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

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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 [FORMULA] can also be found in this way. To see this we introduce the decomposition

[EQUATION]

The limits on this sum are from [FORMULA] to [FORMULA] in contrast to Eq. (3) which uses [FORMULA]. The inhomogeneous boundary condition is carried only through [FORMULA], satisfying

[EQUATION]

and the scalar eigenfunctions [FORMULA] satisfy

[EQUATION]

Here the effective eigenvalues are [FORMULA] (see Eq. (30)), rather than [FORMULA], 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 [FORMULA] 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 [FORMULA] and [FORMULA] from Eq. (13) by [FORMULA] and [FORMULA], respectively, i.e.,

[EQUATION]

and where the radial functions are structured as

[EQUATION]

Thus the vacuum part of the solution, [FORMULA], corresponds to the terms multiplying [FORMULA] and [FORMULA] 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

[EQUATION]

[EQUATION]

[EQUATION]

and upon interchanging a and b, the last three equations provide [FORMULA], [FORMULA] and [FORMULA] needed to determine [FORMULA]. The radial eigenvalues [FORMULA] correspond to the vth root of the nth order transcendental equation

[EQUATION]

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

[EQUATION]

with a complementary expansion obtained upon interchanging a and b. The convergence of Eq. (31) is demonstrated numerically in Fig. 3.

[FIGURE] Fig. 3a-f. The convergence of the special function expansion Eq. (31). For a , c  and e the exact value of the Bessel function combination, LHS of Eq. (31), is plotted across [FORMULA], at [FORMULA], for the modes [FORMULA] respectively; two further lines are added, associated with the RHS of Eq. (31), obtained by summing only the first two terms in the series to [FORMULA], or the first three terms to [FORMULA], but these are nearly coincident because of the rapid convergence of the eigenfunction expansion. The convergence is shown more clearly in b , d and f  by plotting the difference between the exact function (LHS) and the series (RHS) for different partial sums, the sums to [FORMULA] for d  and f , where convergence is rapid, and sums to [FORMULA] in b which again converges but is limited by numerical accuracy. The different forms of a , c and e arise from the proximity of [FORMULA] to the first eigenvalue of each particular mode, i.e. [FORMULA], [FORMULA] or [FORMULA], so that a has an amplified, near eigenvalue, character. Although the radial function is zero at [FORMULA], the solar magnetic field solution includes a complementary function (interchange a and b in Eq. (31)) which is only zero at [FORMULA]. Finally, the [FORMULA] mode vanishes if the entire solar surface is considered.

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

[EQUATION]

and the formula of Laurence et al. Eq. (6),

[EQUATION]

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 [FORMULA], 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 [FORMULA], we start from the scalar eigenfunctions specified by

[EQUATION]

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 [FORMULA], vanish in the coefficient integral and so the partition does not effect the result). The complementary radial eigenfunction term [FORMULA] does not need to be considered because the two are linearly related through Eq. (B.14) and Eq. (B.17) as

[EQUATION]

The field eigenfunctions are obtained from a specialisation of Eq. (2),(see Eq. (A.12)), that

[EQUATION]

which are explicitly

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is found from Eq. (29b), and where

[EQUATION]

[EQUATION]

The scalar function [FORMULA] is the potential part within Eq. (27), Eq. (28) and Eq. (29a), which generates a vector potential [FORMULA] and a potential field [FORMULA] through

[EQUATION]

[EQUATION]

We leave it to the reader to obtain the explicit expression for [FORMULA] and [FORMULA] but recall that any gauge [FORMULA] added to [FORMULA] with no effect on the Laurence et al. formula Eq. (33)). The explicit potential field is

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] are given by Eq. (29a) and

[EQUATION]

where [FORMULA] 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 [FORMULA] given by Eq. (40a,etc.) is

[EQUATION]

[EQUATION]

[EQUATION]

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 [FORMULA]

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 [FORMULA] 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 [FORMULA]. To see this, the series Eq. (42a,etc.) can be consolidated into a half-space sum by combining pairs of eigenvalues [FORMULA] and [FORMULA], noting that [FORMULA] and [FORMULA] are all invariant under the mapping [FORMULA], but [FORMULA], and the quotient [FORMULA] are not invariant. The result is that

[EQUATION]

[EQUATION]

[EQUATION]

that is the half-space series does have a tensor character to its coefficients since some of the terms differ by a factor [FORMULA], except at the (first) eigenvalue [FORMULA] 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 [FORMULA] Eq. (18a,etc.), and finding [FORMULA] by a manipulation of Eq. (3), i.e.,

[EQUATION]

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.

[FIGURE] Fig. 4. Field solution based on the eigenfunction expansion Eq. (42a,etc), Eq. (43a,etc) which can be seen to correspond closely to the benchmark field of Fig. 1, with the same values of [FORMULA], [FORMULA], [FORMULA]. The images here only include the first 7 modes [FORMULA] in the radial series.

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

[EQUATION]

where the superscript [FORMULA] is now implicit. This is related to the corresponding (axisymmetric) vector potential and field by

[EQUATION]

[EQUATION]

The scalar eigenfunctions radial function [FORMULA] within [FORMULA] is obtained from Eq. (29b) and Eq. (38b) (upon converting a to b) and converted to sinusoids as

[EQUATION]

Radial eigenvalues [FORMULA] are found from Eq. (30), with [FORMULA] and the zeroes correspond exactly to the poles of [FORMULA] obtained in Eq. (22). The vector potential eigenfunctions and field eigenfunctions are

[EQUATION]

[EQUATION]

In this case Eq. (48a) already conforms to the requirement that [FORMULA] on S, but often considerable effort is needed to find an appropriate gauge that fulfills this constraint.

Rewriting Eq. (5), the Jensen and Chu coefficients are

[EQUATION]

so that explicitly with an orthogonality condition

[FORMULA] where K is the normalisation constant and the coefficients are found to be

[EQUATION]

where the coefficients [FORMULA] are given by Eq. (29c) (upon converting a to b with [FORMULA]), written with sinusoids as

[EQUATION]

Thus the vector potential series is

[EQUATION]

or written as a half-space paired eigenvalue sum, with characteristic tensor coefficients,

[EQUATION]

The convergence of Eq. (53) to Eq. (23) is shown in Fig. 5

[FIGURE] Fig. 5a-d. Evidence that the (axisymmetric [FORMULA]) vector potential series Eq. (53) converges, up to a vanishingly thin boundary layer, to the benchmark vector potential Eq. (23) using [FORMULA] (but see the discussion in Clegg et al. 2000b). In a and b the components [FORMULA] and [FORMULA] are plotted against radius (open inner boundary at [FORMULA] to the closed outer boundary at [FORMULA]) for the exact and for the series solution including 50 radial modes. In a the two lines effectively coincide. In c and d a zoom view of [FORMULA] is shown [FORMULA], with better convergence using 100 and 500 modes, plotted against the exact solution.

The Jensen & Chu derived series field becomes

[EQUATION]

[EQUATION]

[EQUATION]

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.

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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