Vortex filaments in MHD

Two theorems on the Riemannian geometrical constraints on vortex magnetic filaments acting as dynamos in (MHD) flows are presented. The use of Gauss–Mainard–Codazzi equations allows us to investigate in detail the influence of curvature and torsion of vortex filaments in the MHD dynamos. This application follows closely previous applications to Heisenberg spin equation to the investigations in magnetohydrostatics given by Schief (2003 Plasma Phys. J. 10 2677). The Lorentz forces on vortex filaments are computed and the ratios between the forces along different directions are obtained in terms of the ratio between the corresponding magnetic fields which also equals the ratio between the Frenet torsion and vortex line curvature. A similar relation between Lorentz forces, magnetic fields and twist, which is proportional to total torsion integral, has been obtained by Ricca (2005 Fluid Dyn. Res. 36 319) in the case of inflexional disequilibrium of magnetic flux tubes. This is due to the fact that the magnetic vortex lines are a limiting case of the magnetic flux tubes when the length of the tube is much greater than the radius of the tube. The magnetic helicity equation of the filament allows us again to determine the magnetic fields ratio from Frenet curvature and torsion of the vortex lines. Recently, Schekochihin et al (2001 Phys. Rev. E 65 016305) obtained a similar relation between the ratios of magnetic field components by using a detailed analysis of the statistics of curvature. However, in their work no reference is made to torsion or helical vortex filaments.