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

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

Content Provider	Springer Nature Link
Author	González, A. Ojeda Prestes, A. Laurindo Sousa, A. Nilson
Copyright Year	2016
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.
Starting Page	408
Ending Page	414
Page Count	7
File Format	PDF
ISSN	01039733
Journal	Brazilian Journal of Physics
Volume Number	46
Issue Number	4
e-ISSN	16784448
Language	English
Publisher	Springer US
Publisher Date	2016-05-12
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Grad-Shafranov equation Coulomb gauge Magnetic field symmetry Space plasmas Kinetic theory Physics
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy

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