Edward W. Stredulinsky

Asymptotic Massey products, induced currents and Borromean torus links

We introduce a class of currents which allows a new and very explicit form for the Massey product of a third order link as a line integral. The explicit form permits the introduction of an asymptotic Massey product analogous to that introduced previously for Gauss’s integral by V. Arnold. The average third order asymptotic Massey product is shown to be equal to Berger’s third order helicity for divergence-free vector fields in linked tori.

Two-dimensional magnetohydrodynamic equilibria with prescribed topology

We establish existence of weak solutions to the equilibrium equations of magnetohydrodynamics with prescribed topology. This is carried out in two settings. In the first we consider the variational problem of minimizing total energy in a torus under the assumption of axisymmetry, and prescription of mass and flux profiles. Existence of weak solutions implies that the prescription of topology is a natural constraint. Both the compressible and incompressible cases are considered. In the second setting we adapt examples of B. C. Low and R. Wolfson [13] and J. J. Aly and T. Amari [1, 2, 3] associated with Parkers explanation of the extreme heating of the solar corona and other solar phenomena. Existence of solutions with fixed topology is a first crucial step in rigorously examining the relationship between topology and the existence of current sheets. We use a decomposition introduced in [8, 9, 11] that captures much of the topology of level sets for certain classes of Sobolev functions. This decomposition is preserved under weak limits and so is useful for prescribing topological constraints. The approach is especially suited to the use of domain deformations.

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 coefficients.

A lower bound for the energy of magnetic fields supported in linked tori

Abstract Given a divergence free field B in R 3 , the problem arises in fluids and magnetofluids, of finding lower bounds on the E 3/2 = ∫ | B | 3/2 d V norm of the field which depend only on the topology of the trajectories of the field and fluxes. In this Note, using the notion of third order helicity, we present such a lower bound in the case when it is known that a higher order linking is present, i.e., that the magnetic field has support in (or larger than) three solid tori linked like the Borromean rings, and is tangent to their boundaries.

Optimal regularity in a variational problem for current sheets in ideal magnetohydrodynamics

We characterize local behavior, and establish optimal local regularity, for minimizers of the functional E(ψ)=∫Ω|∇ψ|2 over collections C that are weakly closed in H1(Ω), closed under local smooth domain perturbations, and for which E(ψ) controls ∫Ωψ2. Minimizers ψ satisfy a weak magnetohydrodynamic (MHD) equation and correspond to fields in low density ideal plasmas under cylindrical symmetry where the field component in the direction of the axis of symmetry is zero. We prove that (∂ψ/∂x+i∂ψ/∂y)2 is complex analytic, and locally ψ=f(φ) for some φ, f, with Δφ=0 and f Lipschitz continuous with |f′|=1 almost everywhere, near points where ∇ψ≠0. An analogous but more elaborate characterization is established at points where ∇ψ=0. This characterization forms the basis for a general theory of the existence of current sheets due to imposed topological and boundary constraints. Results carry over to functions that are stationary points of E(ψ) with respect to local smooth domain variations.

AXISYMMETRIC MHD EQUILIBRIA FROM KRUSKAL-KULSRUD TO GRAD

We provide a rigorous proof of the equivalence of the Kruskal-Kulsrud and Grad variational problems and show that minimizers are weak solutions of the associated Euler-Lagrange equations.

Topological rearrangement of a function

Abstract We extend our results on weak diffeomorphism classes and decompositions of Sobolev functions to a more general framework. We introduce a family of decompositions of Sobolev functions W 0 1, p rich enough that we conjecture it allows decomposition of all Sobolev functions, not just the “craterless” ones considered in [7]. The associated weak diffeomorphism classes of a W 0 1, p Sobolev function are weakly closed when p ≥ n .