Definition of the general toroidal angle $\zeta$

In Sec. 5.1, we introduced the local safety factor $\hat{q}$. Equation (179) indicates that if the Jacobian is chosen to be of the form $\mathcal{J}= \alpha (\psi) R^2$, where $\alpha (\psi)$ is an arbitrary function of $\psi$, then the local safety factor is independent of $\theta$ and $\phi$, i.e., magnetic line is straight in $(\theta, \phi)$ plane. On the other hand, if we want to make field line straight in $(\theta, \phi)$ plane, the Jacobian must be chosen to be of the specific form $\mathcal{J}= \alpha (\psi) R^2$. We note that, as mentioned in Sec. 5.2, the poloidal angle is fully determined by the choice of the Jacobian. The specific choice of $\mathcal{J}= \alpha (\psi) R^2$ is usually too restrictive for choosing a desired poloidal angle (for example, the equal-arc poloidal angle can not be achieved by this choice of Jacobian). Is there any way that we can make the field line straight in a coordinate system at the same time ensure that the Jacobian can be freely adjusted to obtain desired poloidal angle? The answer is yes. The obvious way to achieve this is to define a new toroidal angle $\zeta$ that generalizes the usual toroidal angle $\phi$. Define the new toroidal angle $\zeta$ as[6]

$$\zeta = \phi - q (\psi) \delta (\psi, \theta),$$ (235)

where $\delta = \delta (\psi, \theta)$ is a unknown function to be determined. The new local safety factor in $(\psi , \theta , \zeta )$ coordinates, $\mathbf{B} \cdot \nabla \zeta /\mathbf{B} \cdot \nabla \theta$, is written as

$$\frac{\mathbf{B} \cdot \nabla \zeta}{\mathbf{B} \cdot \nabla \theta} = \frac{(\nabla \Psi \times \nabla \phi + g \nabla \phi) \cdot \nab... ... \delta)}{(\nabla \Psi \times \nabla \phi + g \nabla \phi) \cdot \nabla \theta}$$

$$= \frac{(\nabla \Psi \times \nabla \phi + g \nabla \phi) \cdot [\na... ...phi - \nabla (q \delta)]}{(\nabla \Psi \times \nabla \phi) \cdot \nabla \theta}$$

$$= \frac{g \vert \nabla \phi \vert^2 + (\nabla \Psi \times \nabla \p... ... [- \nabla (q \delta)]}{(\nabla \Psi \times \nabla \phi) \cdot \nabla \theta} .$$ (236)

In $(\psi , \theta , \phi )$ coordinates, the above equation is written as

$$\frac{\mathbf{B} \cdot \nabla \zeta}{\mathbf{B} \cdot \nabla \theta} = \frac{g \vert \nabla \phi \vert^2 - \Psi' (\nabla \psi \times \na... ...tial (q \delta)}{\partial \psi} \nabla \psi \right]}{- \Psi' \mathcal{J}^{- 1}}$$

$$= - \frac{g}{\Psi'} \frac{\mathcal{J}}{R^2} - q \frac{\partial \delta}{\partial \theta},$$ (237)

where $\mathcal{J}$ is the Jacobian of $(\psi , \theta , \phi )$ coordinates. Using the expression of the local safety factor given in Eq. (179), the above equation is written

$$\frac{\mathbf{B} \cdot \nabla \zeta}{\mathbf{B} \cdot \nabla \theta} = \hat{q} - q \frac{\partial \delta}{\partial \theta},$$ (238)

To make the new local safety factor independent of $\theta$, the right-hand side of Eq. (237) should be independent of $\theta$, i.e.,

$$\hat{q} - q \frac{\partial \delta}{\partial \theta} = c (\psi),$$ (239)

where $c (\psi)$ can be an arbitrary function of $\psi$. The most convenient choice for $c (\psi)$ is $c (\psi) = q$, i.e., making the new local safety factor to be equal to the original global safety factor. In this case, equation (239) is written as

$$\frac{\partial \delta}{\partial \theta} = \frac{\hat{q}}{q} - 1.$$ (240)

Equation (240) can be integrated over $\theta$ to give

$$\delta (\psi, \theta) = \delta (\psi, 0) + \frac{1}{q} \int_0^{\theta} \hat{q} d \theta - \theta .$$ (241)

Since there is no other requirements for the $\psi$ dependence of $\delta$, the $\delta (\psi, 0)$ can be chosen to be zero. Then the above equations is written

$$\delta = \frac{1}{q} \int_0^{\theta} \hat{q} d \theta - \theta,$$ (242)

i.e.,

$$\delta = \frac{1}{q} \int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta - \theta$$ (243)

Substituting the above equation into the definition of $\zeta$ (Eq. 235), we obtain

$$\zeta = \phi - \int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta + q \theta .$$ (244)

In summary, the field line is straight (with slope being $q$) on $(\theta, \zeta)$ plane if $\zeta$ is defined by Eq. (244). In this method, we make the field line straight by defining a new toroidal angle, instead of requiring the Jacobian to take particular forms. Thus, the freedom of choosing the form of the Jacobian is still available to be used later to define a good poloidal angle coordinate.

[In numerical implementation, the term $\int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta$ appearing in $\delta$ is computed by using

$$- \int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} ... ...si \vert} d l_p = - \int_0^{\theta} \frac{1}{R} \frac{B_{\phi}}{B_p} d \ell_p .$$ (245)

For later use, from Eq. (), we obtain

$$\frac{\partial (\delta q)}{\partial \psi} = - \frac{d}{d \psi} \left( \frac{g}{\Psi'} \right) \int_0^{\theta}... ... \int_0^{\theta} \frac{\mathcal{J}}{R^2} d \theta - \frac{d q}{d \psi} \theta .$$ (246)

This formula is used in GTAW code to calculate $\partial (\delta q) / \partial \psi$, where the derivative $\partial (g / \Psi') / \partial \psi$ is calculated numerically by using the central difference scheme.]

It is ready to see that the function $\delta (\psi, \theta)$, which is introduced above to make $\mathbf{B} \cdot \nabla \zeta /\mathbf{B} \cdot \nabla \theta = q$, satisfies the periodic condition $\delta (\psi, 0) = \delta (\psi, 2 \pi) = 0$. [Proof: Equation (242) implies that

$$\delta (\psi, \theta = 0) = 0.$$

and

$$\delta (\psi, \theta = 2 \pi) = \frac{1}{q} \int_0^{2 \pi} \hat{q} d \theta - 2 \pi = 0,$$ (247)

i.e., $\delta (\psi, 0) = \delta (\psi, 2 \pi) = 0$.]