Explicit expression for the generalized toroidal angle

For the above magnetic field, the toroidal shift involved in the definition of the generalized toroidal angle can be evaluated analytically. The toroidal shift is given by

$$\overline{\delta} = \int_0^{\theta} \hat{q} d \theta,$$ (353)

where the local safety factor $\hat{q}$ can be written as

$$\hat{q} = - \frac{g_0}{R^2} \frac{\mathcal{J}}{\Psi'} .$$ (354)

Using $\mathcal{J} = - R r$ and

$$\Psi' = \frac{1}{q (r)} \frac{g_0 r}{\sqrt{R_0^2 - r^2}}$$ (355)

The local safety factor $\hat{q}$ in Eq. (354) is written as

$$\hat{q} = q \sqrt{R_0^2 - r^2} \frac{1}{R} .$$ (356)

Using this, expression (353) is written

$$\overline{\delta} = q \sqrt{R_0^2 - r^2} \int_0^{\theta} \frac{1}{R} d \theta$$ (357)

The integration $\int_0^{\theta} 1 / R d \theta$ can be evaluated explicitly (using maxima), yielding

$$\frac{2 \arctan \left( \frac{\sin \theta (2 R_0 - 2 r)}{2 (\cos \theta + 1) \sqrt{R_0^2 - r^2}} \right)}{\sqrt{R_0^2 - r^2}}$$ (358)

Then expression (357) is written

$$\overline{\delta} = 2 q \arctan \left( \frac{(R_0 - r)}{\sqrt{R_0^2 - r^2}} \tan \left( \frac{\theta}{2} \right) \right),$$ (359)

where use has been made of $\sin \theta / (\cos \theta + 1) = \tan (\theta / 2)$. Using this, the generalized toroidal angle can be written as

$$\alpha = \phi - \overline{\delta}$$

$$= \phi - 2 q \arctan \left( \frac{(R_0 - r)}{\sqrt{R_0^2 - r^2}} \tan \left( \frac{\theta}{2} \right) \right) .$$ (360)

The results given by the formula (359) are compared with the results from my code that assumes a general numerical configuration. The results from the two methods agree with each other, as is shown in Fig. 27, which provides confidence in both the analytical formula and the numerical code.

Figure: The results of $\overline{\delta} = \int_0^{\theta} \hat{q} d \theta$ computed by using formula (359) and the numerical code agree with each other. The different lines correspond to different magnetic surfaces.