i.e.,
3.3. Simplified boundary conditions: Jensen & Chu formulation
The consistency of the coefficient Jensen & Chu formula Eq. (5) to the previous results is now shown by looking at the simplified case (as for Sect. 2b). As a prelude to obtaining the Jensen and Chu coefficients a simplification of Eq. (27) can be written as
In this case Eq. (48a) already conforms to the requirement that on S, but often considerable effort is needed to find an appropriate gauge that fulfills this constraint.
so that explicitly with an orthogonality condition
The convergence of Eq. (53) to Eq. (23) is shown in Fig. 5
where the square bracketed coefficients of Eq. (54a,etc.) are immediately recognisable from the general field Laurence et al. solution Eq. (42a,etc.).
To summarise, the new Laurence et al. formula for the coefficients Eq. (6), for use in the field eigenfunction expansion Eq. (3), has been used to obtain a solution to the linear force-free field bounded by two spherical shells, Eq. (42a,etc.). The result has been proved correct since it converges to the independently derived, non-eigenfunction, field solution Eq. (18a,etc.), and is identical to a representation obtained by a field transformation Eq. (2) of the scalar series Eq. (27), although the limits of the resulting series have to be reconciled. It was then encumbent on us to relate these results to those using the usual Jensen and Chu coefficient formula Eq. (5). It is found that the Jensen and Chu approach provides identical results, at least for the case of axisymmetry tested in Eq. (54a,etc.), and provided the constituents of its formula are properly constrained . It has been our motivation to demonstrate a formula Eq. (6) that is free from such constraints.
© European Southern Observatory (ESO) 2000
Online publication: October 2, 2000