## 3. Eigenfunction expansion for the benchmark fields## 3.1. Series solution using a scalar parameterThe 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 The limits on this sum are from to in contrast to Eq. (3) which uses . The inhomogeneous boundary condition is carried only through , satisfying and the scalar eigenfunctions satisfy Here the effective eigenvalues are
(see Eq. (30)), rather than
, being independent of and where the radial functions are structured as 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 and upon interchanging 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). The equivalence of Eq. (28) and Eq. (14) requires that the special functions can be expanded with a complementary expansion obtained upon interchanging
## 3.2. Eigenfunction Field expansion using the new method of Laurence et al.A vector eigenfunction expansion is now found using the new coefficient formula Eq. (6). We write the sum Eq. (3) explicitly as and the formula of Laurence et al. Eq. (6), Notice that there is an auxiliary variable structure (see dash
marks), that is, our calculation will show 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 The field eigenfunctions are obtained from a specialisation of Eq. (2),(see Eq. (A.12)), that where is found from Eq. (29b), and where The scalar function is the potential part within Eq. (27), Eq. (28) and Eq. (29a), which generates a vector potential and a potential field through We leave it to the reader to obtain the explicit expression for and but recall that any gauge added to with no effect on the Laurence et al. formula Eq. (33)). The explicit potential field is where and are given by Eq. (29a) and where follows by interchanging
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 that is the half-space series does have a The field lines for the series solution Eq. (42a,etc.) under the transformation Eq. (43a,etc.) are shown in Fig. 4 to compare with Fig. 1 under the same particular boundary conditions.
## 3.3. Simplified boundary conditions: Jensen & Chu formulationThe 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 where the superscript is now implicit. This is related to the corresponding (axisymmetric) vector potential and field by The scalar eigenfunctions radial function
within
is obtained from Eq. (29b) and
Eq. (38b) (upon converting Radial eigenvalues are found from Eq. (30), with and the zeroes correspond exactly to the poles of obtained in Eq. (22). The vector potential eigenfunctions and field eigenfunctions are In this case Eq. (48a) already conforms to the requirement
that on Rewriting Eq. (5), the Jensen and Chu coefficients are so that explicitly with an orthogonality condition where where the coefficients are given
by Eq. (29c) (upon converting Thus the vector potential series is or written as a The convergence of Eq. (53) to Eq. (23) is shown in Fig. 5
The Jensen & Chu derived series field becomes 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
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