On the representation of inhomogeneous linear force-free fields

It is shown that there is a false assumption hidden in the description of a relaxed state with inhomogeneous boundary conditions as the vector sum of a potential field, satisfying the boundary conditions, and a sum of eigenfunctions of the associated eigenvalue problem expanded by certain coefficients. In particular, although the Jensen and Chu formula (1984) can provide the correct expansion coefficients, it contains an implicit paradox in its derivation according to a general vector theorem. The same paradox led Chu et al. (1999) to be concerned about a contradiction obtained by taking the curl of their magnetic field expansion which, if permitted, becomes inconsistent with a current normal to the surface. The assumption that the curl can be commuted across an infinite sum of terms is the mechanism leading to these, apparently paradoxical, conclusions. Two mechanisms for resolving this apparent paradox are possible, one of which will be described in some detail below and the other discussed further in a forthcoming, more theoretical paper (Laurence et al., 2000). The decomposition of the magnetic field above is valid with convergence in the mean squared sense, but a decomposition of the current needs to be reinterpreted in terms of negative Sobolev spaces. To avoid this, and remain in a more easily managable and familiar setting, we derive the expansion coefficients in a way that involves the commuting of the inverse curl (as opposed to the curl) and the series. The resulting series converges in a mean square sense. When this is done the calculation can conform to the general vector theorem and a new gauge-invariant expression for the coefficients is obtained. However the consequence of the non-commutability is nullified in the Jensen and Chu formula, in both simply and multiply connected domains, by the important extra requirement of a boundary condition on the vector potential eigenfunctions; this excludes magnetic field eigenfunctions that carry flux, but there remains a complete set for the expansion and all flux is carried by the potential field. The two formulas are then identical. On a different issue, it is shown that if the general expansion is taken over a half-space, by combining positive and negative eigenvalue terms, then the coefficients are anisotropic, that is they are tensors except when evaluated at the first eigenvalue. A specific example is presented to illustrate the situation and to validate the new method of deriving the coefficien