Normalized poloidal coordinate

The range of $\theta$ defined by Eq. (189) is usually not within $[0 : 2 \pi]$. Define a normalized poloidal coordinate $\overline{\theta}$ by

$$\overline{\theta}_{i, j} = \frac{\theta_{i, j}}{\oint \frac{R}{\vert\mathcal{J} \nabla \psi \vert} d l_p} 2 \pi$$

$$= \frac{2 \pi}{\oint \frac{R}{\vert\mathcal{J} \nabla \psi \vert} d... ...} \int_0^{\mathbf{x}_{i, j}} \frac{R}{\vert\mathcal{J} \nabla \psi \vert} d l_p$$ (191)

which is obviously within the range $[0 : 2 \pi]$. Sine we have modified the definition of the poloidal angle, the Jacobian of the new coordinates $(\psi, \overline{\theta}, \phi)$ is different from that of $(\psi , \theta , \phi )$. Next, we determine the Jacobian $\mathcal{J}_{\ensuremath{\operatorname{new}}}$ of the new coordinates $(\psi, \overline{\theta}, \phi)$. Equation (191) can be written as $\overline{\theta}_{i, j} = s (\psi_j) \theta_{i, j}$ with

$$s (\psi) = \frac{2 \pi}{\oint \frac{R}{\vert\mathcal{J} \nabla \psi \vert} d l_p} .$$ (192)

The Jacobian $\mathcal{J}_{\ensuremath{\operatorname{new}}}$ of the new coordinates system $(\psi, \overline{\theta}, \phi)$ is written

$$\mathcal{J}_{\ensuremath{\operatorname{new}}} \equiv \frac{1}{(\nabla \psi \times \nabla \overline{\theta}) \cdot \nabla \phi}$$

$$= \frac{1}{\{\nabla \psi \times \nabla [s (\psi) \theta]\} \cdot \nabla \phi}$$

$$= \frac{1}{s (\psi)} \frac{1}{(\nabla \psi \times \nabla \theta) \cdot \nabla \phi}$$

$$= \frac{1}{s (\psi)} \mathcal{J}'$$

$$= \pm \frac{1}{s (\psi)} \mathcal{J}$$

$$= \pm \frac{\oint \frac{R}{\vert\mathcal{J} \nabla \psi \vert} d l_p}{2 \pi} \mathcal{J}.$$ (193)

The Jacobian $\mathcal{J}$ can be set to various forms to achieve desired poloidal coordinates (as given in the next section). After the radial coordinate $\psi$ is chosen, all the quantities on the right-hand side of Eq. (191) are known and the integration can be performed to obtain the value of $\theta$ of each point on each flux surface.