Radial coordinate

The cylindrical coordinates $(R, \phi, Z)$ is a right-hand system, with the positive direction of $Z$ pointing vertically up. In GTAW code, the positive direction of $\theta$ is chosen in the anticlockwise direction when observers look along the direction of $\hat{\ensuremath{\boldsymbol{\phi}}}$. Then the definition $\mathcal{J}^{- 1} = \nabla \psi \times \nabla \theta \cdot \nabla \phi$ indicates that (1) $\mathcal{J}$ is negative if $\nabla \psi$ points from the magnetic axis to LCFS; (2) $\mathcal{J}$ is positive if $\nabla \psi$ points from the LCFS to the magnetic axis. This can be used to determine the sign of Jacobian after using the analytical formula to obtain the absolute value of Jacobian.

The radial coordinate $\psi$ can be chosen to be various surface function, e.g., volume, poloidal or toroidal magnetic flux within a magnetic surface.

The frequently used radial coordinates include $\overline {\Psi }$, and $\sqrt{\overline{\Psi}}$, where $\overline {\Psi }$ is defined by

$$\overline{\Psi} = \frac{\Psi - \Psi_0}{\Psi_a - \Psi_0},$$ (202)

where $\Psi_0$ and $\Psi_a$ are the values of $\Psi$ at the magnetic axis and LCFS, respectively. For the last two choices of the radial coordinate, $\nabla \psi$ points from the magnetic axis to LCFS. Other choices of the radial coordinates include the toroidal magnetic flux and its square root, $\overline{\Psi}_t$, and $\sqrt {\overline {\Psi }_t}$, where $\Psi_t$ and $\overline{\Psi}_t$ are defined by

$$\frac{d \Psi_t}{d \Psi} = 2 \pi q, \Psi_t (0) = 0$$ (203)

and

$$\overline{\Psi}_t = \frac{\Psi_t}{\Psi_t (1)},$$ (204)

respectively, where $\Psi_t (0)$ and $\Psi_t (1)$ are the values of $\Psi_t$ at the magnetic axis and LCFS, respectively.

If $\psi = \sqrt{\overline{\Psi}_t}$, then

$$\frac{d \Psi}{d \psi} = \frac{d \Psi}{d \Psi_t} \frac{d \Psi_t}{d... ...si_t}} \frac{d \overline{\Psi}_t}{d \psi} = \frac{1}{2 \pi q} \Psi_t (1) 2 \psi$$ (205)