Knotted linear force‐free magnetic fields‐topological aspects

The problem of computing linear force-free magnetic fields o n a knotted multiply-connected domain is considered. The domain is the support of the current distribution, and the li near force-free fieldproblem reduces to finding an eigenfield of a self-adjoint curl operator. In this context, the GKN Theorem is reformulated in terms of symplectic geometry in order to characterize the self-adjoint extensions of the curl opera tor restricted to solenoidal vector fields. When further res tricted to the isotopy invariant boundary conditions, the self-adjoi nt extensions are parametrized by the Lagrangian subspaces of the symplectic form on the first homology group of the boundary. T his paper discusses some of the topological aspects and gives some pointers for the associated finite element discretizat ion. Copyright line will be provided by the publisher 1 Self-Adjoint Extensions of the Curl Operator Linear Force-free magnetic fields, that is current distribu tions with vanishing Lorentz force and with the ratio of the magnitudes of the magnetic field to current density constant, can be cons tructed from the eigenfunctions of the curl operator[11], [ 3]. A key observation in the context of modeling is that, when vector fields are restricted to be solenoidal and we restrict ours elves to a compact manifold with boundary, the boundary conditions which render the curl and curlcurl operators self-adjoint , are mutually exclusive. Therefore, one cannot directly exploit the experience gained from electromagnetic cavity resonator calculations and apply it to the eigenvalue problem for the curl operator in order to obtain linear force-free magnetic fi elds. The earliest calculations of linear force-free magnetic fie lds by Chandrasekhar-Kendall [2], and later by Moses[12], considered unbounded domains and side-stepped this problem in doing so. Later authors considered compact simply connected regions,where the boundary conditions which render this curl operator self-adjoint can be local on the boundary. Other s considered unknotted multiply-connected regions [14], and made precise prescriptions for a set of boundary conditions which render the curl operator self-adjoint, but no general theor y was given. In more recent work [7], a more general study of self-adjoint curl operators is undertaken within a framework sufficientl y general to account for knotted domains associated with conjectured topological characterizations of force-free magn etic fields [4], [11]. It builds on the reformulation, by Evritt and Mark us [5], of the Glazman-Krein-Naimark (GKN) Theorem. In this reformulation, the boundary conditions which render the curl operator self-adjoint on a suitably chosen space of divergence-free vector fields are in 1-1 correspondence with the Lagrangian s ubspaces of the infinite dimensional symplectic form associ ated with the boundary term in the “integration by parts” formula. A key result of characterizing the boundary conditions which render the curl operator self-adjoint on a compact multiply-connected domain, is that the they are necessarily nonlocal. However the nonlocality can be articulated in terms of a symplectic form on the first homology group of the boundary. In this wa y, the nonlocal aspects can be handled systematically in finite element discretizations.