The toroidal field surfaces in the standard poloidal-toroidal representation of magnetic field

TL;DR

This paper proves that in three-dimensional Euclidean space, the only toroidal field surfaces compatible with the standard poloidal-toroidal magnetic field representation are spheres and planes, clarifying the geometric constraints of such models.

Contribution

The paper demonstrates that only spheres and planes can serve as toroidal field surfaces in the standard PT representation in Euclidean space.

Findings

Only spheres and planes allow standard PT representation.

The curl of a poloidal field remains toroidal only on these surfaces.

Other surfaces do not support the standard PT property.

Abstract

The representation of magnetic field as a sum of a toroidal field and a poloidal field has not rarely been used in astrophysics, particularly in relation to stellar and planetary magnetism. In this representation, each toroidal field line lies entirely in a surface, which is named a toroidal field surface. The poloidal field is represented by the curl of another toroidal field and it threads a stack of toroidal field surfaces. If the toroidal field surfaces are either spheres or planes, the poloidal-toroidal (PT) representation is known to have a special property that the curl of a poloidal field is again a toroidal field . We name a PT representation with this property a standard PT representation while one without the property is called a generalized PT representation. In this paper, we have addressed the question whether there are other toroidal field surfaces allowing a standard PT representation than spheres and planes. We have proved that in a three dimensional Euclidean space, there can be no standard toroidal field surfaces other than spheres and planes, which render the curl of a poloidal field to be a toroidal field.