Topology and existence of three-dimensional kinematic dynamos

Abstract

Curvature and helicity topological bounds for the magnetic energy of the streamlines magnetic structures of a kinematic dynamo flow are computed. The existence of the filament dynamos are determined by solving the magnetohydrodynamic equations for 3D flows and the solution is used to determine these bounds. It is shown that in the limit of zero resistivity filamentary dynamos always exists in the isotropic case, however when one takes into account that the Frenet frame does not depend only of the filament length parameter s, (anisotropic case) the existence of the filamentary dynamo structure depends on the curvature in the case of screwed dynamos. Frenet curvature is associated with forld and torsion to twist which allows us to have a sretch, twist, and fold method to build fast filament dynamos. Arnold theorem for the helicity bounds of energy of a divergence-free vector field is satisfied for these streamlines and the constant which depends on the size of the compact domain $M C R^{2}$, where the vector field is defined is determined in terms of the dimensions of the constant cross-section filament. It is shown that when the Arnold theorem is violated by the filament amplification of the magnetic field structure appears, the magnetic field decays in space.