Discussion About the Magnetic Field Dimensionality, Invariant Axis Condition, and Coulomb Gauge to Solve the Grad-Shafranov Equation

Abstract

We discuss the relationship between the Coulomb gauge, the existence of an invariant axis, and the dimensionality (2-D or 2$\frac {1}{2}$-D) of the magnetic field in a mathematical-physical formalism that leads us to the Grad-Shafranov (GS) equation. In the literature, we found that a 2-D magnetic structure is used as a prerequisite to derive the GS equation from the Vlasov equation. However, other consulted works are based on a 2$\frac {1}{2}$-D (two-and-a-half) magnetic structure as a prerequisite to derive the GS equation from the balance of forces between the pressure gradient and the magnetic force, respectively. We replaced the magnetic vector potential on Ampère’s equation and used the Coulomb gauge to obtain a system of three Poisson equations, one for each component. We also used the same procedure explained above, but without the Coulomb gauge. Comparing z-component in both equation systems, we concluded that there are two possible solutions. We suggest using a 2$\frac {1}{2}$-D magnetic field configuration instead of a 2-D, when working with kinetic theory or magnetostatic equilibrium to derive the GS equation. We clarified that there is no relationship between the Coulomb gauge and the magnetic field dimensionality. In this problem, the invariant axis condition is imposed, which means that $\vec {\nabla }\cdot \vec {A}$ is independent of z, i.e., $\vec {\nabla }\cdot \vec {A}$ could have any value in which an invariant axis is a sufficient condition to obtain the GS equation.