Topological invariants of magnetic fields, and the effect of reconnections

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Abstract
Properties of the second-order topological invariant (the helicity) and the third-order topological invariant for ‘‘the Borromean rings’’ (three linked rings no two of which link each other) are discussed. A fourth-order topological invariant of ideal magnetohydrodynamics is constructed in an integral form. This invariant is determined by the properties of Seifert surfaces bounded by two coupled flux tubes. In particular, for the Whitehead link, it represents the fourth-order Sato–Levine invariant. The effect of reconnections on the topological invariants in the limit of small diffusivity is considered. In this limit the helicity is approximately conserved and the higher-order invariants decay rapidly under the action of diffusivity. The destruction of the higher-order invariants, however, creates helicity fluctuations.