Stationary non-ideal cylindrical flux ropes: orthogonal decomposition, Maxwell locking and magnetic helicity

We ask how a stationary cylindrical magnetic flux rope can sustain a field-aligned electric field while satisfying Maxwell’s equations, and what this implies for its helical structure and magnetic helicity. Since

, the electromagnetic field is algebraically non-degenerate and separates locally into two mutually orthogonal rank-two contributions. For stationary, axisymmetric profiles on a finite annulus, the homogeneous Maxwell equations lock these contributions: the observable magnetic pitch, the azimuthal electromotive period and the axial electric drive determine the complementary-sector amplitude and radial–temporal mixing pointwise. On every regular Lorentzian branch, the principal variables can be reconstructed locally from the two observable magnetic components and the two drive constants. The resulting field carries a helical electric current, and requires material force support, a parallel non-ideal response and external power. A constant-conductivity example demonstrates a smooth regular branch supported by a non-negative isotropic pressure. Only after an additional periodic axial identification do the fixed-radius surfaces become tori and the principal magnetic directions acquire winding classes. On this periodic annulus, the generalised magnetic helicity is the oriented area traced by the accumulated axial and azimuthal fluxes. Its principal-sector contributions are flux-weighted integrals of radial pitch shear, while their mutual contribution reduces, in a formal localised-shell limit, to the torus intersection pairing. Because the magnetic field is stationary, the generalised helicity is time independent; the variation of the raw Chern–Simons integral is compensated by the evolving harmonic periods. This is a balance specific to the periodic annulus, not a general helicity-conservation theorem for fields with

bold italic upper E bold dot bold italic upper B not equals 0 E ⋅ B ≠ 0 $\boldsymbol E\boldsymbol{\cdot }\boldsymbol B\neq 0$

.