Jacobian for equal-volume poloidal angle

The volume element in $(\psi , \theta , \phi )$ coordinates is given by $d V = \vert\mathcal{J}\vert d \theta d \phi d \psi$. If we choose a Jacobian that is independent of $\theta$, then uniform $\theta$ grids will correspond to grids with uniform volume interval. In this case, $\mathcal{J}$ is written as

$$\mathcal{J}= h (\psi),$$ (197)

where $h (\psi)$ is a function independent of $\theta$. Then $\overline{\theta}_{i, j}$ in Eq. (191) is written

$$\overline{\theta}_{i, j} = \frac{2 \pi}{\oint \frac{R}{\vert \nab... ...vert} d l_p} \int_0^{\mathbf{x}_{i, j}} \frac{R}{\vert \nabla \psi \vert} d l_p$$ (198)

and $\mathcal{J}_{\ensuremath{\operatorname{new}}}$ given by Eq. (193) now takes the following form

$$\mathcal{J}_{\ensuremath{\operatorname{new}}} = \pm \frac{1}{2 \pi} \oint \frac{R}{\vert \nabla \psi \vert} d l_p .$$ (199)

Note that both $\overline{\theta}_{i, j}$ and $\mathcal{J}_{\ensuremath{\operatorname{new}}}$ are independent of the function $h (\psi)$ introduced in Eq. (197). ($h (\psi)$ is eliminated by the normalization procedure specified in Sec. 5.3.1 due to the fact that $h (\psi)$ is constant on a magnetic surface.)