A geometrical and physical interpretation of quaternionic generalized magnetic flux tubes

Abstract

In the present paper, we give a generalization for the magnetic flux tubes. Firstly, we define the flux tube via quaternions. We obtain the magnetic flux canal surfaces and flux tubes by quaternion product of a unit quaternion and the unit normal vector of the magnetic field line. Besides, we generate these surfaces by the homothetic motion. Then, we determine the magnetic field components by using the quaternionic representation of the flux tube. Secondly, the flux tube is dictated by the magnetic vector field defined at each point on a generating curve. A local orthonormal basis is attached to every point of the generating curve then the magnetic field is given in terms of the local coordinate directions, leading to a geometric viewpoint of the flux tubes. These provide direct access to the magnetic field line and kinematics equations related to the flux tube. Moreover, the magnetic energy function and magnetic diffusion of the flux tube that has an arbitrary cross-section curve are computed. The impacts of the stretch factor of the flux tube on the magnetic trajectories are investigated. Finally, the analytical solutions of the kinematics equations and several examples of the theory are provided. (c) 2020 Elsevier Ltd. All rights reserved.