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

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2. Bench Mark Solutions

We aim here to solve for the linear force-free field in a spherical shell using the scalar potential approach. This both provides a useful solution in its own right, and also provides a benchmark to test the vector eigenfunction solutions.

2.1. General field in a spherical shell

In spherical co-ordinates, Eq. (2) is explicitly

[EQUATION]

with operator

[EQUATION]

The equilibrium problem Eq. (1), with Eq. (7), reduces to solving a Grad-Shafranov equation

[EQUATION]

and subject to appropriate boundary contraints on P. The base functions obtained by a separation of variables in spherical co-ordinates, are given in Appendix A, but they produce a general solution written

[EQUATION]

where [FORMULA], and [FORMULA] and [FORMULA] are surface spherical harmonics (see for example Sneddon (1979), their Eqs. (23.1) and (23.3))

The particular problem of a field between inner and outer spherical shell boundaries at [FORMULA] and [FORMULA] with given radial field on the boundaries (which is directly proportional to P through Eq. (A.12)) yields conditions [FORMULA] and [FORMULA] which depend only on [FORMULA] and so can be expanded as

[EQUATION]

and using standard orthogonality relations (see for example Sneddon (1979), their Eqs. (23.2),(23.4),(23.5)) the coefficients are found to be

[EQUATION]

where [FORMULA] with [FORMULA] for [FORMULA], otherwise [FORMULA], and [FORMULA] is found from Eq. (12) by replacing a with b.

It is then simply a matter of comparing Eq. (10) at [FORMULA] with Eq. (11) to find an expression for the coefficients [FORMULA] in terms of [FORMULA] and the resulting combinations of the Bessel functions at the boundaries. Following some manipulations, the final solution for P (see also Clegg et al. 1999b) can be written

[EQUATION]

where

[EQUATION]

and [FORMULA] is obtained by interchanging b and a in Eq. (14). Here we have used [FORMULA] defined as a radial function of three arguments and we also introduce a similar function [FORMULA] for use later as

[EQUATION]

[EQUATION]

and abbreviate various combinations of sinusoidal functions like

[EQUATION]

Now the field can be obtained directly by transforming Eq. (13) through Eq. (2), although this includes an assumption on commutability, to give

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

and [FORMULA] is obtained from Eq. (19a) by replacing [FORMULA] with [FORMULA], while [FORMULA] is obtained from Eq. (19b) by interchanging a with b

[EQUATION]

[EQUATION]

[EQUATION]

Field lines obtained from the solution Eqs. (18a,18b,18c) are shown in Fig. 1 for a particular solar magnetic field. In this example, boundary conditions were taken from SOHO-MDI photospheric magnetograms centred on 27 August 1996 when an extended low-latitude coronal hole was present. A uniform radial field strength has been imposed across the boundary at [FORMULA], selected from trial values to ensure that the solution closely resembles SOHO-LASCO observations (as discussed in the introduction). Different magnetic helicities are shown corresponding to different values of [FORMULA] as detailed in the caption. Note that the coronal hole lies alongside the western (right-hand) edge of the N-S aligned arcade which can be seen near the centre of the disc.

[FIGURE] Fig. 1. A sample field solution from Eqs. (18a,18b,18c) modelling the solar corona with boundary conditions from the SOHO-MDI magnetograms for 27 August 1996, a period when a large low-latitude coronal hole developed, extending southward from the north polar hole, to an active region in the southern hemisphere (see Clegg et al. 1999a, 1999b). The solution is shown for [FORMULA], 0, [FORMULA] (where [FORMULA] is related to the helicity content of the field and [FORMULA] is the first eigenvalue discussed in Sect. 3).

2.2. Simplified boundary conditions

The solution Eq. (13) does, with a broadband boundary condition spectrum, involve an infinite series in [FORMULA] and so it is not immediately obvious that, in using Eq. (2), there can be a commutability of the curl across this sum to produce the field Eqs. (18a,18b,18c). However, arbitrary and very simple boundary conditions could be used instead so that the sum can become finite, or even based on a single mode. Such simplifications can properly be described as "exact" and this special case benchmark solution provides an incontravertable test for the eigenfunction expansion scalar and vector field solutions. A suitable set of simplifications is as follows:

  • (i) take the outer shell boundary, [FORMULA], (for now) to be perfectly conducting so that all related coefficients [FORMULA] vanish.

  • (ii) take only axisymmetric fields [FORMULA], [FORMULA] and so coefficients [FORMULA] also vanish,

  • (iii) admit only an [FORMULA] mode inner boundary condition, [FORMULA], specified by [FORMULA].

Now [FORMULA] (see the r-component of Eq. (A.12) in the Appendix), whereas Eq. (13) reduces to [FORMULA] so that [FORMULA], and so the simplified version of Eq. (13) and Eq. (14), written in terms of sinusoids, with [FORMULA] and [FORMULA], is

[EQUATION]

Notice that there is a pole when the denominator vanishes, that is, for values of [FORMULA] where

[EQUATION]

and for this [FORMULA] model with [FORMULA] and [FORMULA], then [FORMULA] and [FORMULA].

The [FORMULA], [FORMULA] vector potential, found from

[FORMULA], is explicitly

[EQUATION]

and similarly the simplified model field is obtained from [FORMULA] which can then be shown directly to satisfy the requirements of Eq. (1) without the need to assume a commutability. A marginally less simplified case can be used to model an idealised corona and is obtained by first inferring the solution with only an inhomogeneous outer boundary condition, by exchanging a and b in Eq. (23), and combining this new equation with some factor k times the original (inhomogeneous inner boundary) equation Eq. (23). Some examples of such a magnetic field are shown in Fig. 2.

The interesting question of how the field structure changes as [FORMULA] crosses the first eigenvalue, and the topology of fields at higher eigenvalues, is discussed at length in Cantarella et al. (2000) (see their Sect. VII and Fig. 4 & Fig. 5) and also in Dixon et al. (1990). Such fields have "islands" of magnetic field detached from the boundary surfaces (such as the photosphere). At an eigenvalue, the field is fully detached but of infinite magnitude. In practice a minimum energy state must always have [FORMULA] below the first eigenvalue because of the infinite magnetic energy barrier there, and it is anyway true that only relatively small (and non-uniform) values of helicity have been measured in the solar corona. It is then better to model the solar corona by assuming a "partially relaxed" magnetic field whereby if enough helicity could be injected it would be a weighted average of [FORMULA] that approaches the first eigenvalue (Kitson & Browning 1990). A final equilibrium (if one exists) then depends on both the initial and boundary conditions and so the model of Fig. 2 is too simplified in practice although it does elucidate the magnetic form. The model does however provide some insight into the paradox, as presented by say Eq. (4), since the curl of this field solution can be compared directly with the partial sums of the right-hand side of Eq. (4) (the coefficients are obtained in Sect. 3.3). It turns out that Eq. (4) is indeed an erroneous expansion but nevertheless the coefficients obtained through Eq. (5), an equation also closely related to the paradox, do provide a correct expansion of the field in Eq. (3) (as discussed in Clegg et al. 2000b and Laurence et al., in preparation).

[FIGURE] Fig. 2a-d. "Generalised" axisymmetric benchmark field formed by the superposition of two solutions: the curl of Eq. (23) corresponding to an open inner boundary, and the curl of Eq. (23) with a and b exchanged to describe an open outer boundary solution, for a  [FORMULA], b  [FORMULA], c  [FORMULA], d  [FORMULA] relative to the first eigenvalue [FORMULA]

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Online publication: October 2, 2000
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