## 1. IntroductionIn 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
An equilibrium field where the Lorentz force vanishes is associated with the well known elliptical problem Eq. (1), in a volume First, the equilibrium problem can be formulated in terms of a
scalar (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
where the eigenfunctions satisfy
with eigenvalues and where
on 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 which can be obtained rigorously. It was obtained by only allowing
an 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 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 © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |