Equilibrium of magnetic fields with arbitrary interweaving of the lines of force i. discontinuities in the torsion

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Abstract
This paper considers the static force-free equilibrium V×B=αB of a magnetic field in which all of the lines of force connect without knotting between parallel planes. The field is formed by continuous deformation from an initial uniform field, and is conveniently described in terms of the scalar function ψ, which is effectively the stream function for the incompressible wrapping and interweaving of the lines of force, and the scalar function θ, which describes the local compression and expansion. Equilibrium requires satisfaction of two independent equations (the third equation defines α), which cannot be accomplished without the full freedom of both functions ψ and θ. It is shown by integration along the characteristics of the equilibrium equations that, when ψ is predetermined by an arbitrary winding pattern, there appear discontinuities in α. Discontinuities in α have discontinuities in the field (i.e. current sheets) associated with them. We expect such discontinuities to be produced in the magnetic fields extending outward from the convecting surfaces of the cooler stars.