Periodic conditions of physical quantity along $\theta$ and $\alpha$ in field-line-following coordinates $(\psi , \theta , \alpha )$

Since $(\psi , \theta , \phi )$ and $(\psi, \theta + 2 \pi, \phi)$ correspond to the same spatial point, a general physical quantity $f$ expressed in terms of coordinates $(\psi , \theta , \phi )$, i.e., $f = f (\psi, \theta, \phi)$, must satisfy the following periodic condtions along $\theta$:

$$f (\psi, \theta + 2 \pi, \phi) = f (\psi, \theta, \phi) .$$ (285)

Since $(\psi , \theta , \phi )$ and $(\psi, \theta, \phi + 2 \pi)$ correspond to the same spatial point, $f$ must satisfy the following periodic condtions along $\phi$:

$$f (\psi, \theta, \phi + 2 \pi) = f (\psi, \theta, \phi) .$$ (286)

Since $(\psi , \theta , \alpha )$ and $(\psi, \theta, \alpha + 2 \pi)$ correspond to the same spatial point, a general physical quantity $g$ expressed in field-line-following coordinates $(\psi , \theta , \alpha )$, i.e., $g = g (\psi, \theta, \alpha)$, must satisty the following peroidic condition along $\alpha$:

$$g (\psi, \theta, \alpha + 2 \pi) = g (\psi, \theta, \alpha) .$$ (287)

However, $P_1 = (\psi, \theta, \alpha)$ and $P_2 = (\psi, \theta + 2 \pi, \alpha)$ are not usually corresponding to the same spatial point. In fact, equation (282) implies, for point $P_1$, its toroidal angle $\phi_1$ is given by

$$\phi_1 = \alpha + \int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta,$$ (288)

while for point $P_2$, its toroidal angle $\phi_2$ is given by

$$\phi_2 = \alpha + \int_0^{\theta + 2 \pi} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta = \phi_1 + 2 \pi q,$$ (289)

i.e., $\phi_1$ and $\phi_2$ are different by $2 \pi q$. From this, we know that $(\psi , \theta , \alpha )$ and $(\psi, \theta + 2 \pi, \alpha - 2 \pi q)$ correspond to the same spatial point. Therefore we have the following periodic condtion:

$$g (\psi, \theta, \alpha) = g (\psi, \theta + 2 \pi, \alpha - 2 \pi q),$$ (290)

which involves both $\theta$ and $\alpha$.