Fundamentals of Plasma Physics 2025, 37 - Topological Interpretation of Magnetic Helicity
Two (numbered) thin linked and untwisted flux tubes with respective fluxes Φi, with tube axes follow respective contours, occupying respective tube volumes. The magnetic field is assumed to vanish outside the tubes. Define the volume helicity K = ∫VA Bd3r, which decomposes into a sum of integrals over the respective volumes. Magnetic flux through a surface S with perimeter C is Φ = ∫S B ds = ∮CA dl. In the case of a fat flux tube, this decomposes into adjacent thin flux tubes.
If the radius of flux tube 2 is shrunk until it encircles flux tube 1, and its cross-section is deformed until its field lines are uniformly distributed over the length of flux tube 1, V2 is like a thin coat on flux tube 1. Then, dK = 2Φdφ. For each layer, add dK = 2Φφ'dΦ, so then K = 2∫0ΦΦφ'dΦ. The flux through the cross section of flux tube 1 is then exactly Φ. The twist of the B-field is equal the winding number of a field line in θ-direction around the ϕ axis. By dϕ = |∇ϕ|dlϕ, dθ = |∇θ|dlθ, define the twist T(Φ) = ψ' = dθ/dϕ, so that K = 2∫ΦT(Φ)dΦ.
In MHD, magnetic flux is frozen into the plasma frame, so the B-field topology is constant. A small amount of resistivity allows violation of this frozen-in condition where velocity vanishes due to symmetry. There, the approximation of MHD Ohm's law fails, and instead defines it to E = ηJ. Reconnection destroys individual linkages between flux-tube, though each destroyed linkage is immediately replaced, conserving total system helicity. The process involves energy dissipation, due to reconnection's finite resistivity.