Fundamentals of Plasma Physics 2025, 38 - Kinking & Magnetic Helicty

A volume V bounded by a surface S where no B-field penetrates S (B ds = 0 over S), and V extending to infinity, the helicity in V is written K_V = ∫_V A Bd³r. The volume V is decomposed into subvolumes with B ds = 0, at all locations on their mutual interface. Any specific field line in V is entirely in one or the other of the subvolumes. Divide into V_tube and V_ext. In the flux-tube, B = B_axis + B_azimuthal. The components can be separated in computation. Cut the tube surface into ribbons and later integrate over the ribbons, to define a poloidal flux ψ(Φ) as the magnetic flux penetrating a subribbon extending inward from the outer surface of the possibly helical flux tube to some given interior magnetic surface Φ. ψ = ∫_subribbonds B = ∮_Cdl A. K_Vtube = K_writhe + K_twist = ∫_Vtube d³r A_axis B_axis + ∫_Vtube d³r(ψ(Φ)/2π ∇ϕ B_axis + A_axis 1/2π ∇ψ×∇ϕ).

Ktwist is a volume integral, so the flux tube may be cut at &vaprhi; = 0 without changing its value. It restricts the range of ϕ between 0 and 2π. The evaluation of the integral is insensitive to the connectivity of the axis. Write B_axis = 1/(2π) ∇×Φ∇θ, Φ = ∫ ds B_axis = Φ/(2π) ∮_Cθ dθ. A_axis = Π/2π ∇θ.

K_writhe uses d³r = dl _˙ ds, so that K_writhe = ∫_Caxis A_{axis ˙} dl_axis∫ dΦ = Φ∫_Caxis A_{axis ˙} dl_axis. In a case where the flux tube axis is not helical occurs at b = 0, which puts the flux tube axis into a plane. C_axis can be slipped through the B_axis field lines to the surface of the flux tube. It follows, K_writhe = Φ∫_C'A_{axis ˙} dl_axis = Φψ_ext. When the axis is helical, and b > a, then the flux tube revolves around the axis of the helix, and as the helical axis closes upon itself, an integral number of periods is necessary. B_axis is parallel to the flux tube axis, so it can be slipped through the B-field lines again for the helix axis. If a > b instead, the flux can be subdivided into an inner core with minor radius r < b, and an outer annular region b < r < a with the remainder of Φ.

A kink instability is governed by ideal MHD, and conserves helicity. The number of turns of the kink is defined by the initial twist condition k _˙ B = 0. The kink will result in a helix with N = n for k_θ = 1/a, and k_ϕ = n/R. For increasing kink amplitude, the flux tube twist will have to decrease.