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The properties of two local reference frames based on the magnetic field and the current density are investigated for magnetized plasmas in toroidal geometry with symmetric angle. The magnetic field-based local frame of reference has been well-studied for example by Dewar and colleagues [Phys. Fluids 27, 1723 (1984)] An analogous frame based on the current density vector is possible because it is also divergence free and perpendicular to the gradient of the poloidal flux. The concept of straightness of a vector is introduced and used to elucidate the Boozer and Hamada coordinate systems. The relationship of these local frames to the more well-known Frenet frame of reference, which specifies a curve in terms of curvature and torsion, is given. As an example of the usefulness of the these formal relationships, we briefly review the ideal MHD theory and their use. We also present a new annihilation operator, useful for eliminating shorter time scales than the time scale of interest, for deriving the inner layer equations of Glasser, Greene, and Johnson [Phys. Fluids 18, 875 (1975)]. Compared to the original derivation that is based on the local frame of reference in terms of the magnetic field, the new annihilation operator that is based on the local frame of reference in terms of the current density simplifies the derivation.

The purpose of this research was to better understand tearing and reconnection in genuinely three-dimensional configurations. We have identified an equilibrium model that should contain the required features. Three papers have been written and a fourth is in preparation. They are listed in the bibliography.

Magnetic field and fluid vorticity share many features. First, as divergence-free vector fields they are conveniently visualized in terms of their field lines, curves that are everywhere tangent to the field. The lines indicate direction and their density indicates field strength. The question arises of the extent to which the evolution of the fields can be treated in terms of the evolution of their field lines. Newcomb (1958) derived the general conditions on the evolution of vector fields that permit the identification of field lines from one instant to the next. The equations of evolution of the vorticity field and the magnetic field fall within Newcomb's analysis. The dynamics of the flows differ between these two systems, so that geometrically similar phenomena happen in different ways in the two systems. In this paper the geometrical similarities are emphasized. Reconnection will be defined here as evolution in which it is not possible to preserve the global identification of some field lines. There is a close relation between reconnection and the topology of the vector field lines. Nontrivial topology occurs where the field has null points or there are field lines that are closed loops.

Bisection methods, which depend on the existence of a criterion for determining whether a root exists within a given volume, may be useful in the evaluation of equation roots since the volume in which the root is known to be located can be steadily decreased. This criterion is presently furnished via topological degree theory, strictly for the case of 3D volumes; usefulness is demonstrated in the location of roots, and the classification of roots as either X- or O-points is discussed.