Before considering the free boundary equilibrium, it is instructive to consider the fixed boundary equilibrium problem, where the shape of the boundary flux surface is given (i.e., the value of Ψ is a constant on this boundary). In dealing with the fixed boundary problem, the curvilinear coordinate system is useful. Specifically, the convenience provided by a curvilinear coordinate system is that the coordinates can be properly chosen to make one of the coordinate surfaces coincide with the given boundary flux surface, so that the boundary condition becomes trivial.
Next section discusses the basic theory of curvilinear coordinates system[4]. Many theories and numerical codes use the curvilinear coordinate systems that are constructed with one coordinate surface coinciding with magnetic surfaces. In these coordinate systems, we need to choose a poloidal coordinate 𝜃 and a toroidal coordinate ζ. A particular choice for 𝜃 and ζ is one that makes the magnetic field lines be straight lines in (𝜃,ζ) plane. These kinds of coordinates are often called magnetic coordinates. That is, “magnetic coordinates are defined so they conform to the shape of the magnetic surfaces and trivialize the equations for the field lines.”
A further tuned magnetic coordinate system is the so-called field aligned (or filed-line following) coordinate system, in which changing one of the three coordinates with the other two fixed would correspond to following a magnetic field line. The field aligned coordinates are discussed in Sec. 11.