Plasma physists love to consider a kind of simplified problem: the fixed boundary equilibrium problem, where the shape of the boundary flux surface is given (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 is that the coordinates can be adjusted to make one of the coordinate surfaces coincide with the given boundary flux surface, so that the boundary condition becomes trivial.
More generally, curvilinear coordinates can be ajusted to make coordinate surfaces coincide with magnetic surfaces (dicussed later). This often simplifies analysis of wave and transport problems, especially when we properly choose the poloidal/toroidal coordinate to make magnetic field lines look like straight lines in terms of these coordinates.
Next section discusses the basic theory of curvilinear coordinates system[4].
In many studies of tokamak plasmas, one need construct a curvilinear coordinate system based on a given magnetic cofiguration in order to make the problem amenable to analytical methods or numerical methods. Specifically, 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. As metioned above, 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. 13.
Next, let us discuss some general properties about coordinates transformation.