In the strongly magnetised plasma of technology and astrophysical systems in equilibrium, it is often a good approximation to assume a linear force-free magnetic field, that is, the state associated with a magnetic relaxation process (Woltjer 1958; Laurence & Avellaneda 1991). For a slightly dissipative plasma, the notion provides a means to avoid consideration of the detailed plasma dynamics and instead invokes a principle, that of minimum energy with conserved global magnetic helicity (Taylor 1974, 1986). In particular, in technology it is thought that a relaxation process can explain the equilibrium configuration found in several plasma magnetic confinement devices, such as the Spheromak (Dixon et al. 1990), Reversed Field Pinch (Taylor 1974; Ortolani & Schnack 1993), and certain other toroidal configurations (Turner 1984; Taylor & Turner 1989; Browning et al. 1993). It is also common to apply the linear force-free model to a diverse range of problems in solar physics and astrophysics (Nakagawa 1973; Chui & Hilton 1973; Barbosa 1978; Rust & Kumar 1994; Browning 1988; Vekstein et al. 1993; Aly 1993; Kusano et al. 1995; Amari et al. 1997). This paper considers a solar (or stellar) corona modelled between spherical boundaries. In fact the linear force-free field model can only provide a weak abstraction of the complex coronal magnetic field, but the solution itself is of theoretical interest and is used to understand better the different solution forms that arise from the methodology. We are concerned here with some more mathematical aspects of the problem, but we do illustrate the theory with a concrete example of the solar coronal magnetic field.
Although the coronal plasma evolves through a sequence of highly dynamical and non-linear interactions, such as magnetic field reconnections, the relaxation hypothesis is that the final equilibrium state is much simpler, that is, the local plasma current becomes directed along the global magnetic field, except possibly at current sheets, so that the Lorentz force vanishes. In this magnetostatic configuration pressure gradients are ignored because of the low coronal plasma pressure (), and the scale height renders the gravity potential insignificant. In addition no sophistication is added to account for steady flows that could support a stationary equilibrium (Tsinganos 1982). However advections continuously change the boundary conditions at the photosphere and so it is appropriate to interpret any equilibrium obtained as an instantaneous state that is destined to evolve quasi-statically. This is largely true for photospheric motions that are slower than the characteristic Alfven and magnetoacoustic times (Heyvaerts & Priest 1984), but for very fast photospheric motions the equilibrium cannot adjust fast enough and so instead waves are launched. The situation where the photospheric forcing time is similar to the Alfven time is worthy of study but this has not yet been properly investigated.
In considering a solar coronal equilibrium, it is somewhat problematic to define a suitable plasma volume, although even a choice with free boundaries is possible within the theory of relaxed states (Browning 1988). Instead, in the global context, it is usual to establish a fixed inner boundary at the photosphere or lower chromosphere, together with an artificial outer (spherical) boundary. An outer boundary of some kind is required since the force-free approximation (and other modelling assumptions) inevitably break down at some distance from the solar surface. Furthermore, force-free fields in an infinite region in general have unphysical properties such as infinite magnetic energy (Aly 1992, 1993). Thus the coronal plasma may be regarded as residing between two spherical shells. It is then a prerequisite to measure or model the flux distribution over those surfaces. In general, comprehensive line-of-sight magnetograms are available for the solar surface (for example the photospheric MDI facility on SOHO, or lower chromospheric magnetograms from Kitt Peak), the problem then being (usually) to transform from the line-of-sight to obtain the normal field component, and to overcome the constraint of access only to the visible disc. Appropriate techniques are described in Clegg et al.(1999a) to extrapolate the boundary condition over the entire solar surface. A more realistic coronal force-free magnetic field model accounts for the variation of current between flux surfaces (see for example Clegg et al. 2000a), representative of the very complex transverse magnetic field gradients at the boundary as recorded on vector magnetograms (Mickey et al. 1996; Sakurai et al. 1995). However, at present vector measurements are only available over strong magnetic field regions where Zeeman triplets can be resolved. Hence, the simplification of a uniform (current/flux) distribution is still widely studied since it is amenable to analysis, and like the even simpler current-free (potential field) approximation it provides a basis to assess the sensitivity of the field to the assumptions used and to compare with solar observations.
For the potential model it is usual to assume a "source surface" outer boundary condition (Schatten et al. 1969), where the field is wholly radial at some radius , but in fact this condition is incompatible with a force-free field in general. Rather it is better to assume that the radial field component is uniform in magnitude at some distance from the solar surface, as discovered by the Ulysses mission (Forsyth et al. 1997), and so an estimate of the field strength there is required. However only sparse data is available here and so our approach is to evaluate the field for a family of radial field strengths and then to select the solution that best reproduces other observations. In particular the LASCO coronographs tend to silhouette the field because of the high plasma conductivity. Such evidence also reveals the large radial extent of plumes and streamers and in choosing a spherical outer boundary it could be argued that it is better to place the outer boundary beyond where the bounding surface is more isotropic with respect to the plasma ( is different between open and closed field regions). However by definition the force-free model takes no account of pressure gradients and gravity potentials and so here the compromise choice of is used, notwithstanding these shortcomings.
in a volume V, solved with appropriate boundary conditions for on S (the two spherical shell boundaries). The scalar is taken to be uniform in this paper which makes the problem linear. This may be justified partly by relaxation theory (see above). There are several strategies to solve Eq. (1), but two approaches are central to this paper:
(Chandrasekhar 1956; Moffatt 1978) where is a vector potential and is a unit radial vector in spherical co-ordinates. The equilibrium problem reduces to a Grad-Shafranov partial differential equation (see Eq. (9)), and a solution can be found in terms of special functions which is used as a benchmark to test the other methods.
Second, a decomposition is often applied where the boundary conditions are accounted for by a potential field to allow for the flux across S, superposed with an homogeneous part, i.e. an expansion of eigenfunctions (where the mode numbers refer to the spherical co-ordinates ) writing
where the eigenfunctions satisfy with eigenvalues and where on S. Such an expansion has been shown to converge by the work of Yoshida and Giga (1990). The problem is to find the coefficients , but there are now two formulae that purport to determine their values.
In our companion paper Clegg et al.(2000b) it was shown that during the derivation of the coefficients a contentious equation can arise. A similar situation is described in Chu et al.(1999) where, upon taking the curl of Eq. (3) and using the definitions of the potential and eigenfunction fields, there is apparently a relationship for the current that
but this is "paradoxical" in the sense that a non-vanishing normal component of current can seemingly be constructed out of a series of terms in Eq. (4) that themselves vanish normal to the boundary. This might be possible in a negative Sobolev space but more usually such an equation cannot converge (given are a set of divergence-free vectors) as proved by a theorem described in Clegg et al.(2000b). The problem has arisen because it has been assumed that the curl differential operator can be commuted across an infinite sum of terms to obtain Eq. (4), but it is known that the derivative of a sum of terms does not necessarily equate to the sum of their derivatives. Our interest is not so much with the particular dilemma posed by Eq. (4) but rather because an often used formula for the coefficients in Eq. (3), i.e. Eq. (5) below, can be obtained through a calculation that includes a similar assumption of commutability, creating a similar erroneous equation as Eq. (4), using the usual orthogonality relations (Clegg et al. 2000b). Hence either the formula Eq. (5) is incorrect or there is an alternative means to justify its usage.
The conventional approach then is to find the coefficients for Eq. (3) by a formula developed by Jensen & Chu (1984), i.e.,
and the formula requires that a particular boundary condition is imposed on the vector potential eigenfunctions that on S, since this choice affects the numerator of Eq. (5). This extra requirement, and the apparent "paradox" described above has led one of us (PL) to develop a new formula for the coefficients (Clegg et al. 2000b; Laurence et al. in preparation) written as
which can be obtained rigorously. It was obtained by only allowing an inverse curl operation to be commuted across the infinite series, and this diminishes each term in size to ensure convergence, before applying orthogonality conditions to find as Eq. (6). The expression is also more convenient than Eq. (5) since any choice can be made for the vector potential constituents within Eq. (6), i.e. there is no implicit gauge constraint. In addition, the term vanishes for simple geometries.
It turns out that the two formulae Eq. (5) and Eq. (6) are fully compliant because Eq. (5) can also be derived as a special case of the rigorous approach Eq. (6) (Clegg et al. 2000b). The formulae should then be identical in both simply and multiply connected situations but for Eq. (5) the vector potential eigenfunctions must always be constrained by on S. This means that in a torus with net flux only fluxless eigenfunctions are included in the expansion, but these do form a complete set, so that it is the potential field in Eq. (3) that must account for the flux and boundary conditions. It is the purpose of this paper to apply each coefficient formula Eq. (5) and Eq. (6) to an expansion Eq. (3), to show explicitly that they are identical (when Eq. (5) is properly constrained), and to validate the eigenfunction solution through an independent, non-eigenfunction approach to the problem Eq. (9). The particular situation considered is a force-free field between two spherical shells (see also Cantarella et al. 2000). The solution technique presented can be used for practical extrapolation problems, and an example is presented using SOHO-MDI data to provide boundary data for the solar coronal field.
In Sect. 2 of the paper the benchmark solution is derived from the Grad-Shafranov approach to yield a linear force-free field in a spherical shell with both general and simplified boundary conditions. The simplification is included because the solution P can be expanded into a direct scalar series (Sect. 3.1) and thence Eq. (2) used to obtain a series for . However this is to assume that the differential operator in Eq. (2) can be commuted across an infinite sum of terms, and so for the avoidance of doubt about convergence here, the simplified case is included having only a finite number of boundary terms. In retrospect, the instances of commutability encountered in this paper are allowed but the simplified form is helpful because of the removal of special functions. For the remainder of Sect. 3, the series eigenfunction solutions are found using the expansion coefficients derived from both Eq. (5) and Eq. (6). The new formula Eq. (6) is shown to be valid, at least for this example, converging to the independently obtained benchmark solutions. The validity of the formulation Eq. (6) has been demonstrated elsewhere for a cylinder (Clegg et al. 2000b). Thus the kernel of this work is a detailed calculation to clarify the inter-relationships of the scalar and direct vector approaches and the consistent solution is then applied to some particular solar magnetic fields.
© European Southern Observatory (ESO) 2000
Online publication: October 2, 2000