Coordinate Systems

Let $X$, $Y$, $Z$ be a conventional right-handed Cartesian coordinate system (where $Z$ measures vertical height).

Let $R$, $\varphi$, $Z$ be the corresponding cylindrical coordinate system, where $R\equiv (X^{2}+Y^{2})^{1/2}$ and $\varphi\equiv \tan^{-1}(Y/X)$. It follows that

$$\vert\nabla\varphi\vert = \frac{1}{R}.$$ (14.1)

Finally, let us define a flux coordinate system: $r(R,Z)$, $\theta(R,Z)$, $\varphi$, where [5,14]

$$(\nabla r\cdot\nabla\theta\times \nabla\varphi)^{-1} = {\cal J},$$ (14.2)

$${\cal J}(R,Z) =\frac{r\,R^{2}}{R_0}.$$ (14.3)

(Note that the $R$, $Z$, $\theta$, $\varphi$ are the same as the corresponding quantities defined in Section 2.10.) Here, $R_0$ is a convenient scale major radius, the magnetic flux-surface label $r$ has units of length, and $\theta$ is is an angular coordinate that increases by $2\pi$ radians for every poloidal circuit of the magnetic axis. Let $r=0$ correspond to the magnetic axis of the plasma, and let $r=r_{100}$ correspond to the last closed magnetic flux-surface. Suppose that $\theta=0$ on the outboard midplane of the plasma. It follows that $0<\theta<\pi$ below the midplane.