On the representation of inhomogeneous linear
force-free fields
J. R. Clegg
Centre for Astrophysics, University of Central Lancashire, Preston, PR12HE
United Kingdom
P. K. Browning
Department of Physics, University of Manchester Institute of Technology, Manchester,
M601QD United Kingdom
P. Laurence
Courant Institute, 251 Mercer Street, New York, New York, 10012 and Universita
`
di
Roma, ‘‘La Sapienza,’’ Piazzale Aldo Moro 2, 00185 Rome, Italy
B. J. I. Bromage
Centre for Astrophysics, University of Central Lancashire, Preston, PR12HE United
Kingdom
E. Stredulinsky
University of Wisconsin Richland, Richland Center, Richland, Wisconsin 53581
͑Received 27 March 2000; accepted for publication 5 May 2000͒
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 asso-
ciated eigenvalue problem expanded by certain coefficients. In particular, although
the Jensen and Chu formula ͑1984͒ can provide the correct expansion coefficients,
it contains an implicit paradox in its derivation according to a general vector theo-
rem. The same paradox led Chu et al. ͑1999͒ 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 ap-
parent 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., 2000͒. The decomposition of the magnetic field above is valid with conver-
gence 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 curl͒ 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 eigen-
functions 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 coefficients. © 2000 American Institute of Physics.
͓S0022-2488͑00͒03909-8͔
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 10 OCTOBER 2000
67830022-2488/2000/41(10)/6783/25/$17.00 © 2000 American Institute of Physics